Explicit Formulas

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich E Takens, Groningen Subseries: Mathematisches Institut der Universit~it und Max-Planck-Institut fiir Mathematik Bonn - vol. 21 Advisor: F. Hirzebruch

1593

Jay Jorgenson & Serge Lang Dorian Goldfeld

Explicit Formulas for Regularized Products and Series

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Authors Jay Jorgenson Serge Lang Mathematics Department Box 208283 Yale Station 10 Hillhouse Ave New Haven CT 06520-8283, USA Dorian Goldfeld Mathematics Department Columbia Unversity New York, NY 10027, USA

Mathematics Subject Classification (1991 ): 11 M35, 11M41,11 M99, 30B50, 30D 15, 35P99, 35S99, 42A99 Authors Note: there is no MSC number for regularized products, but there should be.

ISBN 3-540-58673-3 Springer-Verlag Berlin Heidelberg New York

CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10130182 46/3140-543210 - Printed on acid-free paper

EXPLICIT

FORMULAS

REGULARIZED

FOR

PRODUCTS

AND SERIES

Jay Jorgenson and Serge Lang

A SPECTRAL

INTERPRETATION

OF WEIL'S EXPLICIT

FORMULA

Dorian Goldfeld

EXPLICIT

FORMULAS FOR REGULARIZED PRODUCTS AND SERIES

Jay J o r g e n s o n and Serge Lang

Introduction I A s y m p t o t i c e s t i m a t e s of regularized h a r m o n i c series 1. 2. 3. 4. 5. 6. 7.

Regularized products and harmonic series Asymptotics in verticM strips Asymptotics in sectors Asymptotics in a sequence to the left Asymptotics in a parMlel strip Regularized product and series type Some examples

II Cramdr's T h e o r e m as an Explicit F o r m u l a

3

11 14 20 22 24 34 36 39 43

Euler sums and functional equations The general Cram~r formula Proof of the Cramdr theorem An inductive theorem

45 47 51 57

III Explicit Formulas u n d e r Fourier A s s u m p t i o n s

61

1. 2. 3. 4.

1. 2. 3. 4. 5.

Growth conditions on Fourier transforms The explicit formulas The terms with the q's The term involving The Well functional and regularized product type

62 66 73 78 79

IV From F u n c t i o n a l E q u a t i o n s to T h e t a Inversions 85 1. An application of the explicit formulas 2. Some examples of theta inversions

87 92

Viii

V From Theta Inversions to Functional Equations 1. 2. 3. 4.

The Weft functional of a Gaussian test function Gauss transforms Theta inversions yield zeta functions A new zeta function for compact quotients of M3

V I A G e n e r a l i z a t i o n o f Fujii's T h e o r e m

97

99 101 109 113 119

122 125 128

1. Statement of the generalized Fujii theorem 2. Proof of Fujii's theorem 3. Examples

131

Bibliography

A SPECTRAL INTERPRETATION WEIL'S EXPLICIT FORMULA

OF

Dorian Goldfeld

1. 2. 3. 4. 5. 6. 7.

Introduction Notation Construction of the indefinite space s Spectral theory of s Eisenstein series Cusp forms The zeta function associated to an automorphic form on L2(T) 8. The Rankin-Selberg convolution 9. Higher rank generalizations 10. References

Index

137 139 140 141 142 145 147 148 148 152

153

EXPLICIT

FORMULAS

REGULARIZED

FOR

PRODUCTS

AND SERIES

Jay Jorgenson and Serge Lang

Introduction Explicit formulas in number theory were originally motivated by the counting of primes, and Ingham's exposition of the classical computations is still a wonderful reference [In 32]. Typical of these formulas is the Riemann-von Mangoldt formula log

pn<x

p

=

x -

}; P

x P --

-

1 log(1 - x - 2 ) .

P

Here the sum on the left is taken over all prime powers, and the sum on the right is taken over the non-trivial zeros of the Riemann zeta function. Later, Weil [We 52] pointed out that these formulas could be expressed much more generally as stating that the sum of a suitable test function taken over the prime powers is equal to the sum of the Mellin transform of the function taken over the zeros of the zeta function, plus an analytic term "at infinity", viewed as a functional evaluated on the test function. It is the purpose of these notes to carry through the derivation of the analogous so-called "explicit formulas" for a general zeta function having an Euler sum and functional equation whose fudge factors are of regularized product type. As a result, our general theorem applies to many known examples, some of which are listed in w of [JoL 93c]. The general Parseval formula from [JoL 93b] provides an evaluation of the "term at infinity", which we call the Weil functional. Also, as an example of our results, let us note that even in the well-studied case of the Selberg zeta function of a compact Riemann surface, our computations show that one may deal with a larger class of test functions than previously known. For some time, analogies between classical analytic number theory and spectral theory have been realized. MinakshisundaramPleijel defined a zeta function in connection with the Laplacian on an arbitrary Riemannian manifold [MiP 49], and subsequently Selberg defined his zeta function [Se 56]. In [JoL 93a,b] we developed a general theory of regularized products and series applicable equally to the classical analytic number theory and to some of these

analogous spectral situations. In particular, we proved the basic properties of the Weil functional at infinity in the context of regularized products and series, with a view to using the functional for the explicit formulas in this general context. A f u n d a m e n t a l class o f z e t a f u n c t i o n s . In [JoL 93c] we defined a fundamental class of functions to which we could apply these properties and carry out analogues of results in analytic number theory. Roughly speaking, the functions Z in our class are those which satisfy the three conditions: there is a functional equation; the logarithm of the function admits a generalized Dirichlet series converging in some half plane (we call this Dirichlet series an Euler sum for Z); - the fudge factors in the functional equation are of regularized product type. The precise definition of our class of functions is recalled in Chapter II, w The explicit formula can be formulated and proved for functions in this class. In Chapter II, w we discuss the extent to which this class is a much broader class than a certain class defined by Selberg [Se 91]. Furthermore, certain applications require an even broader class of functions to which all the present techniques can be applied. We shall describe the need for such a class in greater detail below. Just as we did for the analogue of Cram~r's theorem proved in [JoL 93c], we emphasize that the explicit formula involves an inductive step which describes a relation between some of the zeros and poles of the fudge factors and some of those of the principal zeta function Z. Such a step can be viewed as a step in the ladder of regularized products, because our generalized Cram~r theorem insures that a function Z in our class is also of regularized product type provided the fudge factors are of regularized product type. If Z is a function in our class, and, for Re(s) sufficiently large, the expression log Z(s) -- ~ q

is the Euler sum for log Z(s), with a sequence {q} of real numbers 1 tending to infinity, and complex coefficients c(q), then such q play the role of prime powers. However, readers should keep in mind cases when q does not look at all like a prime power. For example, the general theory applies to the case when Z(s) is a general

Dirichlet polynomial, up to an exponential fudge factor; a precise definition is given in Chapter II, w Such polynomials contain as special cases the local factors of more classical zeta functions and L-functions. In examples having to do with Riemannian geometry, log q is the Riemannian distance between two points in the universal covering space. The general version of Cram6r's theorem in [JoL 93c] was carried out for the original Cram6r's test function Cz(s) = r One can also view this version as a special case of an explicit formula with more general test functions. This is carried out in Chapter II. In [JoL 93c], w we gave a number of examples for our Cram6r-type theorem. To these we are adding not only the general Dirichlet polynomials as mentioned above, but also Fujii-type L-functions, obtained from a zeta function by inserting what amounts to a generalized character as coefficient of the Dirichlet series defining the zeta function (see the papers by Fujii listed in the bibliography). In Chapter V we show both how to recover Fujii's theorems for the functions he considered, namely the Riemann zeta function and the Selberg zeta function for PSL(2,Z), as well as an analogous theorem for the general zeta functions in our class, all as corollaries of our Cram6r's theorem. Similarly, a result of Venkov, which relates the eigenvalues of the Laplacian relative to PSL(2,Z) to the classical von Mangoldt function, will be generalized to any non-compact finite volume hyperbolic Riemann surface in [JoL 94]. The generalization involves another inductive type argument, using the fact that the fudge factor in the functional equation of the non-compact Selberg zeta function involves the determinant of the scattering matrix, which itself is in our class of functions since it has an Euler sum and a simple functional equation with constant fudge factors. In this case, the Euler sum exists whereas a classical Euler product does not. Thus, the general theory simultaneously contains previous results and gives new ones which were not proved previously by authors using such tools as the Selberg trace formula. A n a l y t i c e s t i m a t e s for t h e p r o o f . In addition to the Parseval formula of [JoL 93b], the proof of the general explicit formula relies on certain analytic estimates for regularized harmonic series, including the logarithmic derivatives of regularized products in strips. We gave such estimates already in [JoL 93a,b], but we need further such estimates which we present in Chapter I of the present work, using the technique of our generalized Gauss formula. Hard-core analytic estimates having thus been put out of the way, the rest of the work is then relatively formal. It is noteworthy that to each regularized product we associate naturally two non-

negative integers determined directly from the definition. Then the fundamental estimates of Chapter I show that the order of growth of the logarithmic derivatives of such products in strips is determined by these two integers. In the application to the evaluation of certain integrals involving test functions, one can then see that the order of decay of these test functions, needed to insure that the integrals converge, is also determined by these two integers. Our systematic approach both improves known estimates for the Selberg zeta function (cf. Chapter I, w and provides estimates for functions in our class which had not been considered previously. T h e t a i n v e r s i o n s . We shall postpone to still another work the application of explicit formulas to the counting of those objects which play the role of prime powers. Here we shall emphasize an entirely different type of application, obtained by taking Gaussian type functions as the test functions instead of other test functions which lead to the counting. Applying the general explicit formula to such Gaussians gives rise to relations which are vast generalizations of the classical Jacobi inversion formula for the classical Jacobi theta function, where t on one side gets inverted to lit on the other side of the formula. The classical Jacobi inversion formula is the relation 1 2rr E

n=--OO

1 ~e_(2,~,)2/4t, 4v/-~t

e-"~'-

n=--

which holds for all t > 0. Here, log q = 27rn where n is a positive integer. The zeta function Z(s) giving rise to the above theta series is essentially the special Dirichlet polynomial sin(

is)

= -

1

Ir,(1 _ e_2rts).

Thus, the most classical theta series appears in a new context, associated to a "zeta function" which looks quite different from those visualized classically. The general context of Chapter IV and Chapter V allows a formulation of a theta inversion when the theta series is of type

•

ake-,xk*

k

with various coefficients ak. Theta inversion applies in certain cases when the sequence {Ak} is the sequence of eigenvalues of an operator. For example, as we will show in Chapter V, w such an

inversion formula comes directly from considering the heat kernel on the compact quotient of an odd dimensional hyperbolic space which has metric with constant negative sectional curvature. For certain manifolds, the theta inversion already gives rise to an extended class of zeta functions, which instead of an Euler sum may have a Bessel sum. For manifolds of even dimension, the class of functions having an Euler sum or Bessel sum is still not adequate, and it is necessary to define an even further extended class, which we shall describe briefly below. At this moment, it is not yet completely clear just how far an extension we shM1 need, but so far, whatever the extension of the fundamental class we have met, the techniques of [JoL 93a,b,c] and of Chapter I apply. In [JoL 94], we show how the general explicit formula also applies to the scattering determinant of Eisenstein series. Here, the Euler sum exists, and scattering determinants are in the fundamental class.

An additive theory rather than multiplicative theory, and a n e x t e n d e d class o f f u n c t i o n s . The conditions defining our fundamental class of functions are phrased in a manner still relatively close to the classical manner, involving the functions multiplicatively. However, it turns out that many essential properties of these functions involve only their logarithmic derivative, and thus give rise to an additive theory. For a number of applications, it is irrelevant that the residues are integers, and in some applications we are forced to deal with the more general notions of a regularized harmonic series (suitably normalized Mittag-Leffier expansions, with poles of order one) whose definition is recalled in Chapter I, w In general, the residues of such a series are not integers, so one cannot integrate back to realize this series as a logarithmic derivative of a meromorphic function. Even for the Artin L-functions, although they can be defined by an Euler product, it was natural for Artin to define them via their logarithmic derivative, and at the time, Artin could only prove that the residues were rational numbers. It took many years before the residues were finally proved to be integers. The systematic approach of [JoL 93a,b,c] in fact has been carried out so that it applies to this additive situation. The example of Chapter V, w shows why such an additive theory is essential. Thus we are led to define not only the fundamental class of functions whose logarithmic derivative admits a Dirichlet series expression as mentioned above, but an extended class of functions where this condition is replaced by another one which will allow appli-

cations to more situations, starting with applications to various spectral theories as in [JoL 94]. Nevertheless, we still defined the fundamental class of functions having Euler sums, and we phrase some results multiplicatively, partly because at the present time, we feel that a complete change of notation with existing works would only make the present work less accessible, and partly because the class of functions admitting Euler sums is still a very important one including the classical functions of algebraic number theory and representation theory. However, we ask readers to keep in mind the additive rather than multiplicative formalism. Many sections, e.g. Chapter I and w and w of Chapter V, are written so that they apply directly to the additive situation. Functions in the multiplicative fundamental class are obtained as Mellin transforms of theta functions having an inversion formula. Functions in the extended additive class are obtained as regularized harmonic series which are Gaussian transforms of such theta functions. For example, the (not regularized) harmonic series obtained from the heat kernel theta function in the special case of compact quotients of the three dimensional, complete, simply connected, hyperbolic manifold is essentially

Ck(x)r k

Observe how the presence of s ( s - 2) in the series formally insures a trivial functional equation, that is invariance under s ~-* 2 - s. Conversely, given a function in our extended additive class, one may go in reverse and see that the original theta inversion is only a special case of the general explicit formula valid for much more general test functions. The existence of an explicit formula with a more general test function will then allow us to obtain various counting results in subsequent publications. Finally, let us note that many examples of explicit formulas using various test functions involving many examples of zeta functions have been treated in the literature, providing a vast number of papers on the subject. Most of the papers dealing with such explicit formulas are not directly relevant for what we do here, which is to lay out a general inductive "ladder principle" for explicit formulas in line with our treatment of Cram6r's theorem. For instance, Deninger in [Den 93] emphasizes the compatibility of an explicit formula for the Riemann zeta function with a conjectural formalism of a Lefschetz trace formula. Such a formalism might occur in the

presence of an operator whose eigenvalues are zeros of the zeta function. Our inductive hypotheses cover a wider class of functions than in [Den 93], and our treatment emphasizes another direction in the study of regularized products and series. Factors of regularized product type behave as if there were an operator, but no operator may be available. We also mention Gallagher's attempt to unify a treatment of Selberg's trace formula with treatments of ordinary analytic number theory [Ga 84]. However, the conditions under which Gallagher proves his results are very restrictive compared to ours, and, in particular, are too restrictive to take into account the inductive ladder principle which we are following. A c k n o w l e d g e m e n t : During the preparation of this work, the first author received support from NSF grant DMS-93-07023. Both authors benefited from visits to the Max-Planck-Institut in Bonn.

CHAPTER I A s y m p t o t i c e s t i m a t e s of regularized harmonic series. The proof of the general explicit formulas for functions whose fudge factors are of regularized product type will require a number of asymptotic estimates of general regularized harmonic series. The purpose of this chapter is to establish and tabulate these estimates in convenient form. The main definitions and results of this chapter are stated in w and w These asymptotic formulas are needed just as one needs the asymptotic behavior of the gamma function and the zeta function in classical analytic number theory (see, for example, Chapter XVII of [La 70]). However, classical arguments which estimate this behavior cannot be applied in general, and must be replaced by more powerful tools, such as our extension of Cram~r's theorem, proved in [JoL 93c], as well as our systematic analysis of the regularized harmonic series, given in [JoL 93a] and [JoL 935]. Following the notation of [JoL 93a] and [JoL 93b], we let R(z) be the regularized harmonic series associated to the theta function 8(t) = ~ ake-Xkt; in other words

R(z) -- CTs=ILM~9(s,z) where L M is the Laplace-Mellin transform, and CTs=I is the constant term of the power series in s at s = 1. As is shown in w of [JoL 93a], the function R(z) has a meromorphic continuation to all z E C whose singularities are simple poles located at z - --Ak with corresponding residue ak. In what we call t h e s p e c t r a l c a s e , meaning ak E Z for all k, we have a regularized product D(z) which is a meromorphic function defined for all z C C and which satisfies the relation

R(z) = D'/D(z).

12

However, in this chapter, we will work in the more general situation by considering a regularized harmonic series which is not necessarily the logarithmic derivative of a regularized product. Our basic tool for estimating R(z) is our general Gauss formula, which we shall recall at the end of w1. From the general Gauss formula, we shall determine the asymptotic behavior of R(z) as z ~ oo in each of the following cases: 1) in a vertical strip obtained by restricting Re(z) to a compact interval; 2) in a sector IIm(z)l 1. Under condition D I R 1 we delete from the complex plane C b e horizontal half lines going from - ~ to -Ak for each k, together, when necessary, the horizontal half line going from - o o to 0. We define the open set: U L =

the complement of the above half lines in C.

If all .kk are real and positive, then we note that UL is simply C minus the negative real axis R 0 such that for all k sufficiently large, we have 7r

- - ~ -4- ~ < arg()~k) < --

--

7r m 2

m

~~

15

We will consider a t h e t a s e r i e s or t h e t a f u n c t i o n , which is defined by oo

OA,L(t) = O(t) = ao + Z ake-X*t' k----1

and, for each integer N >_ 1, we define the a s y m p t o t i c e x p o n e n t i a l p o l y n o m i a l s by N-1

QN(t) = ao + Z

ake-X*t"

k=l

We are also given a sequence of complex numbers {p} = {pj} with Re(p0) < Re(p1) < . . . < Re(pj) < . . . increasing to infinity, and, to every p in this sequence, we associate a polynomial Bp of degree np and set

bp(t) = Bp(logt). We then define the a s y m p t o t i c p o l y n o m i a l s a t 0 by

Pq(,) =

~

bA,)e.

Re(p) to.

A S 2. For every q, we have

O(t) - Pq(t) = o(tRe(q) llogt[ re(q)) for t --~ O, which shall w r i t t e n as

o(t) ~

b,(t)t,. p

A S 3. Given 5 > 0, t h e r e exists an a > 0 a n d a constant C > 0 such t h a t for all N and 0 < t < 5 we have 1 0 ( t ) - QN(t)[ 2T

ak w

+ Ak)-+l "

Since y lies in a compact interval, we have the estimate

lu+wl ~T, so we have the bound

lakl

O~r(2T)(,, 9O) = 0

)

E (IAkI _ T),~+, ]~kl>2T

i

Using Lemma 4.2(a) and an integral comparison, there exist positive constants Cl and c2 such that

o~nzw(2T)(~; ~ 0) =o

l ~

x ~~ (c2x11~, _ T ) , + , ) d x ) \

x

~ol~l-(~+1)l~l dx I

/ =0

(Tao+al-(n+l)).

30

To finish one integrates n-times. An extra power of l o g T occurs precisely in the case when a0 + al E Z C

lr

all nk.

Equivalently, there is a positive constant c a n d a sequence {n/c} o f positive integers w i t h n/c --+ oc such that

I '-1

b)

foralln/c.

Let rn = I ~ l ~ ( l o g I .1) There is a positive constant c a n d a sequence {nk } o f positive integers w i t h n/c ~ o~ such that r n ~ + l - - rn~ ~ C for all nk. Equivalently, there is a positive constant c and a sequence {n/c} o f positive integers w i t h n/c ~ oo such t h a t -

:/,:,,-i (log n/c) -S~/~

I:,,,, I _> c 9r,/c

forallnk.

Proof. For t h e first s t a t e m e n t in (a), one has, by (6), the b o u n d k = o(r/c), which would be contradicted if no such constant c or subsequence {n/c} would exist. T h e second assertion in (a) and b o t h assertions in (b) are established similarly. []

By a s s u m p t i o n D I R 3, there is a constant C such t h a t for all k, we have the inequalities Re(Ak) < IX/Cl < CRe(A/C). One t h e n has results analogous to L e m m a 4.2 and L e m m a 4.4 for the sequence {Re(),k)}. Specifically, we shall need the following version of L e m m a 4.4. L e m m a 4.5. A s s u m e the n o t a t i o n as above. a) L e t rn = Re(An) ~1 9 T h e r e is a positive constant c arid a sequence {nk} Of positive integers w i t h nk ~ oc such t h a t for all nk, rn~+l

--rn~

~_ C.

32

Equivalently, there is a positive constant c and a sequence {nk } of positive integers with nk ~ oo such that for d l nk,

Re(/~nk+l) -- Re(/~nk)

>_ c . n ~ /'rl - 1

b) Let rn = Re(A,~)~(logRe(kn)) - z . There is a positive constant c and a sequence {nk} of positive integers with nk such that t'or a11 nk, r n k + l -- rnk ~ C.

Equivalently, there is a positive constant c and a sequence {nk} of positive integers with nk --* ee such that for a11 nk, Re(,~n~+l) - Re(,~n~) ~ c" rtl/a-l(log~tk)-~/a.

The proof of Lemma 4.5 is identical to that of Lemma 4.4, hence will be omitted. The following proposition estimates the finite sum in (1) for a particular sequence Tn of real numbers with T,, ---+ ~ so that the finite sum grows as slowly as possible. With the proposition, the proof of Theorem 4.1 is complete.

P r o p o s i t i o n 4.6. There is a sequence of real n u m b e r s T~ with Tn ~ c~ such that for all y in any compact interval os R, the following a s y m p t o t i c relations hold. a) In the notation of D I R 2, we have > ~, "

IAkl~2rn

ak -Tn+iy+Ak

=o(T~n,+'~

b) Let rn = rn(po) + 1. Then, in the spectral case, we have > ~, -"

ak = 0 (TnRe(p~ - T n + iy +,~k

33

Proof. For any z with Re(z) sufficiently large and

Izl

#

I kl,

we

can estimate the finite sum in (a) by ak E z+.~k IXkl0. In the notation used above, we have -Re(p0) = 1 and re(p0) = 0, hence the gamma function is of regularized product type of reduced order (0, 0). Theorem 2.3 yields the classical Stirling's formula for log F(s). As for asymptotics in vertical strips, Corollary 2.2 yields the equally classical asymptotic formula

r'/r(s)

= log Isl + o(1)

as s ~ c~ in any vertical strip of finite width. This result is an important ingredient in the proof of the classical explicit formula for zeta functions of number fields (see, for example, Chapter XVII of [La 70]).

Example 2: Dirichlet polynomials. We define a Dirichlet polynomial to be a holomorphic function of the form N P ( s ) = ~-'~anb~ n=l

where {an } is a finite sequence of complex numbers, {bn } is a finite sequence of positive real numbers. In Chapter II we will apply the general Cram@r theorem from [JoL 93c] to show that P is of regularized product type of reduced order (0, 1).

Example 3: The Riemann zeta f u n c t i o n . Let CQ(S) be the Riemann zeta function and consider the sequences A§ = {p/i E C l 6~(p) = O, 0 ~ Re(p) ~ I,

Im(p) > O}

and

A_ ={p/(-i) EC ] CQ(P)=O, O 0, then the series

V(z) : E ~ converges for Im(z) > 0 and has a singularity at the origin of the type log z/(1 - e-z), by which we mean that the function

F(z) = 2 . i V ( z )

log z 1_ r

which is defined for Ira(z) > 0, has a meromorphic continuation to where n ranges over all C, with simple poles at the points • the integers, and at the points :t: logp m, where pm ranges over the prime powers. In [JoL 93c] we proved an analogous theorem for any meromorphic function with an Euler sum and functional equation whose fudge factors which are of regularized product type. In this chapter we will prove a Cram6r-type theorem by considering the same contour integral as analyzed in [JoL 93c] for a more general class of test functions. Specifically, we consider the contour integral

1f r

2rci

over a semi-infinite vertical rectangle Ts which is assumed to contain the top half of the "critical strip" of Z. The test function r is assumed to be holomorphic on the closure of Ts and to have reasonably weak growth conditions, essentially what is needed to make the proof go through. In the Cram6r theorem from [JoL 93c], the test function r depended on a complex parameter z, namely

r

= r

= ~sz

44 For Im(z) > 0, the function $z has exponential decay when s lies in a finite strip and Ira(s) ~ oo. As a result, the analysis needed in [JoL 93c] required a very weak growth result, which we proved for general meromorphic functions of prescribed order (see, in particular, Lemma 2.1 of [JoL 93c]). In w we recall the fundamental class of functions which we have defined, and discuss its relation to the Selberg class of functions defined in [Sel 91]. The definitions of w are used throughout, but the subsequent sections may be logically omitted for the rest of this work. Taking into account the asymptotic estimates of Chapter I, our method of proof from [JoL 93c] applies to the more general class of test functions considered here. In w we establish notation and state the main result of this chapter. The proof is given in w and various applications are discussed in w As remarked on page 397 of [JoL 93c], our proof does differ from the original proof due to Cram6r, which is one of the reasons why we can easily generalize the theorems to the class of functions which have Euler sums and functional equations with fudge factors which are of regularized product type.

45 w

Euler sums and functional equations.

We shall say that the functions Z and ,~ have an E u l e r s u m and f u n c t i o n a l e q u a t i o n if the following properties are satisfied: 1. M e r o m o r p h y . The functions Z and Z are meromorphic functions of finite order. . E u l e r S u m . There are sequences {q} and {(t} of real numbers > I that depend on Z and Z, respectively, and that converge to infinity, such that for every q and ~, there exist complex numbers c(q) and C(~l) and o~ > 0 such that for all Re(s) > cry,

log z ( s ) = ~

c(q)

and

log 2 ( s ) = ~

q

c(O)

0

The series are assumed to converge uniformly and absolutely in any half-plane of the form Re(s) > 0~ + ~ > a~. 3. F u n c t i o n a l E q u a t i o n . There are functions G and G, meromorphic and of finite order, and there exists 00 with 0 0 such that 00 + a > a 0 as in w below. We dealt with

46 the scattering determinant in [JoL 93c], in the context of Cram6r's theorem. The Euler sum for Z implies that Z is uniformly bounded for Re(s) _> a~ + e for every e > 0. Notice that Z has no zeros or poles for Re(s) > a~, and all zeros and poles of Z in the region Re(s) < - a agree in location and order with poles and zeros of as. R e m a r k 2. A Dirichlet series expression is assumed only for Z and Z, so the fudge factors do not occur symetrically for the zeta function in the above conditions, although they might appear to do so in the functional equation. For example, in the most classical case of the Riemann zeta function, the fudge factor is essentially the gamma function, which does not have a Dirichlet series expansion but is of regularized product type. We define a triple (Z, 2, ~) to be in t h e f u n d a m e n t a l class if Z, 2 have an Euler sum and functional equation, and the fudge factors are of regularized product type. Selberg has defined a "Selberg class" of functions in analytic number theory (see [Se 91], [CoG 93]). Our class is much wider than Selberg's class in several major respects: 1) Selberg's fudge factors are of gamma type, i.e. ['(as +/~). 2) Selberg assumes a Ramanujan-Petersson estimate on the coefficients of the Euler sum, but we do not. 3) Selberg's Euler sum involves ordinary integers and ordinary prime powers. We allow arbitrary positive numbers {q}. Our conditions allow for a much wider domain of applicability in spectral theory as illustrated by our varied examples, and many more to be treated in subsequent papers. For example, our conditions allow the fudge factors to include F2, Fd for general d, or for and L functions themselves, or any function of regularized product type. R e m a r k 3. Even so, the Euler sum condition is still not sufficiently general for our purposes and will be ultimately be generalized to a Bessel sum condition. For further comments on this point of view, see w and w of Chapter V. In the same vein, the functional equation will also be replaced by an additive relation where the additive fudge factor will be assumed to be a regularized harmonic series, or regularized harmonic series type, in analogy with regularized product type.

47

w

The general Cram6r formula.

Let a > 0 be such that a0 + a > a~. We define the following regions in the complex plane: ~ + = semi-infinite open rectangle bounded by the lines Re(s) = - a ,

Re(s) = cr0 + a,

Im(s) = 0.

7E+(T) = the portion of g + below the line Im(s) = T. W e allow Z and 9 to have zeros or poles on the finite real s e g m e n t

[ - a , a0 + a], but we assume that 9 and Z have no zeros or poles on the vertical edges with Re(s) = - a and Re(s) = cro + a and Im(s) > O. Let r be any function which is holomorphic on the closure of the semi-infinite rectangle 7~+, and let H be a meromorphic function on this closure. We axe interested in studying the (formal) sums (1)

div+, a ( r

E

VH(Z)r

zETr +

where VH(Z) = ordH(z)

is the order of the zero or pole of H at z, so the sum (1) is actually over the divisor of H which lies in Tea+. Such sums do not converge a priori, so we need to define them as limits in a suitable sense, and for suitable functions r On a space of functions decaying sufficiently fast (depending on H), the divisor div+, ~ = E

VH(Z)(Z)

gives rise to the functional defined by the sum (1). The functional itself may be denoted by [div+,z] to distinguish the functional from the divisor. To determine such a space of functions, we proceed as follows. Let H i , . . . , Hr be a finite number of functions which are meromorphic on the closure of 7r + , and let r be a function which is holomorphic on this closure. We say that a sequence {Tin} of positive

48 real n u m b e r s t e n d i n g to infinity is J - a d m i s s i b l e for {H1,. 9 9 Hr} if for any k, Hk has no zero or pole on the segment

Sm = [-a + iTm,ao + a + iTm] and

qS(s)H~/Hk(s) --+ 0

for s C Srn with rn --+ oo.

W h e n { H 1 , . . . , Hr} is the set of functions {Z, z~, ~}, we say simply that {Tin } is a d m i s s i b l e . W i t h respect to a ~b-admissible sequence {Tin}, we define the divisor f u n c t i o n a l for H, which we denote by to be the limit

(2)

div+, (4)

= li~moo

VH(Z)C~(z),

E

zen+(T.) if such a limit exists. In particular, for H = Z, we let {p} = set of zeros and poles of Z in T~+ so the sum (2) can be written as

div+,a(q~)=~]i~rnoo E

vz(p)~(p).

We define other functionals as follows. Here we do not try to give subtle conditions on what a m o u n t s to a half-Fourier transform, so we simply assume t h a t the derivative $'(s) is in L 1 of each vertical half line [ - a , - a + ioo] and [~r0 + a, ~r0 + a + ic~]. We define the positive C r a m 6 r f u n c t i o n a l for q, which we denote by Cr+,a($), to be t h e integral ~o+a+ioo

Cr+,~(~) =

/

q~'(s)q-'ds.

ao+a

Similarly, the n e g a t i v e C r a m ~ r f u n c t i o n a l for q is defined by the integral ~oq-a =

/

~o+a-ioo

~'(~o - s)q-Sds.

49

Also, with respect to an H-admissible sequence {Tin}, where H is a meromorphic function which is holomorphic on Re(s) = - a , we define the functional -a+iTm

f r

U+,_a(r = Jim

--a

When all the above functionals are defined, we then consider The Cram~r formula. 2~ridiv+,=(r

: Z a0-ba

+ U;,_a(r

+

+ r

+

I

r

--a

(log Z(o'0 + a) - log Z ( - a ) ) .

Such a formula is derived formally by considering the contour integral

1 i r 2~ri o7r which can be evaluated in one way by using the residue theorem, and in another way by using the Euler sum and functional equation for Z. We are interested in conditions on r for which the Cram6r formula holds. For this purpose, we consider the following growth assumptions on r G R 1. r 1 6 2 1 6 2

is in L' on the vertical ray [ - a , - a + ic~].

G R 2. The derivative r [--a, --a + icx~] and

is in L 1 of each vertical half line

[ao + a, ao + a + icxD].

The first condition compares the decay of r with the exponential growth of r As in [3oL 93c], one often considers the situation

50 when 9 is of regularized product type. In this case, the above growth conditions can be verified as follows. P r o p o s i t i o n 2.1. a) If 9 is of regularized product type of reduced order (M, m) and the ray [ - a , - a + i o o ] lies in an admissiblestrip (defined in Chapter I, w then

O'/O(s) = O(blM(log Isl) m) for Isl ~ ~ and s on the ray I--a,--a + i ~ ] .

b) If for some 6 > O, r has the decay

r

= O(1/IslM+~(loglsl) m+l+a) for Isl--+ ~ on the ray, then G R 1 is satisfied.

The hypotheses in the above criterion have been shown to be satisfied in several cases which are of direct interest in our theory of regularized products, for instance Theorem 5.2 of Chapter I. The next theorem asserts that the divisor functional [div+,,] is defined on the vector space of functions satisfying G R 1 and G R 2, and satisfies the Cram6r formula. T h e o r e m 2.2. Assume r and 9 satisfy the two growth conditions G R 1 and G R 2. Then a11 the functionMs div+,a(r Ug,_a(r , Crq+,a(r and Crq, a(r &re defined, and the following formula holds: 27ridiv+,a(r = E

c(q)Cr~, a(r --E

c(q)Cr+, a(r

q

+ u+,_o(r

+ f

r

z(~)ds

--a

+ r

(log Z(cr0 + a) - log Z ( - a ) ) .

As stated above, the proof of Theorem 2.2 will be given in the following section, and various applications of the theorem will be discussed in w

51

w

P r o o f of t h e C r a m 4 r t h e o r e m .

The pattern of proof of Theorem 2.2 follows w of [JoL 93c], which, as we shall remark below, contains one significant technical improvement over the proof of the original theorem given by Cram@r [Cr 19] for the Riemann zeta function. Choose an e > 0 sufficiently small so that Z has no zeros or poles in the open rectangle with vertices --a,

--a + ie,

ao + a + ie,

ao + a

or on the line segment [ - a + i e , ao + a + i e ] . Note that the function Z may have zeros or poles on the horizontal line segment I - a , a0 + a]. For T sufficiently large, we shall study the contour integral --a+ie

27riVz( z, e; T ) =

o'o+a+ie

f

+ f

--a+iT

--a-l-ie

o'o+a+iT

(1)

-a+iT

+ f

f

o'o+a+i~

o'o+a+iT

We may assume that Z has no zeros or poles on the line segment connecting the points - a + i T and Oo + a + i T , because we will pick T = T m for m sufficiently large. Let: 7~T(e) = the finite rectangle with vertices - a + iT,

- a + ie,

ao + a + ie,

ao + a + iT.

By the residue theorem, we have

2.iyz(z,

;T)=

v(p)r peaT(e)

Theorem 2.2 will be established by studying each of the four integrals in (1). For simplicity, let us call these integrals the left, bottom, right, and top integrals, respectively. We begin with the top integral which will be shown to be arbitrarily small upon letting T = Tm approach infinity.

52

L e m m a 3.1. Let {Tm} be an admissible sequence relative to Z. T h e n we have -a+iT, n

i

lim m---~ oo

r

=0.

"o+a+iTm

The proof of Lemma 3.1 follows directly from the definition of an admissible sequence. To continue, we have, from the growth assumption G R 1, the limit --a+ie

lim

m-...+ ~

--a+ie

r

i

r

= f --a+ioo

-a+iTm

and ao+a+iTm

ao+a+i~

ao+a+ie

ao+a+ie

By combining these equations with Lemma 3.1, we have the following preliminary result. P r o p o s i t i o n 3.2. W i t h notation as above, we have 2~riVz(z,e)=

lim 2rciVz(z,e;Tm)

m --+r

--a-J-ie

= f

o'o+a+ie

+ f

r

--a+ioo

r

--a+ie

ao+a+ioo

+ f

r

ao+a-J-ie

As before, let us call the integrals in Proposition 3.2 the left, bottom and right integrals, respectively. By the above stated assumption on e, we have

Y~(z, ~) = Y~(z).

53

To continue our proof of T h e o r e m 2.2, we will c o m p u t e t h e three integrals in Proposition 3.2 using the axioms of Euler sum and functional equation. After these computations, we will let e approach 0, which will complete t h e proof. Let us use the functional equation to re-write the left integral as the s u m of three integrals involving Z and (I). Specifically, we have -a+ie

--a+ie

f r

f

--a+ioo

,.

--a+ic~ --a+ioo

(2)

= f

r

-a+ie aoTa--ie

+ f r

(3)

ao+a--ioo

After we let e --+ 0, the integral in (2) appears in the s t a t e m e n t of T h e o r e m 2.2 as the functional U+(r Note t h a t letting e --~ 0 is justified since (I) was assumed to be holomorphic and non-zero on the vertical lines of integration. As for (3), we can re-write this integral using the Euler s u m of Z, yielding o'oTa--ie

f r ao+a-ioo ao+a--ie

--~(a~ - s)l~ z~(s);i:;-;i -J- i

r

(ro+a--icx)

---- r

+

it)log 2(ao +

a-

ie) ao+a--ie

+~c(~) f (1 aoTa--ioo By

the Euler sum

condition

and

r

- s)q-'ds.

the fact that a > 0, we can let r

54 approach zero to get the equality O'o+a

f r

ao+a--ioo

(4)

= r

log Z(a0 + a) + E

c(q)Cr~, a(r

Both terms in (4) appear in the statement of Theorem 2.2. In the same manner as above, the right integral can be re-written using the Euler sum of Z, yielding ao+a+ioo

f

r

ao+a+ie o'o+a+ioo

=r

log z(~) ~0+o+-~~ -

/

r

Z(~)ds

ao+a+ie

= -r

+ a + ie) log Z(ao + a + ie) ao+a+ioo

--E

c(q) q

r

/ o.o+a+ie

Again, we can let e approach zero to obtain the equality o'o+a+ioo

r O" O -JF a

(5)

= -r

+ a) log Z(a0 + a ) - E

c(q)Crq+, a(r

q The second term in (5) appears in Theorem 2.2. The first term in (5) does not appear in Theorem 2.2 because this term cancels with a term that appears in the evaluation of the bottom integral, as we shall now see. In the evaluation of the bottom integral, we see the importance of choosing e > 0 before integrating by parts. By the choice of e,

55 Z has no zeros or poles on the line segment [ - a + ie, ~0 + a + ie], so we have tro+a+ie

r --a+ie

(6)

= r

log

Z ( 8 ) ao+a+ie --a+ie

-

ao+a+ie /

r

log Z(s)ds.

--a+ie

Now let e ~ 0 to get the equality O'o--ba

f

--a

= r

+

a)logZ(ao -4-a) -r

~to-~a

(7)

- / r --a

To complete the proof of Theorem 2.2, simply combine equations (2) through (7). Note the cancellation of one term in (5) with a term in (7). R e m a r k 1. The value of log Z(-a) is obtained by the analytic continuation of the Euler sum of Z along the horizontal line segment [~0 + a + i e , - a + ie], followed by the continuation along the vertical line segment I - a , - a + ie], which is equivalent to the analytic continuation along the top of the horizontal line segment I - a , ~0 + a] for small e. To be precise, one should write the integral in (7) as

f

--a ao-ba

(s)

=f

--a

ao-}-a

f

--a

56 R e m a r k 2. In the case that Z(s) is real on the real axis, t h e n arg(Z(s)) is a step function on [-a, ao + a] and takes on values in Z 9~ri, except at the zeros and poles of Z, where t h e a r g u m e n t is u n d e t e r m i n e d . In this case the integral with arg Z(s) in (8) can be evaluated directly and trivially, as an elementary integral.

57

w

An inductive theorem.

When comparing our work with that of Cram6r in [Cr 19], the reader should note that we have overcome a point of substantiM technical difficulty that Cram& encountered when proving Theorem 2.2 for the Riemann zeta function ~Q(S). By choosing a suitably, we have avoided having to consider the convergence of the Euler sum of Z on the line Re(s) = cr0' . Cram& used the fact that ~q(s) does not vanish on the vertical line Re(s) = 1 as well as specific knowledge about the distribution of prime numbers, namely

1 ~ < i - = p O(loglogx) as x--+oo

and the Landau theorem which states that the limit 1

li~Ino~E pl +it p 0 such that for all s with Re(s) > a~ we have

n=l

an \ bn ]

----hi

~1


a~ we have logZ(s) = Z q

c(__q)q8 and

log..7(s) = Z c(ocl~) el

59

If we set

V(s) = aNbSN a n d

G ( s ) = albl s,

t h e n the trivial relation P(s) = Q ( - s ) can be written as

G(s)Z(s) =

d(-s)2(-s),

so we also have a functional equation with a0 = 0. Notice that the functional equation implies that all the zeros of P lie in some vertical strip. Further, we can apply T h e o r e m 4.1 to conclude that the Dirichlet polynomial P is of regularized product type, with reduced order (0, 1). As a result, the estimates from Chapter I, specifically T h e o r e m 6.2, hold for any Dirichlet polynomial. Finally, observe that the local factors of the more classical zeta functions are Dirichlet polynomials. Indeed, such factors are of the form Polp(p - s ) + l where Polp is a polynomial with constant t e r m 1, and p is a prime number. For the R i e m a n n zeta function, this local factor is simply Polp(T) = 1 - T, so we have Polp(p - s ) = 1 - p-~. In the representation theory of GL(n), the polynomial Polp has degree n. For representations in GL(2) associated to an elliptic curve, say, we have Polp(T) = 1 - apT + pT 2, so in terms of p - S the local factor is

1 - app -s -4- pp-2S

Thus the local factors of classical zeta functions are themselves of regularized product type.

C H A P T E R III Explicit formulas under Fourier A s s u m p t i o n s The classical "explicit formulas" of analytic number theory show that the sum of a certain function taken over the prime powers is equal to the sum of the Mellin transform taken over the zeros of the zeta function. Historically, only very special functions were used until Weft pointed out that the formulas could be proved for a much wider class of test functions (see [We 52]). We shall give here a version of these explicit formulas applicable to a wide class of test functions in connection with general zeta functions which have an Euler sum and functional equation whose fudge factor is of regularized product type. As a result, our general theorem contains the known explicit formulas for zeta functions of number fields and Selberg type zeta functions as well as new examples of explicit formulas such as that corresponding to the scattering determinant and Eisenstein series associated to any non-compact finite volume hyperbolic Riemann surface. Various facts from analysis which we shall use in this chapter have been proved in our papers [JoL 93a] and [JoL 93b], as well as Chapter I. As a result, most of the steps taken here are relatively formal. We carry out the steps by integrating over a rectangle in the classical manner, but one aspect of this classical procedure emerges more clearly than in the case of classical zeta functions, namely the inductive procedure arising from a functional equation of the type =

2(

0 -

with zeta functions Z and z~ and fudge factor (I) which is of regularized product type. For instance, for the Selberg zeta function of compact Riemann surfaces, these factors involve the Barnes double g a m m a function, and for the non-compact case, these fudge factors may involve the Riemann zeta function itself at the very least. Ultimately, arbitrarily complicated regularized products will occur as fudge factors in such a functional equation.

62

w

G r o w t h conditions on Fourier t r a n s f o r m s .

We shall consider growth conditions on Fourier transforms and logarithmic derivatives of regularized products, and we begin by estimating Fourier transforms. Following Barner [Ba 81], we require the test functions g to satisfy the following two basic Fourier conditions. F O U 1. g e BV(R) A LI(R). F O U 2. g is n o r m a l i z e d , meaning 1

g(x) = ~ (g(x+) + g(x-))

for all x C R.

These will be the only relevant conditions in this section, but in the next section to apply a Parseval formula, we shall consider a third condition at the origin, namely: F O U 3. There exists e > 0 such that g ( x ) = g(O) + O(Ixl ~)

for x ~

0.

If we let N be any integer > 0, then we say that g satisfies the basic F o u r i e r c o n d i t i o n s t o o r d e r N if g is N times differentiable and its first N derivatives satisfy the above three basic conditions. L e m m a 1.1. Assume g satisfies F O U 1 and F O U 2 to order

M. Then

gn(t) -= O(1/Itl M+I)

for ltl ~ c~.

Proof. We integrate by parts M times to give g^(t)

-

-~

g(M)(x)e-itZdx. --

00

To finish, note that for any h C B V ( R ) A L I ( R ) , we have the Stieltjes integration by parts formula O0

hA(t) _ i

1

1 / --

00

e_it~dh(x) '

63

from which we o b t a i n the e s t i m a t e 1

R(h), Ih^(t)l a for which we have the b o u n d

(I)

tF(x)l _ - log ~1 - 1 , we have oc)

e-tdt o

where

0(3

o(t) =

1 -- e -t" n=0

From this, and Theorem 4.3 of [JoL 93b], for any f which satisfies the four basic Fourier conditions and a > - 1 , we have

ff"(t)r'/r(a+it+l)dt T

lim

T--~

1

-T

e - x dz.

This is Weil's formula as reworked by Barner (see [Ba 81], page 146).

CHAPTER

IV

From Functional Equations to Theta Inversions The classical aacobi theta inversion formula for the Riemann theta function of one variable states that for all t > 0, we have the equality 1 ~--~n=-oo E e- n 2 t If we set

1

v~

n=E-ooe-(2rtr02/4t

OO

then the Jacobi inversion formula can be stated as the equality

e(u) = ~ e ( 1 / u )

for u > 0.

SpectrMly, the inversion formula can be viewed as expressing a sum over all the eigenvalues of the Laplace operator on the circle (namely the squares of the integers) as equal to another similar sum, with the inversion t ~-+ 1/t. We give the following very simple spectral interpretation of the Jacobi inversion formula. Let X = 2 r r Z \ R be the circle. The heat kernel for the usual Laplacian on R is

KR(x,t,v) =

1 _(x_v)2/4t 4vqTi~

The heat kernel on X is the 27rZ periodization of the heat kernel KR on R. On the other hand, the eigenfunction expansion of the

86 heat kernel K x can be easily computed. When we equate this periodization with the eigenfunction expansion of the heat kernel, we obtain what amounts to a theta inversion formula, namely 1

oo

oo

1

r~=--oo

e--nZt einX e--iny.

271" n = - o o

In Theorem 1.1 below, we show how the above classical theta inversion formula admits a vast extension to much more general theta functions, essentially formed with the sequence of zeros and poles of functions in the fundamental class as defined in Chapter II, w Specifically, inversion formulas follow from our general explicit formula when using Gaussian type test functions. In this context, the Jacobi inversion formula comes from the explicit formulas associated to the sine function e--~rs

sin(zris) --

i

2i

e ~s

elrs

--

2i

(1 - e -2'~8)

which, when written in this form, can be seen to have an Euler sum and functional equation with cr0 = 0 and a simple exponential fudge factor. We will prove our general inversion formulas in w and give various examples in w We will show conversely in Chapter V how inversion formulas for theta functions satisfying AN 1, AS 2, and A S 3 yield Dirichlet series with an additive functional equation.

87

w

A n a p p l i c a t i o n of t h e explicit f o r m u l a s .

We shall apply the general explicit formula of Chapter III, to the test function ft defined for t > 0 by (1)

ft(u)=

1,~--e-O~ x/ 41rt

Ft(x)-

so

_ ~ e--x2/4t . 4~_

Note that Ft is the heat kernel on R. It is immediate that Ft satisfies the four basic Fourier conditions needed in the proof of the explicit formulas. By a direct calculation, we have (2)

M~o/2ft(s)=eCs-~~

and

FtA(r)= Vr~ 1 e _r2,

For example, to derive the first formula in (2), write OO

M~'~

= f f'(u)uS-~'~ 0 O0

_ =

1

f e_z214t+z(s_aol2)dx

e(S-ao/2) 2t.

With this, the explicit formula yields the following result, which we call a t h e t a inversion formula. T h e o r e m 1.1. Let (Z, 2, ~) be in the fundamental class, and assume that 9 has no zeros or poles on the line Re(s) = a0/2. Let {p) and {~} be as in Chapter m , w Then ~or ~I t > 0 we h ave

1

E -c(q) log q e_(logq)2/4t

4v~7 +-- 1

qao/2

q

E el

+

E~(t),

-c(~)log ~o12

qe_(log ~1)2/4t

88

where

1

Er

= ~

O0

f

+ it)dr.

~-r%'/r

--CX)

R e m a r k 1. As in Chapter III, sums over p, a and the integral for Er are to be understood as limits of sums and integral from -Tin to Tin, taken over a suitably defined sequence {Tin}, depending on Z and (I). R e m a r k 2. If (I)(s) =

G(s)/G(ao - s),

then

E,(~) : Ea(~) + Ea(~) where O0

Ea(,-) =

1 / e_r2tG,/G(ao/2+ir)dr. -- 00

In particular, if G = G, then E r

=

2EG(r).

Let us establish some notation in order to write the formula in Theorem 1.1 in a form fitting AS 1, AS 2 and AS 3. Let L = {#k} be the set of numbers

r = {~k } = { - ( p - ~0/2) 2, - ( ~ - ~0/2)2}, ordered as a sequence, with integral multiplicities

{a(~k)} = {v(p),v(~)}. Define the associated theta function

(3)

0L(~) = ~

v(p)~ (.-~0/~)~' + ~ p

v(.)~ ("-~o/~)2' c~

Similarly, let L v be the family of numbers L V = {#vk} = {(1ogq)2/4,(l~ with (not necessarily integral) "multiplicities" {a(#~)} = { - c ( q ) l o g q -c((])1og 61 qc,ol2 ' ~ao/2 j,

89

and define the associated theta function --c(q)log qe_[(,og q)2/4]t qao/2

OLV(t) = ~ q

+

(4)

et With this notation, we can now write Theorem 1.1 in the following form. T h e o r e m 1.2. With notation as above, we have the inversion formula 1 On(t) .~_~:8nv(1/t)+ E , ( t ) . x/ 41rt In general, we define a theta inversion formula to be a relation between two theta functions, such as (3) and (4), with an additional term such as Ev(t) which we require to satisfy the asymptotic condition AS 2. T h e o r e m 1.3. With notation as above, the theta function

p

satistles the asymptotic conditions AS 1, AS 2, and AS 3. Proof. By the Cram6r theorem, we have that the sequences {p/i:Z(p)=O

with Im(p) > 0

and

-a - a .

Then for M1 t > 0 we have p

_1~

+~

[~q--C(q)logqe_OOgq)2/4t]

4~o/~

+ E~(elo./e~o.)(O + E-o(e',i~h,/e~h,)(t).

91 We note that in Theorem 1.4 both theta series satisfy the asymptotic conditions A S 1, A S 2, and A S 3. This assertion is true for the sum over ~ since we assumed ~ was of regularized product type, because we choose - a sufficiently far to the left, and so we can apply Theorem 7.6 and Theorem 7.7 of [JoL 93a]. As for the sum over p, one applies Cram~r's theorem and Theorem 7.6 and Theorem. 7.7 of [JoL 93a]. R e m a r k 3. In the present application, and the subsequent ones in this chapter and the next, we apply the explicit formula to the simplest types of Gaussian test functions. Even for such test functions, one can give examples where instead of (47rt) 1/2 or (4~rt) 3/2 (as in Chapter V, w one takes (4~rt) n/2 for an odd integer n, and in addition one also has an arbitrary polynomial as a coefficient. If one uses (4~rt) n/2 with an even integer n, then one gets Bessel series instead of Dirichlet series. All these cases deserve to be treated systematically, since they apply to several important situations of spectral analysis for classical manifolds, and we shall do so elsewhere. Here we selected the simplest cases to serve as examples.

92

w

Some examples of theta inversions

To emphasize the significance of the theta inversion formula given in Theorem 1.1, let us now discuss a few specific applications of the theorem. E x a m p l e 1: T h e s i n e f u n c t i o n . Let us write the sine function as 6 -Trs

sin(Tris) --

__ e 7rs

elrs

2i

= a(4Z(s)

2i

where G(s) = - e ' ~ ' / 2 i and Z(s) = 1 - e -2'~'. The fact that the sine function is odd trivially yields the functional equation = -a(-4z(-,),

so a0 = 0. Further, Z satisfies the Euler sum condition since log Z(s) = log (1 - e - 2 ~ ' ) ---- ' ~ n.~--I

le-2'~ns, n

whence {q} = {e 2rrn } for n > I and c(e 2rrn) = 1/n. Since sin(Tris)is zero only when s E Z, and then with multiplicity one, the inversion formula Theorem 1.1 specializes to the equality oo

1

(X)

E e - " h = 2 " - -4x/-~tE 2~re-(2'~n)~/4t+ Ee(t),

(1)

n----1

n.~----O0

where the factor of 2 appears since the sums over {q} and over {~} coincide in this example. Also, we have (2)

rr

v'~ "

Combining the terms in (1) and (2), we obtain the classical Jacobi inversion formula, which is the relation I

oo

27r n=--c~ E 6-n2t --

I

~

oo

E

e--(21rn)2/4t

for t > O.

93

Similarly, one can derive other classical formulas using the cosine function and hyperbolic trigonometric functions. E x a m p l e 2: D i r i c h l e t p o l y n o m i a l s . Recall the definition of a Dirichlet polynomial given in w of Chapter I. We saw in Chapter II that any Dirichlet polynomial P can be written as

P(s)=aNbSN [ 1+ ~1 aN a'--2-n\bN,] J =aNbsN'Z(s)' where Z has an Euler sum and functional equation with fudge factor ~(S)=

aN ( b N ~ s a, \ - C 1 /

Hence, one can directly evaluate the Weil functional, yielding the formula O0

E,(t) =

1 / e_r2t(log(bN/bl))dr_ log(bN/b1 ) --00

Therefore, the associated theta inversion formula is simply v(p)e" 2* _

log(bN/bl) x/47rt

~

1

~

c(q)(log q)e -(l~

a)~/4t

q 1

~ C(~l)(log (])e -O~ eO'/4t. el

The specific case of N = 2 with b2 = b~-1 = e '~s

and

a2 = - a l =

i/2

yields the Jacobi inversion formula. We find this example particularly interesting for the following reason. In Example 1, the Jacobi inversion formula is a relation involving two well-known sets of data, meaning that both theta functions in the inversion formula involves the squares of integers. However, in the case of a Dirichlet polynomial, we have one wellknown set of numbers, namely the sets {q} and {(t} which are

94 explicitly and simply expressed in terms of the initial set of numbers {aj, bj}, and one unknown set of numbers, namely the set of zeros.

Example 3: Zeta functions of number fields. In w of Chapter III we gave an evaluation of the Weil functional associated to the classical g a m m a function. Hence, there is a theta inversion formula associated to any zeta function of a number field as in Theorem 1.1. The term Ev or EG is given explicitly as follows. If =

,

then, by page 146 of [Ba Sl], see also w of [JoL 93b], we find

EG(t) --

log A 1/2

-2-~

i e-~h dr --OO

i r'/r(l/2+ it)e-r2tdr --OO

OO

+ 2---T-

0

_rl/2 _

__ __ 2e-X2/4te3x/2O(2x)] e-2Xdx 0

where

oo

n---~0

1 1 - e -x

It is important to note that to zeta functions, we are associating theta functions which admit inversion formulas but are different from the theta functions used in Hecke's proof of the functional equation and meromorphic continuation (see, for example, Chapter XIII of [La 70]).

Example 4: A connection with regularized products. In spectral theory, one meets the situation when a certain operator has the sequence of eigenvalues L = {~k } with integer multiplicities {ak}. In such a situation, when the theory of regularized products

95

applies (namely our axioms from [JoL 93a], recalled in Chapter I, w one may have the additional relation DL(~(8--ao)) = Z(s)G(8).

This occurs for instance for the Selberg zeta function and its analogues. Then (pk - * 0 / 2 ) 2 = - x k + o0 /4. Thus we have the simple relation v(pk)e (ph-a~ k

--

e(ao2/4)t ~"~ake-~

t --_ e(a~/4)teL(t),

k

expressing the theta series formed with the squares (Pk - a0/2) 2 in terms of the theta series with the eigenvalues. As stated above, in the case of the operator d 2/dx 2 on the circle S 1 = R/2~rZ, this relation reduces to the Jacobi theta series formed with )~k = k 2 and a0 = 0. In both cases we are dealing with the eigenvalues of the positive Laplacian on some Riemannian manifold.

CHAPTER

V

From Theta Inversions to F u n c t i o n a l E q u a t i o n s In this chapter we shall carry out the inverse construction of the preceeding sections; that is, from a theta series satisfying an inversion formula, we derive, by means of a Gauss transform, a Dirichlet series satisfying an additive functional equation. Of course, Riemann's proof of the functional equation of the Riemann zeta function also relied on a theta inversion formula, but in the present situation, our use of theta inversion is different from that of Riemann because we take a Laplace transform, with a quadratic change of variables, of the regularized theta series instead of the Mellin transform of the theta series. Hence we construct new types of zeta functions which are essentially regularized harmonic series. In case the residues are integers, such series are logarithmic derivatives of functions in the fundamental class. Thus, we see that the general theory requires that we consider the additive class rather than the more restrictive multiplicative class. Let R be of regularized harmonic series type. In w1, we analyze the transform E~ defined by u+iT,~

E,,R(t) = 2ira

1 J e(~_,,o/2)2tR(z)dz. u-iT.

In w we carry out the properties of the Gauss transform of theta These propseries, inverted theta series, and the transform erties are shown to imply the functional equation of the new zeta function in w In w we work out an example to obtain a zeta function for any compact quotient of the three dimensional hyperbolic space Ma. This zeta function lies in our additive class but, in general, not in our multiplicative class. For extensions of this example, see the remarks at the end of w

E,,R.

Throughout the remainder of this chapter, unless otherwise specified, we use the following basic conditions.

98

T h e basic conditions We let ~r0 be a real number > 0. We let R be a function of regularized harmonic series type. We let {#k} and {ak ) satisfy D I R 1, D I R 2, and D I R 3, and we assume that the corresponding theta series oo

O(t) = OL(t) = ~

ake - ~ `

k=O

satisfies AS 1, AS 2, and AS 3. We let (Mo, m) be the reduced order of the theta series 8. Note that we have used #k instead of Ak in the notation. This is because in a subsequent application to spectral theory, there will be an operator with eigenvalues /kk which are translates of #k, namely

#k -- Ak - a2o/4. We let R0 be the regularized harmonic series associated to the theta function 8, as defined in Chapter I, w that is,

Ro(z)=CT,=ILMO(s,z). We let {q} be a sequence of real numbers > 1 converging to infinity. We let {c(q)} be a sequence of complex numbers, and we let c(q)

=Z

q

qS

be the associated Dirichlet series, which we assume converges absolutely in some right half plane. Thus, JI'(s) = ~ q

- c ( q ) log q q~

We let OLv (t) : ~ q

--c(q)log qe_[(logq),/4], q#o/2

99 w

T h e WeU f u n c t i o n a l of a G a u s s i a n t e s t f u n c t i o n .

In this section, we are concerned with a function R of regularized harmonic series type. For each ( E C we let:

aa(~) = a(~) = residue of R at 4Observe that if R = Re = ~ / ~ , where ~ is of regularized product type, then a(4) = v(() is the order of 9 at ~. Any function R of regularized series type can be expressed as a sum of two such functions, each of which has poles only in a half plane to the left or a half plane to the right. Let us write such a decomposition as R ( z ) -~ Rleft(z) + Rright(Z). Assume that A > 0 is chosen so that all the poles of Rle~t lie in a left half plane of the form Re(z) < A - 5, for some ~ > 0. Similarly, assume that a > 0 is chosen so that all the poles of Rright lie in a right half plane of the form Re(z) k - a + 5, for some ~ > 0. Assume that Rleft has reduced order (Mleft,mleft) and Rright has reduced order (Mright, mright) As a direct application of Theorem 6.2(a) of Chapter I, we obtain the following result. L e m m a 1.1. W i t h notation as above, we have E a R l e f t ( t ) = 0(• -(Ml*rt+2)/2)

for $ ---+0

and E-aRright(t) = O(t -(M~'ght+2)/2)

for t -~ O.

Proof. By Theorem 6.2(a) of Chapter I, we know that on the line Re(z) = A the function Rlea has polynomial growth. In fact, the integral giving EARle~t can be coarsely estimated by

with some number b > 0. The first estimate asserted then follows from the standard change of variables y = x/tu. The second estimate is proved similarly. [] The proof of Lemma 1.1 also applies to prove the following asymptotic bounds.

100 Lemma

1.2. W i t h notation as above, we have

EARleft(t) = O(t -(Ml'ft+2)/2)

for t --~ co

and E-aRright(t) =- O(t -(M~ight+2)/2)

for t ---~ co.

Note t h a t the b o u n d given in T h e o r e m 6.2(a) of C h a p t e r I is stronger t h a n w h a t is used in the proof of L e m m a 1.1 a n d L e m m a 1.2 above. However, t h e above results are sufficient for our purposes. In fact, one can easily improve the exponent to - ( M + 1)/2 + e for any e > 0. T h e b o u n d s in L e m m a 1.1 and L e m m a 1.2 are e n o u g h to allow us to deal with the Gauss transforms of EARle~t a n d E-aRright in the next section.

101

w

Gauss transforms.

We shall need some analysis concerning Bessel integrals and what we call the Gauss transform, which we carry out in this section. Throughout we let r, t, u, x be real variables with t, u, x > 0. We start by recalling the collapse of the Bessel integral Ks(x, u) under certMn conditions. L e m m a 2.1. Let oo

f

dt

K~(x, u) = I g-x2, e-u2/tts a

7"

Then: a) In the case s = 1/2, we have the evaluation Kll2(X, u12) = Vl-~e-uz.

X

b) Let ~[~ and Z[~ be the differential operators

~I~ -

10 2x Ox

10 and

2lu=-~uuOuu.

Then we have the relations ( 2 I , ) " K s ( x , u ) = Ks+n(x,u)

and (2l~)"Ks(x,u) = K , _ n ( x , u ) . For further properties of the Bessel integral, including a proof of Lemma 2.1, the reader is referred to [La 87]. W a r n i n g . For our purposes, we use a normalization of the Bessel integral slightly different from the classical one. If, for instance, K ~ ( c ) denotes the g - S e s s e l function which one finds in tables (see Magnus, Oberhettinger, Erdelyi, Whittaker and Watson, etc.), then we have the relation

2K (2c) = Ks(c)

102

where OO

g,(c)

=f 0

The classical normalization gets rid of some factors in the differential equation, and our normalization gets rid of extraneous factors in the above integral. For a suitable function f of a single real variable, we recall the Laplace-Mellin transform is defined as

o

We shall deal with f = /9, where 0 is a t h e t a series of the sort considered previously. We shall put s = N + 1, where N is an integer > 0, sufficiently large to cancel the singularity at 0, and we shall also make a change of variables with z = (s - a0/2) 2. W i t h this, we obtain an integral operator which we call the G a u s s transform. More precisely, let N be a positive integer and let or0 be a real n u m b e r > 0. For a suitable test function f , we define OO

Gauss(~oN~2( f ) ( s ) = ( 2 s -

ao) / f(t)e-(S-a~

dtt

o

Then

Gauss(No]2(f)(s) = dz. ds

L M f ( N + 1 , ( s - a0/2)2).

Because of the change of variables from z to s, we are led to consider the differential operator :Ds_ao/2 -- - - -

d

1

o

ds 2s - ao"

Let x = s - a0/2. For every integer N > 0, we t h e n get the formula 1 __DN

2x" 21 N o 2x - -

s-~0/2

~

103 G a u s s t r a n s f o r m o f a n i n v e r t e d t h e t a series

We now shall prove inversion formulas for Gauss transforms of certain series. Such series occurred in the previous section in connection with functions in the fundamental class. Here we start with series for which we assume only the basic conditions. Directly from Lemma 2.1, we have the following theorem. T h e o r e m 2.2. Let G(t) -

1

~-, - c ( q ) log q e--(log q)2/4t.

4X/4' X/4' X/4' X/~4' X/q4~

qa0/2

Then for s r e M and sufl]ciently large, and N >_ O, we have.

GausS~o~]2(a)(s) = vf_,,o/2.n'(s ). Proof. The proof follows immediately from the definitions and andu=logqand Lemma2.1. For e a c h q , we put x = s - a o / 2 compute the transform of each individual term gq(,)

1

=

e

_u2/4t

as follows:

r

_

2x ~KN+l/2(z,u/2) 2x

- f---v,qlI.

N

9

g,12(x, u12)

- ~-~llx \2x = ,DN e-OOg ,:l)~

]

= DN_a0/2 (q-Sqa~ from which the formula in Theorem 2.2 is now clear.

, []

104

Gauss transform of a theta series

Next, let us apply the Gauss transform to a theta series itself; that is, let us evaluate CX~

~_ (28 -- (TO)/ O(t)E--(s-a~ 0

d-~

where OO

O(t) = E

ake-~kt"

k----O

Let Pk be complex numbers, all but a finite number of which have positive imaginary part, and such that (Pk - a0/2) 2 = --#k.

T h e o r e m 2.3. For any positive integer N s u ~ c i e n t l y large, and s t e a / a n d sut~ciently large, we have O0

/ O(t)e_(s_~o/2)~ttN+ 1 dt 0

k=0

Ek=0a k

r(N +

oo [( ~ -

1)

pk)(~ + ;~ - ~~

r ( g + 1) [(~ - ~ 0 / 2 ) ~

- (pk - ~ 0 / 2 ) ~ ] N + ~ '

hence the series gives the meromorphic continuation of the integrad to adl s C C with poles at the points s = Pk and s = cro - Pk. Proof. This identity follows routinely by integrating term by term, using the change of variables t ~ t / ( s - ao/2) 1/2. There is no problem with the square root since s was assume to be real and sufficiently large. []

Next we obtain the inversion analogous to Theorem 2.2, but for the theta series instead of the q series.

105

T h e o r e m 2.4. For any positive integer N sut~ciently large, a n d s real and sumciently large, we have

Gauss 0 /)2(O)(s) :

v f o/2

Proof. As a function of s, (2s - ao)Ro((s - a 0 / 2 ) 2) is a merom o r p h i c function whose singularities are simple poles at t h e points s : Pk a n d s ---- cr0 - Pk w i t h residue ak. For any individual k, we have - - -~Z

Z "gy ~ k

- - ( Z -~- ~ k ) N - b 1 "

So, N

2s --- a0

1

(~

-

--[(8 - r

~o/2)2

-

(pk

-

~o/2)2

r ( N + 1) 2 - ( P k -- (Yo/2)2] N-b1'

which can be w r i t t e n as 1 2 8 - - O"0

~N s--ao/2

2s - a0

(s - o o / 2 ~ -- (-pk - o 0 / 2 ) 2

]

J

"

T h e result now follows from T h e o r e m 2.3 after m u l t i p l y i n g by the t e r m 2s - a0 a n d t h e n s u m m i n g over k, which is valid since for N sufficiently large the series is absolutely convergent. [] R e m a r k 1. F r o m T h e o r e m 7.6 a n d T h e o r e m 7.7 of [JoL 93a], one can show t h a t it suffices to take N > (Mo + 2)/2, w h e r e has r e d u c e d order (Me, m). In fact, this r e q u i r e m e n t can be easily i m p r o v e d to N > (M0 + 1)/2 + e for any e > 0. R e m a r k 2. In the d e g e n e r a t e case w h e n Pk = a0/2, t h e regularized h a r m o n i c series w i t h the q u a d r a t i c change of variables has a double pole at s = a 0 / 2 w i t h zero residue, which is consistent with t h e above evaluation of residues.

106

I n v e r s e G a u s s t r a n s f o r m of a r e g u l a r i z e d h a r m o n i c series t y p e Finally, we deal with the Gauss transform applied to the terms

EARleft E-aRright which arose in Chapter IV. The next lemma evaluates this Gauss transform, and amounts to an inversion formula, showing that up to taking derivatives, the Gauss transform is the inverse of the EA transform. T h e o r e m 2.5. With notation as above, we have, for N sutl]ciently large and s real and sufficiently large, the equalities Causs(0N]2(EARleft)(s) = 7)Ea0/2Rleft(s ) and

GausS~o~/2 (E--a ~right )( S ) = "D;Lc~o/2 ~right (0.0 -- '~).

Proof. Let F(s) = / EARleft (t)e -(s-a~

ZttN+l dtt

0

which is defined for N sufficiently large by Lemma 1.1, and s real and sufficiently large by Lemma 1.2. Then

oo A+icx~ (1) F(s) -- 1 f / e(Z_ao/2)2te_(S_ao/2)2tRleft(z)tN+ldzdt 2~i ---t" 0 A--ioo By changing the order of integration, which is valid by Theorem 6.2(a) of Chapter I, we may write (1) in the form

,/

A+ioo r ( N + 1) [(8 - 0-0/2) 2 - (z - 0-0/2)2] N + I Rleft

(2) F(~)- ~ i

A-ioo Let us write

r(N + 1) [(s - ~

= ~,-~o/~

- (z - 0.0/2)2]N+,

(~ _ ~ o / 2 ) ~

-

(z - ~o/27

'

(z)dz.

107

which allows us to express (2) as

F(s) (3) A+ioo

27ri

~I~-~~

(s--

A-/oo

o0/2)5

(z

-

-

~0/2)2

Because of the s y m m e t r y in s and z, we can write (3), as

F(s) A+ic~

-

1 i 21ri

(~z-~0/2 )N

(-1) g

(~

- o0/27 : ~ - ~0/2)~ ]

R,e, O.

> l(max{MRl~ft,MR~ight,Mo}

-t- 1) +

e,

110

T h e o r e m 3.2. There is a p o l y n o m i M P ' os degree < 2N - 1 such that for all real s sumciently large, we have

(2s - a0)RoL

((s

-

a0/2) 2)

= JI'(s) + Rleft(s) + nright(G0 -- S) + P'(s). Proof. Let r ( ~ ) = (2~ - o0)RoL ((~ -- ~ 0 / 2 ) 2) -- a ' ( ~ ) -- R l e f t ( s ) - Rright(O'0 -- 3),

and let us write the formula in Theorem 3.1 as O 1 ) ~)N--1 Os 2s r ~_~o/2F(s) = O.

Hence, there exists a polynomial P1 of degree < 1 such that DN-1 s _ , o / 2 F ( s ) = Pl(s). The proof finishes by continuing to unwind the differential operator ~DN - 1

s_ero/2 9

[]

C o r o l l a r y 3.3. T h e Dirichlet series JU(s) has a m e r o m o r p h i c continuation such that under that m a p s ~-~ ao -- s the function J-I'(8) + Rleft(8) + Rright@r0

-- 8) + P ' ( s )

is odd.

This is immediate from Theorem 3.2 and the formula in Theorem 2.3. Upon integrating, the functional equation obtained in the present case is what we call an additive functional equation, to distinguish it from the multiplicative functional equation satisfied by functions in the fundamental class. We have now concluded the process from which a theta function with an inversion formula leads to a Dirichlet series having an additive functional equation with fudge terms which are of regularized harmonic series type.

111

E x a m p l e 1. If we apply the above construction to the Riemann zeta function, then we have

R(~) = ~1 r /,r ( , / 2 )

+ ~r'/r((1 - ~ ) / 2 ) - log ~,

hence we have R l e f t ( s ) ~-~

~r'/r(~/2)-log

7r,

and

Rright(S)

=

-~r'/r((1 - ~)/2).

It is easy to see that JI'(s) = 2(~/(Q(S) and M = 0, hence the above theorems apply with N = 1. Therefore, Corollary 3.3 reconstructs the fact that the function

2r162

= 2~/r

+ r'/r(~/2) - log~r

satisfies the functional equation ~ / ~ Q ( S ) = - ~ / ~ Q ( 1 - s). R e m a r k 2. It may seem odd that whereas we started with a triple (Z, Z, if) in our fundamental class with functional equation which is not symmetric, we derived a function in Corollary 3.3 which has a symmetric functionM equation up to an additive polynomial factor. The reason for this is the following. Since the test function to which we apply Theorem 2.3 of Chapter III is even, we can not distinguish between elements in the set {q} and elements in the set {~}. Therefore, we actually ended up considering the zeta function H i ( s ) = Z ( s ) Z ( s ) which satisfies the symmetric functionM equation Hl(s)~(s)

= Hx(ao - s ) ~ ( a o - s).

One could equally well use the odd test function Ft(x)-

-x/2t ~ e -

~/4" . "

so then M ~ o / 2 f ( s ) = - ( s - cro/2)e (~-~~

112 After carrying through the computations as above, we end up with a formula of the form ( 2~ - ~o ) n o ( ( ~ - ~o / 2 )2 )

= ,n'(~) -(~

-0-0/2

[n~oft(,~) - .n~ht(o-o - s)] + P'(,~).

This equation corresponds to the functional equation H2(s)H2(.o

- ~) = 1

where H~(~) - Z(.~)= .~(s). Z(~) Then, combining the two functional equations above, we recover the original functional equation (up to a sign) (z(.~),i,(~)) ~ = (2(0-o - ~))~, in other words z(~),I,(~) = +2(o-o - ~). E x a m p l e 2. The functional equation for the function //2 in Remark 2 is like that of the scattering determinant associated to any non-compact hyperbolic Riemann surface of finite volume (see [He 83]).

113

w176A new zeta function for compact quotients of M3. In this section we will give, what we believe, is a new example of a zeta function. The approach is that of the previous sections, namely to every theta inversion formula there is a corresponding zeta function. As we will see, the "zeta function" has a Dirichlet series in a hMf-plane, but is a regularized harmonic series whose singularities are simple poles whose residues are not necessarily integers. Hence the function is not the logarithmic derivative of a regularized product type. Let M3 be the simply connected three dimensional Riemannian manifold whose metric has constant sectional curvature equal to - 1 ; see page 38 of [Ch 84] for a precise model. On M3 there is a transitive group of isometries, so the heat kernel relative to the Laplacian acting on smooth functions, which we denote by KM3, is a function of distance r and t. In fact, let

(1)

h3(r,t)-

1 _r2/4te_ , r (47rt)3/2 e sinh r"

Then gM3(X,t,~)

=

h3(dM3(~,~/),t).

For details of this computation, see page 150 of [Ch 84] or page 397 of [DGM 76]. The result is basically due to Millson. Let X = F \ M 3 be a compact quotient of M3. Using the spectral decomposition of the heat kernel K x on X, one has the expression (2)

Kx(x,t,y) = E Ck(X)r k=O

where {r is a complete system of orthonormal eigenfunctions of the Laplacian and {Ak } is the corresponding system of eigenvalues. Choose 2 and ~ in M3 lying above x and y in X. Then

(3)

Kx(x,t,y) : E KM~(2,t,7~). yEF

View the pair (x, y) as fixed, and for each 7 C F define logq-y(x, y) = logq7 = dM3(~,TY).

114

Thus we obtain a sequence {q~}eEr, which we reindex by itself as simply {q}. Suppose x # y. Then by (1), we have (4)

Kx(x,t,y)-

2 --t e--(log q)Z/at (4~r~)3/2 ~ q _lo.__g_q_ q-1 q

which we call the F - e x p a n s i o n or the q - e x p a n s i o n . Combining (2) and (4), we immediately obtain the theta inversion formula OO

k----0

(5)

2

-

(4~rt)3/2

e- t

E q _lo_g.q q-1 q

e_(log q)2/4t.

Observe that the factor e -t could be brought to the other side. In the notation of w cr0 - 2 and #k = Ak -- 1. We note that (5) holds for any points x ~ y on X, and we have not taken a trace of the heat kernel, although we have taken the trace with respect to the infinite Galois group F, i.e. the fundamental group of X. Formula (5) simply reflects a combination of the existence and uniqueness of the heat kernel on X together with the spectral expression (2) and the group expression (3) for K x . As in the previous sections, any theta inversion formula can be used to obtain a zeta function with additive functional equation. Let us carry through the computations in this case. We are still assuming x ~ y. From (5), let us derive two expressions for the function OO

(6)

Fx,y(s) = (2s

-

2) / Kx(x,t,y)e-s(s-2)tt2dt. 0

Using the spectral expansion (2), the integral in (6) yields the equality (7)

F,,y(s) = (2.s - 2) E

k----0

(s(s - 2) + Ak)a'

which can be shown to converge uniformly and absolutely on X by combining standard asymptotic formulas for eigenvalues on compact manifolds (see [BGV 92], for example) and Sogge's theorem on sup-norm bounds for eigenfunctions (see page 226 of [St 90]).

115

On the other hand, we shall also take the Gauss transform of the right side of (5), for which we need the collapse of the Bessel integral OO

K3/2(s - 1,log q)= f e -(s-1)2te -(l~ q)2/4tl~3/2d_.~ 0

--

2s

-

2 0s

q-(~-l)

s

)

by Lemma 2.1. Therefore, taking the Gauss transform of (3), or the right side of (5), we obtain a Bessel sum for Fx,v which collapses to a Dirichlet series, for Re[(s - 1)2] > 0, namely 2 F,,~(s) - (4~r)3/2 E

q_l~

(2s - 2)K3/2(s - 1, log q)

q

which becomes

(s)

1 a ( -1 logq -s~ F',y(s)-2rOs 2s--2E1_q-2 q )" q

Using a simple argument involving volumes of fundamental domains, one can show directly that the Dirichlet series (8) converges for Re(s) > 2. Therefore, we have for s in this half-plane, the formula 1 0 ( Fx,v(s)-27rO s

-1

logq

)

2s_2 E 1Zq-2q-" q

oo

= (2~

-

2r162

2) Z (~(~ 21 + ~1~ _

k----O

From (7) it follows that the Dirichlet series in (8) has a meromorphic continuation to all s E C with the additive functional equation

F,,~(~)=-r,,~(2-~). Further, the meromorphic continuation has singularities precisely when s(s - 2) = --Ak, that is (9)

s ---- 1 :k iL/Ak -- 1

116

each singularity being a double pole with zero residue. U p o n integrating with respect to s, we obtain the Dirichlet series (10)

$Ix'y(S) = E

1 q-S 1_ q-2 = E E

q

q-S-2k,

q k=O

which satisfies the relation ~Ds_l JItz,y(8) -~ -27rFx,y(._q ).

The function JI = dI:~,y plays the role of the logarithm of a zeta function in the f u n d a m e n t a l class, and would be such a l o g a r i t h m if the residues of JI ~ were integers, but they are not in general. Theorem i) ii) iii) iv)

4.1. The function dI has the following properties:

The series for dI(s) converges for Re(s) > 2; JI ~ has a meromorphic continuation to edl s C C; dI ~ is odd u n d e r the m a p s ~ 2 - s; The singularities of the meromorphic continuation of the function ( 2 s - 2 ) d I ' ( s ) are all simple poles at the points (9), with corresponding residue 2~rr162

P r o p e r t y (iv) should be viewed as a type of R i e m a n n hypothesis for the zeta function (10). We conclude with some remarks in the case x = y. We pick = #. T h e n the sum in (1) must be d e c o m p o s e d into the t e r m with 7 = id and the other terms. We have qid = 1 and q.y > 1 for all 7 ~ id. In the present case, the t e r m with 7 = id is easily computed, and instead of (4) we t h e n obtain

(11)

Kx(x,t,x)

1

-- (4rot)3~2 e

-t +

2 (47rt)3/2

e-t E q

log q q _ q-1

e-(l~

q)2/4 t

where the sum is over q = q'r for 7 # id. Thus the identity t e r m must be h a n d l e d separately. The integral transformation considered in (6) is sufficiently simple so t h a t we have oo

2 s - 2 / e_(~_l)htl/2dt 1 (4-zr)3]~ t --4Ir" 0

117 As a result, when considering the series (10) for x = y, summing over q.~(x, x) = dM~ (x, "y:~) with -y r id, all properties (i) through (iv) hold with property (iii) being changed to allow an additive factor which is a polynomial of degree three in s. The coefficients of this polynomial can be determined by considering the limit as s ~ oc and using the asymptotic formulas from Chapter I. R e m a r k 1. Computations similar to the above hold for compact quotients of odd dimensional hyperbolic spaces, since, for these manifolds, one can obtain simple expressions similar to (1) for their heat kernels; see page 151 of [Ch 84]. For non-compact quotients, one must take into account the appearance of the Eisenstein series in the spectral decomposition of the heat kernels, and the subsequent appearance of other terms in the additive functional equation. Examples of such formulas, as well as the more complicated situation of even dimensional hyperbolic manifolds, will be treated in a future publication. R e m a r k 2. Now that we have constructed a zeta function with additive functional equation and a Dirichlet series, we can apply the methods of [JoL 93c] to study the function (11)

~

Cn(X)r

z'iv/-~-I ,

n~0

defined for Ira(z) > O. As in [JoL 93c1, there is a meromorphic continuation of (11) to complex z with singularities at the points {+ log q}. This should be viewed as a Duistermaat-Guillemin type theorem, as in [DG 75], since (11) is a wave kernel. We shall deal with this situation at greater length elsewhere, especially since it requires a systematic exposition of the additive fundamental class and its corresponding Cram6r theorem and explicit formulas. R e m a r k 3. The above example shows the necessity of an "additive class" of zeta functions, as discussed in the introduction, by which we mean functions which are meromorphic with simple poles, have Dirichlet series in a half-plane, and have an additive functional equation with additive fudge factors expressible as linear combinations of regularized harmonic series types. The above example is not expressible in the fundamental multiplicative class since the residues, as determined in (iv) above, will certainly not be integers.

118 R e m a r k 4. Readers will note the distinction between the zeta function defined by Minakshisundaram-Pleijel in IMP 49] by the Dirichlet series OO

r162

and the new zeta function JI~,y.I The Minakshisundaram-Pleijel zeta function is essentially the one obtained as the Mellin transform of the theta series satisfying the basic theta conditions, whereas the new zeta function is obtained as the Gauss transform of the theta series.

CHAPTER

VI

A g e n e r a l i z a t i o n of Fujii's t h e o r e m Let L~ denote the set of zeros of the Riemann zeta function with positive imaginary part, meaning L~={pEC:fQ(p)=0

and

Ira(p) > 0 } .

Write p = fl + i7 for any p C L~. The zeta function

v(p)

(1)

,(p)

defined for Re(s) > 1, was studied in [Del 66] and [Gu 45], and it was shown that (1) admits a meromorphic continuation to all s E C with explicitly computable singularities, including a double pole at s = 1. Building on these results, Fujii considered the zeta functions

,(p) sin(o~)

(2) pEL +

and

~ ,(p) cos(a')') 7s

(3)

pEL +

for non-zero a e R and Re(s) > 1. It was shown in [Fu 831 that (2) admits a holomorphic continuation to all s E C for any non-zero a, and (3) admits a mermorphic continuation to all s C C with a simple pole at s = 1 having residue equal to -(27r)-lA(e~)e -a/2, where

(4)

A(x)

S logp,

t

0,

if x = p k w h e r e p i s a p r i m e a n d k C Z > 0 otherwise.

120

Instead of functions formed separately with sine and cosine, one may as well consider what we call the Fujil f u n c t i o n eiC~7

=

v(p) ~/--T-

and its meromorphic continuation. We should note that Fujii's proof of the meromorphic continuation involves a very detailed study of many integrals arising from a generalization of Delsarte's work [Del 66] involving various integral transforms of the classical Riemann-von Mangoldt formula. As remarked on page 233 of [Fu 83], one can prove analogous results for the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular group PSL(2, Z). These theorems are given in [Fu 84b] and are as follows. Let {Ak} be the set of eigenvalues of the hyperbolic Laplacian on the space PSL(2, Z)\h and set Aj = 1/4 + rj with rj > 0. For any a E R +, Fujii considered the function sin(arj) ,

U

3

which is defined for Re(s) > 2. Through a rather lengthy and involved application of the Selberg trace formula, it was proved in [Fu 84b] that (5) has an analytic continuation to all s E C to a holomorphic function. Again, one could consider the Fujii function

rj

for all o~ E R and study its meromorphic continuation, thus capturing many of the results obtained by Fujii in the papers [Fu 84b] and [Fu 88a]. At this point, one could envision a series of articles in which one would define and study a Fujii function associated to every special zeta function, such as zeta functions and L-functions from the theory of modular forms, zeta functions and L-functions of number fields, spectral zeta functions constructed from the eigenvalues

121 associated to the Laplacian acting on any non-compact arithmetic Riemann surface, to name a few examples, with each example yielding a new paper. Such a case-by-case extension of the classical Cram6r theorem [Cr 19] has begun to appear in the literature. However, in [JoL 93c], we gave a vast generalization of Cram6r's theorem containing all previously known special cases and many more. Similarly, in this chapter, we obtain a generalization of Fujii's theorem which applies to any zeta function with Euler sum and functional equation whose fudge factors are of regularized product type. This generalization is simply a corollary of the generalized Cram6r theorem, and, in particular, applies both to the zeta functions arising from algebraic number theory and to those arising from spectral theory. In w we will state the generalized Fujii theorem, and the proof will be given in w To conclude this chapter, we will give various examples of the generalized Fujii theorem in w

122

w

S t a t e m e n t o f t h e g e n e r a l i z e d Fujii t h e o r e m .

Let us assume the notation of the previous chapters. With this, we can state the following result, which we call the generalized Fujii t h e o r e m . T h e o r e m 1.1. Let (Z, 2 , ~) be in the f u n d a m e n t a l class. Let a be such that ao + a > a'o, and let {p} be the set of zeros and poles of Z in the open infinite rectangle ~ a with vertices at the four points - a + ioo,

-a,

ao + a,

ao + a + icxD.

Let v(p) = ordpZ and set {A} = { p / i } . Then:

i) For any non-zero a E It, the Fujii function eia)~

Fz, o(s;

v(p)

= pE'R=

A8

has a m e r o m o r p h i c continuation to all s E C.

ii) For any non-zero a E R , the continuation of the Fujii function Fz,a(s; a) is holomorphicfor aJ1 s E C except for simple pole at s = 1 with residue

--(27/') lc(q) log q

if a = log q

-- (2~') -1C(~)(log ~)~--ao

if a = - log

0

otherwise.

In [Fu 83], Fujii considered the zeta function formed with the imaginary parts of the non-trivial zeros of the Riemann zeta function, assuming the Riemann hypothesis. The following theorem generalizes this result. T h e o r e m 1.2. W i t h notation as in Theorem 1.1, assume there is a real constant flo such that p = flo + i7 for all p C ~ a . Then: i) For any non-zero a C R , the Fujii function e i oL"y

RH

v(p) V

Fio (s; pE'R=

123 has a m e r o m o r p h i c c o n t i n u a t i o n to M1 s C C.

ii) For a n y non-zero a E R , the c o n t i n u a t i o n o f the Fajii function F~,~ RH (s; a) is h o l o m o r p h i c for all s C C except for simple pole at s = 1 w i t h residue

--(27r) lc(q)(log q)q-~O

if a = log q

--(27r) -1 c(q)(log cl)q -a~176

if a = - log

0

otherwise.

One obtains the Fujii theorem by considering the above series with a and - a since, for example, if Z = ~Q, we have

pEL+

7s

Also, as remarked on page 23 of [Fu 84a], we obtain a meromorphic continuation of the series

cos( 7)

2

-

Fz,a(s;

+

pEL~

both with and without a R i e m a n n hypothesis type assumption. Thus what appeared up to now to be a p h e n o m e n o m associated to more or less arithmetic situations, for instance the location of poles at the logs of prime powers or their analogues for the Selberg zeta function, is now seen to be quite a general property of our broad class of functions. Directly from Theorem 1.2, we have the following corollary. C o r o l l a r y 1.3. In addition to the above conditions, a s s u m e Z = Z , a n d a s s u m e all coet~cients c(q) are reM. I f all zeros o f Z in Tea lie on a v e r t i c a / l i n e Re(s) =/30, then we necessarily have =

o/2.

Finally, let us note t h a t the case of a = 0 is handled by our Cram@r theorem, specifically Corollary 1.3 of [JoL 93c], and our results from [JoL 93a], specifically Theorem 1.8 and Corollary 1.10. For completeness, let us list this theorem and refer to the above mentioned references in our work for a proof.

124

T h e o r e m 1.4. W i t h notation as in Theorem 1.1, assume that 42 has reduced order ( M , rn ). Then the zeta function

v(p)

1

pE'R.=

which is defined for Re(s) > M + 1, has a m e r o m o r p h i c continuation to all s E C. If there is a constant/3o such that p =/3o + i7 for MI p E 7~a, then the zeta function

pERa

which is defined for Re(s) > M + 1, has a m e r o m o r p h i c continuation to M1 s E C.

The asymptotic expansion in w of [JoL 93c] and Corollary 1.10 of [JoL 93a] combine to give an explicit description of the poles of the zeta functions in Theorem 1.4. We will not state these results here, but will simply remark that the location and order of poles of the zeta functions in Theorem 1.4 are determined by the asymptotic expansion near t = 0 of the theta function associated to the fudge factor 42.

125

w

P r o o f of F u j i i ' s t h e o r e m .

Let z = a + it for any non-zero a E R, and, with notation as above, let p = iA. In [JoL 93c] we proved various analytic properties of the Cram~r function

Vz(z) = ~ , v(p)~-, pE~

which is defined when Im(z) > 0. In particular, Theorem 1.1 of [JoL 93c] and subsequent discussion imply that the function aoWa

(1)

2~iVz,a(z) - ~o~

f

~-sZ~,l~(~)d ~

ao+a--ioo

has a meromorphic continuation to all z E C, whose only singularities are simple poles at the points log q and - l o g q. The residues of these poles are given on page 390 of [JoL 93c]. Now assume that 9 is of regularized product type of reduced order (M, m). By combining Lemma 3.1, Proposition 3.2, Lemma 3.3, and, quite importantly, Lemma 4.2 of [JoL 93c], we conclude that the integral in (1) has a holomorphic continuation to any nonzero z E R. Therefore, for a ~ 0, the function

v.,z,o(t)

=

vz, o(.

+

it)=

~

v(p)ei"*e-'~

pET~

has an asymptotic behavior of the form oO

V~,z,a(t) "~ E

(2)

cn(a)tn

as t ~ O,

n-~--I

for some constants ca(a) which depend on a. Further, from the formula in Theorem 1.1 in [JoL 93c], we have

(3)

C _ I (Ot~) =

-(27r) 1c(q) log q

if a = log q

-- (27r)- lc(~1) (log ~1)~1-~~

if a = - log Cl

0

otherwise.

126

By applying the Mellin transform, we conclude from (2) that the function O0

eie~ s = / VZ,a(~ + it)t" v(p)

r(,) ~ PE?~-a

= r(,)MV~,z,a(,)

0

has a meromorphic continuation to all s E C whose only singularities are simple poles at the points s E Z IRe(s)l. Let S denote the test function space on F \ A x , and define K to be the space of all s m o o t h functions r \ A • - ~ C.

T h e n for g E S, G E l:2(T), and F C K, we consider the continuous linear functional

< F,G > = ~

F(x)g(llxll) dx IIxll'

\Ax

which defines F as a distribution. In this m a n n e r for g E S and G E T associated to g define dx




=

fr\Ax E(x,~)g(llxll)

Ilxll

I [~+~~176('(1 4~rij~,_~

-('2

+A)s--~+

which is valid for 0 < a < 89 a > a distribution.

)

r (,2-I -s+A

]Re(s)l. This

9 ~(A) dA,

defines

E(x,s)

as

144

Now, for f E S, define

EF(X) = 1 [+iooE ( x , s ) f ( s ) d s , 47ri ., - i ~

the projection of F onto the space of Eisenstein series. Then EF(x)=O(1),

so that EF is in s

We compute

< EF, G > -

1

9 :

-

1)

- s +--s + s 47ril ., -ioo[+i~f ( s ) . [ ~ + i ~ ~ ( ~ ) . ( -s + o'--io0

~(s163

~(~)](~)~(~) d~,

2hi ~ ~ - i ~

which is valid for 0 < a < 7" 1 It follows that

< EF, G > = < F, EG > = < E G , F > . 1, a > iRe(s)l,

In the same manner, forO < a


-

1 / "+~~176 2~ri.,~_ioo

(1 s+,k

1) -s+,k

=/(~>(s) = <E, EF>. It follows that

=

O,

< E, ( F - EF) > = O, and finally

< ( G - Ea), ( F - EF) > = < G , ( F - E F )

>

= = < ( F - EF),(G - EG) >.

145

The above computations establish the fact that F - E F lies in the orthogonal complement of the space of Eisenstein series. To recapitulate, the Eisenstein series

E(x,s)

=

[

-logTr+

lr' ( 89 ~-2

1F' (~___~)] + 5~ (11=11=+ Ilxll-*)'

may be defined as a distribution on s

which satisfies

Ae(x,s)=s2E(x,s),

(11~11"+ IlZll -~)

M~E(x, ~) =

E(x, ~),

and, hence, determines the continuous spectrum of these operators. 1 1 3 The Eisenstein series has poles at s - 2, 2, 2, 32, 52, 52, "'" with residues given by a constant multiple of

IIx[I ~ + Ilxll -~, for

w

n -

1 2,

1 2,

3 2,

3 2,

5 2,

5 2,'"

Cusp Forms:

Let ~(s) = ~--~ F (2) C(s) 9The functional equation of the zeta function may be written in the form

-#-(s) ---- ~(1--s). Let f be a symmetric Schwartz function. A consequence of the functional equation of the zeta function is the well known explicit formula

~ . (." + =-.) i(p) = 2~i ~ _ , ~

r

+ ~) (x~ + ~-~') ](~)d~

+ ~.1 [ ~ + ~ _ [~' 1 ~' ~-,~ ~(~ + :,) + ~(~1

A)] x ~ ](~)d:~,

146

1 T h e sum on the left in the above formula runs over where c > 7" complex n u m b e r s p which satisfy

r

+

p) = 0, ~.

Each zero or pole has multiplicity m o (taken negatively if ~ ( 71 + p ) = 0). T h e only pole is a simple pole at p = ~1 so that m 1 = +1. T h e only difference between this formula and Weil's [W] explicit formula is that it involves the Mellin transform instead of the Fourier transform. Of course one can easily pass from one to the other by a logarithmic transformation. Let F E Y be associated to f. Replacing x by [[x[[, the n o r m of an idele (in the above formula), we may rewrite the explicit formula in the form

F(x) =

~

mp/(p) (ll~ll p + Ilxll -~)

-

El(x).

~( 89 It now follows from the results of w that the cusp forms are given by the functions

Ilxll ~ + Ilxl1-0, where 71 + P is a zero of the zeta function. In fact these functions can be constructed explicitly by forming F ( x ) - E F ( X ) , where F is associated to I and f(A) vanishes at all the zeros or poles of the zeta function with one exception. Every F C Z:2(T) can be expressed as a linear combination of cusp forms plus an integral of Eisenstein series plus a multiple of

Ilxll + + Ilxll-+, which is a residue of the Eisenstein series. This establishes the spectral decomposition of the space/32(T). It is not hard to show that a cusp form [[x[[~ + [[xl[ - ~ has n o r m zero if and only if p is not pure imaginary. This establishes the fact that the R i e m a n n hypothesis is equivalent to the positivity of the inner product < > on the cuspidal spectrum.

147 w T h e z e t a f u n c t i o n a s s o c i a t e d to a n a u t o m o r p h i c f o r m on s Let F(x) e / : 2 ( T ) be associated to f E S. Then

C'

1 / a+i~

A)](A)(xX+x-X)dA,

1 for a > ~. We define the zeta function associated to F by the formula

CF(~) - 2~-i .,~-,oo

+ ~

C

](A) ->,-+~ +

~ ~- ~

d~,

which is the Mellin transform _~(s). To see this, we note that the Mellin transform of x A + x -A (denoted by (~+~-~-~) (s)) is defined as a distribution by the formula

l/~+i~

-(~+~-=~) (s)~(~) d~ = 27ri ~ a - i o ~

-- 2~Fi ,,~_ioo

/oo~ (x ~ + x_~)g(x ) dx x

s - A + ~s+ )~ ~(s) ds

= 2~(~), which defines a continuous linear functional on [0, ~ ] with invariant multiplicative measure d_~. We may thus interpret _L~( 89+ A)fi(A) as the Ath Mellin coefficient of F(x). This is analogous to the Fourier coefficient in the classical theory of automorphic forms where the expansion is taken with respect to the additive group. The zeta function associated to F satisfies the functional equation

CF(~) = - C F ( - ~ ) . The zeta function associated to a cusp form IIx]]p + such that ((1 + p ) = 0, will be - -

s

1 -p

-31-

1 s+p

[]xl]-P with

p

148

w

The Rankin-Selberg convolution: Let

~ = I1~11~ + I1~11-< ~ , = I1~11r + II~ll - r

((89

be two cusp forms where ((1 + p) = + p,) = 0 . Let F C s be associated to f E S. We form the inner product

< F, ~-W~r~,>

=

o ~ f(llxli)(llxllP

+ I1~11-~ (llxllr + I1~11-r

dx

= 2i(p + p') + 2/(p - p') The above formula can be further generalized by enlarging the allowable class of functions F to include distributions. If we consider the case

r(x)_

E(~,s) ~(~) ,

and recall the previously proved fact that the Mellin transform N

(x~+~-~)

(s) -

a-

1

--'

~ +

1

we obtain the Rankin-Selberg convolution

< E ( , s), ~,~,, >

= 2

[1 s+p+

pl +

1 s -p-

pt +

1 s+p-

pl +

1 ]pl

s-p+

"

This may be interpreted as the zeta function associated to an automorphic function for a certain group acting on (A x )2 in analogy with the Gelbart-Jacquet (see [G-J]) lift. These ideas will be briefly explained ill the next section.

w

H i g h e r rank g e n e r a l i z a t i o n s :

For n > 2 let Wn denote the Weyl group of G L ( n ) (which consists of all n • n matrices with zero entries except for exactly one

149 1 in each row and column). For an arbitrary multiplicative group K let 7?n(K) denote the diagonal group of all matrices of the form

kn "'. k2 kl )

with

ki 9

(l