Uniform Distribution of Sequences of Integers in Residue Classes

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1087 Wtadysfaw Narkiewicz

Uniform Distribution of Sequences of Integers in Residue Classes

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author

Wtadys{aw Narkiewicz Wroc{aw University, Department of Mathematics Plac Grunwaldzki 2-4, 50-384 Wroc~'aw, Poland

AMS Subject Classification (1980): 10A35, 10D23, 10H20, 10H25, 10L20, 10M05 ISBN 3-540-13872-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13872-2 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding : Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210

To

my

teacher

Professor

Stanis~aw

on his s e v e n t i e t h

Hartman

anniversary

INTRODUCTION

The aim of these notes, given by the author

which

at various

form an e x t e n d e d

places,

is k n o w n

about u n i f o r m d i s t r i b u t i o n

classes.

Such

when

sequences

L.E.Dickson

tional

i.e.

with respect

We shall

standard

weakly

uniformly

After shall

example

uniform distribution

sequences

and sequences

shall

functions,in

polynomials arithmetical ~-function

f(pk)

in chapter

those,

= Pk(p)

II-IV.

which

are

for primes

functions,

like the number T-function.

to the classical

of the theory of a l g e b r a i c

also

defined

p

and

lead

numbers.

theory

of P.Deiigne,

In such cases we shall

a proper

cular we shall denote

we

i.e.

satisfy

suitable

classical Euler's

to certain

and include

In c e r t a i n

wh i c h will

needed with

func-

ques-

of polynomials.

like the theorems

We shall use n o t a t i o n

with

or sum of divisors,

on m o d u l a r

result

star-

recurrent

by m u l t i p l i c a t i v e

consider

H.P.F.Swinnerton-Dyer

function.

we

arithmetical

k ~I

This will

to

is

of sequences, linear

use more r e c e n t work,

of R a m a n u j a n ' s

results

"polynomial-like",

number

forms,

which

In the fast two c h a p t e r

we shall

the v a l u e d i s t r i b u t i o n

Our t o o l s ' b e l o n g

types

mea-

prime

N.

general

of a d d i t i v e

of sequences

In p a r t i c u l a r

and R a m a n u j a n ' s

of r e s i d u e

(mod N),

integer

certain

of c e r t a i n

P1,P2, . . . .

tions c o n c e r n i n g

mentals

for every

by v a l u e s

distribution

particular

the c o n d i t i o n

of p e r m u t a -

of sequences,

of all primes,

and c o n s i d e r i n g

defined

This will be done study u n i f o r m

a permutation

classes

sequence

(mod N)

sequences

in residue

of this century,

study

distribution

in the first chapter,

polynomial

a thorough

in r e s i d u e

is the

distributed

proving,

consider

weak u n i f o r m

here

of integers

the b e g i n n i n g

inducing

of lectures

a survey of what

prime.

distribution

ting with

tions.

polynomials

also c o n s i d e r

N. The

since

thesis m a d e

to a fixed

ning by that u n i f o r m

of sequences

studied

in his Ph.D.

polynomials,

classes

were

version

is to p r e s e n t

funda-

places we shall J.P.Serre

be used

explicitly

and

in the study state

the

reference.

which

the number

is standard of d i v i s o r s

in number of

n

by

theory.

In parti-

d(n) , o(n)

will

Vl

denote

the

powers,

sum of d i v i s o r s

only p o s i t i v e

of a set

A

will

for primes

(except when

by

Z and

residue

classes

factor

ring

the text,

(mod N),

of W r o c ~ a w

lemmas

wroc%aw,

February

1984

into account. letter

The ring

group

through

to Mrs for

problems

k-th cardinality

be r e s e r v e d

of integers

of invertible

open

The

p will

and p r o p o s i t i o n s

Certain

University

of the typescript.

the sum of their

the

the group

my gratitude

paration

taken and

a word).

consecutively

to express

of M a t h e m a t i c s

i.e.

Theorems,

~!A

ok(n)

be the m u l t i p l i c a t i v e

in each chapter.

numbered

I wish

will

and

being

by

inside

G(N)

Z/NZ.

n

divisors

be denotes

denoted

vely n u m b e r e d

of

will be

of r e s t r i c t e d

elements

of the

will be c o n s e c u t i will

be stated

in

all chapters.

Dambiec the

from the D e p a r t m e n t

patient

and careful

pre-

CONTENTS

I.

GENERAL

RESULTS

I.

Uniform

2.

The

3.

Weak

4.

Uniform

distribution

sets

I

Permutation Generators

3

Hermite's Examples

Consequences

LINEAR

N)

. . . . . . . . . . .

of

sequences

. . . . . .

. . . . . . . . . . . . . . . . . . .

the

group

of

and

Fried's

polynomials

(mod

. . . . . . . . . . . . N)

of

polynomials

....

. . . . . . . . . . . . . . . . . . .

properties

9 11

12

14

polynomials

permutation

theorem

SEQUENCES

8

12

permutation

. . . . . . . . . . . . . . . . . . . . . . . . . .

RECURRENT

I 4

. . . . . . . . . . . . . . . .

of

distribution

comments

I

of

. . . . . . . . . . . . . . . . . . . . . . . .

uniform

15 18 21 23 25 26

. . . . . . . . . . . . . . . .

28

. . . . . . . . . . . . . . . . . .

28

I.

Principal

2.

Uniform distribution ( m o d p) o f s e c o n d - o r d e r linear recurrences . . . . . . . . . . . . . . . . . . . . . .

32

3.

General

. . . . . . . . . . . . . . . . . . . .

38

4.

Notes

. . . . . . . . . . . . . . . . . . .

48

modulus and

Exercises

IV.

. . . . . . . . . . . . . .

(mod

characterization

4

Weak

N)

systems

polynomials for

5

Notes

of

SEQUENCES

2

6

distribution

distribution

POLYNOMIAL

7

(mod

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

Exercises

III.

M(f)

uniform

Exercises

II.

. . . . . . . . . . . . . . . . . . . . . .

ADDITIVE

comments

. . . . . . . . . . . . . . . . . . . . . . . . .

FUNCTIONS

I~

The

criterion

2.

Application

3.

The

4.

Notes

sets

Exercises

and

of of

M(f)

. . . . . . . . . . . . . . . . . . . .

51

Delange

. . . . . . . . . . . . . . . . .

51

tauberian

54

Delange's for

comments

50

additive

functions

theorem

. . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

58 59 60

VIII

V.

MULTIPLICATIVE I

Decent

3

The

number

functions

4

The

vanishing equality

5

The

6

Ramanujan's

7

Notes

and

of

2.

An

3.

Applications

4.

The

5.

Notes

ADDENDA

of

and

the

Am(N)

sum

=

Euler's (5.2)

G(N)

comments

Generating

INDEX

62

~-function

71

. . . . . .

. . . . . . . . . . . . .

algorithm

85 88

. . . . . . . . . . . . . . . . . . .

94 95

. . . . . . . . . . . . . . . . .

the

set

of

values

of

a

polynomial

96 .

. . . . . . . . . . . . . . . . . . . . . . ot

functions and

by

the

ok

comments

for

79

. . . . . . . . . . . . . . . .

FUNCTIONS G(N)

77

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

I.

REFERENCES

62

. . . . . . . . . . . . . . . . . . . .

divisors

T-function

POLYNOMIAL-LIKE

Exercises

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

2

Exercises

VI.

FUNCTIONS

Dirichlet-~D

study k z3

of

M*(f)

. . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

96 102 104 107 112

. . . . . . . . . . . . . . . . . . . . . . . . .

113

. . . . . . . . . . . . . . . . . . . . . . . . . .

114

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

125

CHAPTER GENERAL

§ I. U n i f o r m

I. If then

N

the distribution

the modulus

F(k)

N

all

limits

is c o n s t a n t ,

here

{an}

this notion

is

in c o m p a c t

however

problems

most

situation

and

In the

that

sequel

the

tion

for

f(n)

has

(mod N).

{an}

of t h i s

f

PROPOSITION

equal

here

to

have

nothing

more

certain N)

says

shortly

in t h e r a t h e r

respect

to

function

that

UD(mod

the N).

of u n i f o r m

easy

finite

to d o w i t h

convenient

the

case,

abstract

which

to c o n s i d e r of course

advantages.

if t h e

the

of n o help.

of s e q u e n c e s ,

UD(mod

one

of t h e n o t i o n

is u s u a l l y

sometimes

case when

I/N,

(mod N), case

groups

in p l a c e

criterion

however

and only if for

with

integers,

(k=O,1,...,N-l)

In the p a r t i c u l a r

but presents is

sequence

of

formula

approach be

a sequence

We

sequence

arith-

does

shall

hence

f ( 1 ) , f ( 2 ) ....

this property.

a sequence

lim x -I X -~°°

arising

essence

From Weyl's

mediately,

abelian

it w i l l

a function

its v a l u e s

exist.

the g e n e r a l

functions

change

say

F(k)

by the

is a p a r t i c u l a r

distribution

of

and

uniformlEdistributed

sequence

not

integer

thus necessarily

Obviously

metical

distribution

function

is d e f i n e d

RESULTS

= lira x -I # { n -<x: a n ---k(mod N) } x-~oo

provided F(k)

is a p o s i t i v e

I

the

following

to b e u n i f o r m l y we

1.1.

prefer

A sequence

Z exp(2Zianr/N) n -<X

distributed

to g i v e

r=1 ,2 .... ,N-!

necessary

a simple

{a n }

and

sufficient

(mod N)

direct

of integers is

results

condiim-

proof:

UD(mod

N)

if

one has = 0

•

(1.1)

Proof.

Put

N-I fr(t)

for

-I

=

if

hence

N-I Z r:O

fr (J)

UD(mod

then

of

for

=

N-I ~2 r=1

I/N + (Nx)-I

r=1,2,...,N-1

of t h e

sequence

r = I , 2 .... ,N-I

I : n~x an: j (rood N)

N-I ~ fr(J) + 0 ( x ) j=O

-I

fr (an-J)

=

for

for

N-I ~ fr (J) j =0

n N O(n)

T(n) (n) 2-n I ~ X(x k ) I < T (n) k=1

and

N-R

I~ k=1

] N O (S+ I )

x(x~1+s') I < [

if

N-R N 1 (j) , in

Observe n i t y and

that

the

N o (1+s)

(N-R) 2 - s - 1 / N

same

_

applies

T (s)

to

tends

to z e r o

No(S+1)/N

for

because

N

tending

to

infi-

of

2-s


T(1)

+T(2)

+ 0(I)

+ ... + T ( s )

we

get

N

I

lira ~ ~ N÷~ m=1

The

~ T(n) 2 - n n=1

same

X(Ym ) = O .

approach

shows,

with

the use

of

(1.6)

that

for

j=I,2,...

the r a t i o

N m~l

does

not

having

XN'j (Ym)

tend

to zero,

in m i n d

trivial

on

not divisible

every

of sum

later

integer

a sequence

1),

now

it s u f f i c e s

non-trivial form

to a p p l y

additive

exp(2~ir/N)

Proposition

character

with

1.I,

of the

a certain

integers,

integer

r,

N.

shown

positive

(i.e.

every

is of the by

2. It w a s

mN

that

NZ

and

by H . N I E D E R R E I T E R

N

one

prescribes

([75],

a positive,

a o ( N ) , a 1 ( N ) , .... aN_I(N)

and a s s u m e s

that

if

M

th.4)

divides

N

that

normed

of n o n n e g a t i v e then

for

if for measure reals

j=O,I,...,M-I

one has

aj (M) =

then N ~1

~ ai(N) imodN i~j(modM)

one c a n and 1

find

a sequence

j=O,I,...,N-I

one

lim x # { n -<x: Un - j ( m o d X-~co

Ul,U2,...

with

has

N) } = aj(N)

.

the p r o p e r t y

that

for all

If,

1.2.

It is w o r t h

by N i e d e r r e i t e r

I. We

shall

tribution

~D

uniform

also

(i)

study

set

the

{n:

following

prime

terms

to

then

one o b t a i n s

constructed integers.

(mod N)

notion

of u n i f o r m

of a g i v e n

sequence

N. To be p r e c i s e ,

distributed

two c o n d i t i o n s

= 1}

N, {u n}

set of all

weaker

those

following

(an,N)

and

of the

weakly uniformly

{a n }

j

sequence

distribution

only

are r e l a t i v e l y

provided

The

the

in w h i c h

all

the

§ 3. W e a k

which

N)

for

that

a permutation

a sequence mod

noting,

is a l w a y s

(mod N)

considered call

a~ (N) =I/N

in p a r t i c u l a r ,

Theorem

we

(mod N)

are

disare

shall (shortly

satisfied:

is i n f i n i t e ,

and (ii)

For

every

~ { n ~x: lim ~ {n ~x:

j

prime

to

an ~ j ( m o d N) } (an,N) = I }

N

one has

I ~(N)

X ~

The ling

first

condition

a sequence

buted

(mod

The that

like

of w e a k

of u n i f o r m

groups

E n~x

where The 1.1

and the

to the

1.3.

of integers

Dirichlet

XO

general

following

If

distribution

in the g r o u p

N

x(mod

theory

analogue

the p o s s i b i l i t y

weakly

uniformly

of cal-

distri-

coincides

of r e s t r i c t e d

of u n i f o r m

distribution

of P r o p o s i t i o n

integer,

with residue in

1.1:

then the sequence

if and only if for every non-principal

N)

N).

(mod N) G(N)

is a positive

~D(mod

one has

= 0( ~ Xo(an)) n~x

is the principal

proof

and we

is

character

X(an)

uniform

distribution

(mod N) leads

PROPOSITION {a n }

to e x c l u d e

3).

notion

classes

is i n t r o d u c e d

1,2,0,0,O, .... ,O,0,...

can be c a r r i e d

leave

character out

along

it to the reader.

0

(mod N).

the

lines

of t h a t

of P r o p o s i t i o n

2. F o r between

another

uniform

It is s h o w n

proof

namely,

integral

distributed

(mod N).

L e t us d e n o t e is

WUD(mod

c

N).

the p o s s i b l e case

we propose

the

is

UD(mod

may

in t h e

those

style

is e s t a b l i s h e d .

N)

if a n d o n l y uniformly

obvious).

integers

be posed,

here

a relation

is w e a k l y

is o f c o u r s e

set of a l l

M*(f)

also

distribution

{un + c }

if" p a r t

the

distribution,

N

how one

of T h e o r e m

the a n s w e r

for w h & c h

can charac-

1.2.

Contrary

is u n k n o w n

and by

following

Prove

I.

{u n]

the q u e s t i o n

sets

of u n i f o r m

[68], w h e r e

uniform

sequence

"only

M*(f)

Again

to t h e

PROBLEM

the

(The

by

terize

analogy

S.UCHIYAMA and weak

that a sequence

if for e v e r y

f

cf.

distribution

that if

X

then there exists a function

is a subset of the positive

f

such that

X =M*(f)

integers,

if and only if

X

has the f o l l o w i n g p r o p e r t y ~ If

n eX

and

prime divisors

One ver

sees

of

d

is a divisor of

n,

then

immediately,

the proof

of

b y dr

I.RUZSA,

divides

the

set of a l l

then

it is p o s s i b l e

bers

of

A

but

in p r o v i n g

subset

of o d d

To

the

I. T h e n o t i o n

say t h a t

we

NI,...,N r

let

system

provided

lira x -1 #{n

M*(f)

sequel.

of systems

distribution

(mod N)

and

in the

that of weak to c o v e r of

r

integers.

We

with

respect

cj,...,c r

systems

sequences shall to

one has

j = I , 2 ..... r} = I / N I . . . N r .

10

Similarly, ted is

with

infinite

(cj,Nj)

the

respect and

=I

It

will

every

integers

for = 1

that

UD

S

is

weakly

uniformly

distribu-

(a~ I)

a n(r) ,N) = I }

Cl,...,c r

satisfying

has

(a n~I) . . . a n(rJ ,N)

if

called

t h e. s e t . {n:

of

one

an(j} - c j (rood N i)~

clear

be

if .

choice

j = 1 , 2 ..... r

~,{ n < x:

is

S

N I . . . , N. r

for

for

lira #{n-<x: X÷oo

system

to

I

j = 1 , 2 ..... r}

~ ( N I) . . . ~ ( N r)

with

respect

to

N~,...,N

r

then

m

the

sequence

2. of

The

a n~i)

first

of

is

related

introduced

to

by

these

the

case

N i)

notions

two

(mod

if

N)

notion

[74a] a n d of

evident:

the

L.KUIPERS,

H.NIEDERREITER

two

for

of

for

is

i=I • 2 s . . . .,r

a particular

J.S.SHIUE

r72c]

the

an,

bn

and

passage

i,j=O,1,...,N-1

all

to

a n -i(mod

N) } :

ai ,

l i m x -I # { n x-~oo

_<x:

b n _=i(mod

N) } :

Bi

a n -i(mod

N) , b n _= j ( m o d

and

moreover

UD

with

UD(mod

N)

is

The sition

and

following

to

N,N,...,N

furthermore

criterion

can

the be

of later

define

by

the

the

ceneral

are

said

[77]

seqnences, L.KUIPERS,

last

case

to be

notion

being

independent

,

Yij =aiBj"

has

notion

N)}

One if

system proved

sees

and

easily

only

is in

= Yij

if

all

independent the

same

way

that its

a system members

(mod as

our

N). Propo-

1.1:

PROPOSITION N I , N 2 ..... N r

satisfying

2.4.

(i]

The system

S

is

UD

with respect to

if and only if for every choice of integers b l , . . . , b r ( i = 1 , 2 ..... r) and not all vanishing one has

O sb i I:

The

series

~ p-1 P dJf(p)

diverges,

D

condition

for

two c o n d i t i o n s

UD

of ad-

for every

53

and

(Bd)

For all

T H E O R E M 4.2. f(n)

is

either

(Ad) or

for every

the number

2f(2k)/d

if and only if for every divisor

(Bd)

(1.1)

r=I,2, .... N-1

the f u n c t i o n

rf(n)

d >I

of

N

holds.

By P r o p o s i t i o n

zero mean-value.

is an odd integer.

(H.DELANGE [69]). An i n t e g e r - v a l u e d additive function

UD(mod N)

Proof.

tion

k zl

f

will be

UD(mod N)

if and only if

gr (n) = e x p ( 2 ~ i r f ( n ) / N )

A p p l y i n g the C o r o l l a r y

to P r o p o s i t i o n

has a

4.1 to the func-

we obtain that this will take place if and only if either

the series

p-1

(4.3)

N~rf (p)

diverges,

or,

for all

k a l, the number

2rf(2k)/N

is an odd integer.

Now we shall t r a n s f o r m these conditions.

For

N =d(r,N),

C l e a r l y the conditions

and

r =r1(r,N)

d~f(p)

if the condition = 2rlf(2k)/d 2f(2k)/d integer.

with

(d,r I) =I.

are equivalent,

r e [1,N-1]

hence the series

(Ad) holds for

d =N/(r,N).

is an odd integer,

then

m u s t be an odd integer.

(4.3) diverges Further,

2rf(2 k)

if

(Bd)

if

N

odd and so

2f(2k)/N

is for

by

N~rf(p)

if and only

2rf{2k)/N =

m u s t be even, r I

Conversely,

It follows that the c o n d i t i o n

lent with the d i v i s i b i l i t y of

d

write

is an odd

d =N/(r,N)

equiva-

and the oddness of the

r e s u l t i n g ratio. The T h e o r e m results now from the o b s e r v a t i o n that if the integers

1,2,...,N-I,

d i s t i n c t from unity.

then

d =N/(r,N)

f

is

(Ap) holds for every prime divisor

(Ap) holds for all odd prime divisors f(2),f(22),..,

runs over N

D

C O R O L L A R Y I. An additive function if either

r

covers all divisors of

p

of

are all odd and in case

41N

UD(mod N)

if and only

p

of

N,

the numbers

N, or

the condition

N

is even.

(A 4) holds

as well. Proof.

Assume, of

thet

divisor

p

follows.

Hence

N. T h e n

4/2f(2)

the c o n d i t i o n

f

is

UD(mod N)

(Bp) ~ u s t hold,

(B 2) holds and so

f(2 k)

but

thus

(Ap) fails for a prime 2f(2)/p

is odd for all

(B 4) fails and we see that

is odd and

p =2

k-> I. Since

(A 4) must hold.

54

TO then sors may

get

(A D) d

the converse holds

on

invoke

N

except,

the

theorem.

2. If

COROLLARY

constant value Proof.

c,

If

implication

as w e l l ,

thus

f

N)

follows

UD(mod

N)

but

Since

(Ap)

2/f(2)

=c,

does

then

f

from

not c

that

d :2,

but

if

(Ad)

holds

imply

(Ad)

for all d i v i -

then

is an additive function, will be

then the

(N,c) # I

but

observe

assumptions

possibly,

(N,c) = I ,

UD(mod

our

(Ad)

holds

theorem.

and

hold,

UD(mod

let

the p r e v i o u s

is d i v i s i b l e

by

d i N , d #I

conversely

so w e

implies

and h e n c e

by

2,

(N,c) = I .

hence

that

prime divisor

corollary

(N,c)

diD,

attaining at primes a

for all

be any

and

if and only if

N)

Assume

p

(B 2) h o l d s

and

f of

i£ (N,c).

p =2

and

so w e

get a

contradiction.

COROLLARY indicated

3.

(S.S.PILLAI

a proof,

The functions divisors of

details

and

~(n)

[40].

of w h i c h

with their multiplicities,

3 to t h e

last

section theorem

only classical

tools.

we

are

UD(mod

N)

of Delange's

shall

without The

given

N :2

H.v.MANGOLDT

by E.LANDAU

[09],

the number of prime factors of

§ 2. A p p l i c a t i o n

I. In t h i s

were

functions,

however

in t h i s w a y

one cannot

obtain

show that

the use

same

of a d d i t i v e

[98] 8 167.)

which give the number of distinct prime

~(n)

n, repectively

In t h e c a s e

tauberian

one

works

the reader

may

theorem

can obtain

the Corollary

theorem,

also

utilizing

for a l a r g e r

convince

of Theorem

counted

N.

of W i r s i n g ' s

approach

a proof

for all

n,

4.2

himself

in its

full

class that gene-

rality. The main actually

a

tool

special

PROPOSITION

f(s)

=

here will

~ n:]

4.3.

an n-s

case

be

of it,

(H.DELANGE

t h e Ikehara-Delange which

[54]).

we now

Let

tauberian

states

a >0

and let

theorem,

55

be a Dirichlet

serie~ with non-negative

open half-plane functions with

and assume

s >a

#0

a real number

b

and complex constants

b. If for

that there exists an integer

not equal

to zero or

a

negative in-

with real parts smaller

al,...,a q

q ~1,

Re s ~ a

than

Re s >I

f(s)

gO(s) (s_a)b

then for

convergent in the

regular in the closed half-plane

go(S) .... ,gq(S)

go(a)

teger,

Re

coefficients,

x

q ~ j=l

+

-aj gj (s) (s-a)

tending to infinity

an =

(C + 0 ( 1 ) ) x a ( l o g

,

one has

x) b-1

,

n_<x

with

C =go(a)a-IF-1(b),

For

the p r o o f

special

case

of the

we r e f e r

or to c h a p t e r

III

~(n). of

distribution

Let

z,

Fz(S)

which prime

f

=

~ n=1

Fz(S)

either

m

F-function. theorem,

of w h i c h

to the o r i g i n a l

for

Re

s >I.

paper

is a

of D E L A N G E

[83a3.

4.3 ~ f o l l o w i n g

all

or

N,

of the

H.DELANGE

function

R, and c o n s i d e r

[56]) the

~(n)

for a fixed

and value

series

Since

Z

f (n)

is m u l t i p l i c a t i v e

(I + z p -s +

~ j=2

z f C P O ) p -is)

: g] ( s , z ) ~

p

(1 + ( ~ k=2

zf(pk) p-kS) (] + z p - S ) -I

(I + z p -s) p

where

= ~

and

we have

p

g] (S,Z)

this

zf (n)n -s

f(p) :I,

= ~

either

from Proposition

the D i r i c h l e t

converges p

the r e a d e r

the

tauberian

(mod N) , for

denote

Izl sl,

general

of W . N A R K I E W I C Z

2. N O W we d e d u c e uniform

denoting

F

)

.

,

for

56

LEMMA

hes at

only

s=l

Proof.

I( ~ k=2

with

The f u n c t i o n

4.4.

If

~

in the case,

denotes

when

the real

zf(pk)p-kS) (1+zP-s)-II

a certain

constant

is regular

g1(s,z)

B,

and

f ~

part

~ ( V k=2

for

and

of

Re

and vanis-

s >~

z =-1.

s, t h e n

P-k~) (l-P-°) -I = (P~-I)-2 ~ BP -2~

since

the

series

a >½

the

first

assertion

g1(1,z)

vanishes

p-2O P

converges

almost

To obtain

uniformly

for

the

second,

observe

that

nishes

one of the

factors

in the

product

factor

has

its a b s o l u t e

positive

for

and only

if

p ~2.

Now

value the

equal

factor

defining

to at

least

corresponding

results. only

it h o w e v e r

I - ( p - l ) -2 to

p =2

if v a the

p-th

which

is

vanishes

if

f (2 k ) z

I

k=2

and

2k

this

+ ~ z + I = 0

is c l e a r l y

z =-I.

Since

we have

for

3. T h e

LEMMA

possible

d(2 2) = 2 , z =-I

next

4.5.

indeed

lemma

For

(I + z p -s)

the

deals

Izl 7

=-I

k zl.

factor

in

and

for

and f =

D

(4.4).

one has

= g 2 ( s , z ) ( s - l ) -z

P

where

g2(s,z)

Proof.

For

is regular

Re

s > I

for

Re

s zl

and does

not vanish

P

s =~.

we have

oo

(l+zp -s) = e x p

at

~ l o g ( 1 + z p -s) = e x p ( ~ ( z p -s + ~ P P P k=2

(_I) k + s

z k p -ks) ) .

57

PZ p-S =log(s1__~) ÷ g ( s )

Since

with

g(s)

regular

for

Re

s al

and

k+1

(-1)

I ~ k=2

for

Re

zkp-kS I ~

I

k

p2a _ pa

s =o >½

we o b t a i n

that

for

Re s >I

(1 + z p -s) = ( s - 1 ) - Z e x p { z g ( s ) }g3(s,z) P

with

g3(s,z)

regular

For

COROLLARY.

Fz(S ) =

where For

Izl ~I,

the function

s >½

z ~-I

and n o n - v a n i s h i n g

and

Re

these

a positive

there.

D

one can write

s >I

integer that

N,

w e can

put

for

Re

Re

now return

z r =exp(2~ir/N)

Hz(1

s-> I.

to our m a l n for

#O.

D

task.

r=O,1,...,N-1,

Re s > I

N-1 I = N r~O Fzr ( s ) e x p { - 2 ~ i j r / N }

-s n

and

s >-I

is regular for

F z (S)

preliminaries

j, and o b s e r v e

E

is regular for

Hz(S)

the function

4. A f t e r

fix

Re

(s-1)-ZHz(S)

z =-I

Given

for

.

n f(n) -j (rood N)

Indeed, and we

this

follows

interchanging

the

by

substituting

summations.

here

Using

the

the

series

Corollary

for

Fzr(S)

to the

last

Lemma

obtain

-s

I N

n

HI(S) s-1

+

n

N-I H z (s) ~ __r exp{-2wijr/N} r=1 (s-l) zr

f(n)~j(rm:d N) Since for the

Re z r 0} = A(P)

pEP or

X =

{2 a

Proof. follows

~ peP

p P: a

Observe

that

if

first

f

is

M,N

a r e even,

then

same

corollary

implies

UD(mod sults

2 m) now

for

functions

If

P

for

f(n)

P

for w h i c h

f(n) = 0

the C o r o l l a r y

if

and

UD(mod

UD(mod

MN).

is

UD(mod

f

The necessity

M(f)

if

n=O

[

if

21n

of a l l

set o f all

primes

primes

P'

in

and

X =A(P)

satisfying

will

or

I to T h e o r e m

N)

4.2

and n o t b o t h

Moreover 4),

note,

then

of t h e

it

numbers

that

the

it is a l s o

stated

condition

re-

primes,

the

If

form stated

it is e m p t y ,

holds

then

Let

for the

3 =Pl

+f(b)

A(P) P

nor

=B(P)

additive

in the T h e o r e m .

then

function

0, 0 O

Tp(S)

not v a n i s h

g1(s,X)

a(X)

has

Re s a l / m

can

regular the p r o p e r -

we h a v e

oo

ITp(S) I -> I -

hence

for t h e s e

From

~ k=1

p's

the p r o d u c t

to t h o s e

primes

p,

(f(pJ),N)

=I.

converges

and h e n c e

regular

for

By our

Re

the r e m a i n i n g

T

by

(s) = I +

(5.4),

hence

Tp(S) peA

with

a certain Putting

H T p ( S ) we p>2 TM for w h i c h t h e r e assumption the

s >O

the

separated

and d o e s

primes

P

does

p >2

TM

= exp

~ peA

function

everything

series

of

which

not v a n i s h

Re s > I / m

for now

is an i n d e x

part,

(whose

~ X ( f ( p k ) ) p -ks k=m

for

separate

(5.4)

at

Re

s al/m.

the p a r t

corresponding

I ~j ~m-1

inverses

of t h o s e

we d e n o t e s =I/m

set we d e n o t e

by

such

by

due

to

is

(5.4).

For

A) we h a v e

we can w r i t e

= exp{

~ X ( f ( p m ) ] p ~ms + g 3 ( s , X ) } peA

g3(s,X)

regular

together

we arrive

for at

Re s >_I/re.

p's

g2(s,X)

# O

log Tp(s)

that

66

F(s,X) = (s -I/m) ~(x) g1(s,X) g2(s, X) exp{ ~ X(f(pm))p peA

T M

+g3(s,X) ] ,

thus putting

g4(s,X)

= exp{- ~ X(f(pm))p -:ns} p~A

and g(s,X) =g1(s,X) g2(s,X)exp(g3(s,X) tion. D

+g4(s,X))

we obtain our asser-

Since for the principal character

Xo(mod N)

one has

F (S,Xo)

n-S

=

n

(f(n) ,N)=I we obtain that prime to

N

f

will be

D-WUD(mod N)

if and only if for every

j

one has

I

I

w(N) ~ X(j) F(s,X) lira X

_

~(N)

s-l+0

= ~ I

+

F(s,X o)

_

I

li4n (I + ~ ~(j) F(s,X) X#X o F(S'Xo)

~(N) s~+0

I Z X(j) lira F(s,X) ~(N) X#Xo s. ml+0 F(s,X o)

and using the lemma and the obvious equality is equivalent

a(X) = 0

we see that this

to

Z X(J----~g(I/m'X)s+l+0 lira (s- ~) I X~Xo

(X) exp I

k~ (X(k)-I)p~A~ ~ p-mS 1 =0

(5.5)

(k,N)=I holding for all

j

prime to

N. However,

since the matrix

(X(j))X#Xo (j,N)=I is of rank

o(N)-I

it follows that

(5.5) holds if and only if for all

67

X #X o

one has

lira

W

~

~÷~-+o [ (k,)=1 ~

(Re X(k)-1)

~

I

p-Sin +a(X) l o g ( s - I ) ~

~

=-

(5.6)

)

p~A k

A s s u m e now that for every n o n - p r i n c i p a l character which on

A

there exists a prime

then

a(X) al

and since

p

such that

Re X(k) ~ I

is not trivial on, then we may select such

r, Re X(r) I))

(rood q2). classes

at

least

For

r(mod

one

diverge given qk)

and

which

of them,

the p r o d u c t

k->2

say

fix satisfy

ri,

j=jl...jt

I -3

8),

D

is c o n n e c t e d

with

N) . We prove:

PROPOSITION

which is

The

is d u e

If

5.3.

WUD(mod

Proof. which

one

5(mod

hence

f

is a multiplicative,

then it is also

N)

assertion

essentially

results

D-WUD(mod

immediately

to R . D E D E K I N D

943,

from who

i n t e g e r - v a l u e d function N).

the

following

treated

Lemma,

the c a s e

a n =I.

LE~LMA 5.4.

series with have

a

Let

n=l

non-negative

an

resp.

f(s) = ~

b

,

and

bn

for their abscissas

and assume further, b

an n-s

that

over real values

> b.

[f(s) I

g s) = ~

r~l

arbitrary complex numbers, of convergence,

tends to infinity,

Put further

A(x)

= E an , n_<x

lim B ( x ) / A ( x )

=

I

X-~

then

a =b

lim s÷a+O

and

f(s)/g(s)

= 1 .

be two D i r i c h l e t

bnn-S

with

when

which

0 < a _a

1 ~ bnn-Sl n~x

assertion

and

=

I ~ B(n) (I n~x nS

_< C I ~ A(n)( n 1s n x which holds

=

=

from the f o l l o w i n g

constant

coincide

~ ann - s n_<x

C, and once we k n o w that the a b s c i s -

we use the e q u a l i t i e s

k A(n)( I .... I s ) = ~ A(n) n=1 nS (n+1) n=1

k s n= I

majoration

I )+B(x)[x]-Sl (n+l) ~

1(n+1) s ) + A ( x Y [ x ] -s 1 = C

with a certain

sas the c o n v e r g e n c e

f(s)

results

x z2

log (n+1) f e-StA(et)dt log n

s

log(n+1) -stdt S e = log n

= s 7 e-StA(et)dt O

(5.7)

and

g(s)

to get,

g(s)

and if

= s 7 e-st B(et )dt O with

R(x) =B(x)

= f(s)

-A(x) = o ( A ( x ) ) ,

+ s f e-stR(et)dt O

IR(x)/A(x) I <e

holds

I~ e - S t R ( e t ) d t l - < e s o

7 xo

for

x ZXo,

then w i t h

suitable

CI

log xo

implying

g(s) =f(s)

COROLLARY.

If

function for which it satisfies

e-stA(et)dt

+SOl

e - S t d t < esf(s) +0(I) o

:Log

+o(f(s)).

N ~3

and

m(f,N)

the conditions

f

is an integer-valued multiplicative

is defined and which is given in Theorem 4.1.

WUD(mod D

N),

then

70

4.

It s e e m s

that

which

is

D-WUD(mod

Hence

the

following

PROBL~4

no exal~ple N]

Are

III.

for

is k n o w n

a certain

t~e notions

of a m m l t i p l i c a t i v e

N, w i t h o u t

WUD(mod

being

and

N)

function

WUD(mod

D-~D(mod

N) .

equi-

N)

valent ? One

can

notions of all

however

coincide. those

find

The

large

first

multiplicative

E

classes

class,

of f u n c t i o n s

which

functions

we

shall

for w h i c h

for w h i c h

denote

the

5y

these FN,

consists

series

I/p

P (f(p) ,N)#I

converges.

For

namely

and we

is

N,

this

class

shall

our

two n o t i o n s

deduce

If

N ~3

this

is a given integer

PROPOSITION

5.5.

WUD(mod

if and only if it is

Proof.

N)

Observe

not n e c e s s a r i l y

first,

lying

that

in

FN,

if

f

and

~j(mod

N) }

-

D-WUD(mod

e(N)

for one m o d u l u s proposition

and

4.1.

then

f c FN,

f

N).

is a n y m u l t i p l i c a t i v e

(j,N) =I

I

#{n _<x: f(n)

coincide

from Wirsing's

function,

then

Z X'(mod N) X(j)

X(f(n))

,

thus

] - j (nod N) } - e(N)

dj = x÷~lim x -1 # {n _<x: f(n)

where The ence

M(g)

existence

of u n i t y

value

(mod N),

4.1,

M(Xo(f)),

and

with by

the

Xo(f(2k))

the m e a n

values since

of b o u n d e d

is n o n - z e r o

is v i o l a t e d tive.

as b e f o r e ,

of the m e a n

of P r o p o s i t i o n

or r o o t s mean

denotes,

occuring

Xo

of

Note

denoting

same

now

the

that

for

since d =

g.

is a c o n s e q u -

are

principal

that

function

formula

X(f(n))

Proposition,

It f o l l o w s

of the

in this

the v a l u e s

degrees.

~I.

value

~ N) X ( j ] M ( X ( f ) ) X(m0d

either

f e FN

zero

the

character

the c o n d i t i o n El dj (j ,N)=I

(4.1)

is p o s i -

71

Applying

Lemma

5.3 to the

fj (s) =

~ n -s n f(n) ~.j~nod N)

series

and g(s)

=

~ n -s n (f (n) ,N) =I

and o b s e r v i n g we o b t a i n

that

their

con/mon a b s c i s s a

the

limit

in

that

(5.1)

lira fj(s)/g(s) = l i m ( # { n ~ x : s+1+O x÷~

hence

D-WUD(mod

N)

however

that

deciding

whether

a given

since

one

can

of T h e o r e m

do this

5.1

this

N)

coincide

Proposition

function

applying

(see E x e r c i s e

much

larger

second and

in the n a t u r e ,

class

of

contains like

f

of d i v i s o r s ,

Ramanujan's

This

consist

f

which

class have

the

can write for

most

N ~3

Re

s >I

FN

is

D

useful

WUD(mod

N)

for

or not,

4.1 w i t h o u t

the

use

functions

which

we

shall

now consider

of the m u l t i p l i c a t i v e ~-function,

T-function

r=1,2,...,T

particularly

~roposition

the

functions

is

occuring

s u m of d i v i s o r s ,

the

number

etc.

of all m u l t i p l i c a t i v e

following

For every integer perty that for

from

case.

,

3).

functions

Euler's

unity,

(f(n) ,N) )=1} = d j / d

in this

is n o t

directly

§ 2. D e c e n t

I. T h e

equals

equals

f(x) ~j(mod N]}: #{n~x:

and W U D ( m o d

Note

of c o n v e r g e n c e

integer-valued

functions

property:

there exists an integer and for all integers

T

with the pro-

j, prime

to

N

one

72

p

-~

I

: a(r,j)log

s-~1 + gr,j

(s)

P

with

a(r,j)

and

~0

regular in the closed half-plane

gr,j (s)

Moreover at least one number Such

functions

of

T

(possibly

at

N.

In d e a l i n g

sufficient ver we

A broad

about

subclass

f

p

f(pJ)

If t h i s

over

= Pj(p)

condition

~-function

(with

Pj(x) = 1 + j ) , f

there

one has

f

is s a t i s f i e d

Let

Let

x I ..... x k

which

are prime

by

5.6.

say t h a t

E(f))

it w i l l

k, or

P

primes

multiplicative

over

p

Z

we

such

shall

exist

let

be all

solutions

to

if t h e r e

N,

polynomials

for the E u l e r ' s function

(with

order. T h e exact order

largest

integer

(or

~)

k.

and

of t h e exist

p Pr(p)~j(~d N)

it h a p p e n s

function

f

is decent and

~E(f).

N

~

as

as the

of order

is

that

speak

one has

for the d i v i s o r

Every polynomial-like N

be howe-

the polynomial-

by

if t h e r e

is of infinite

f

f

(5.8)

is d e f i n e d

r S E ( f ) , and

p-S=

p f(pr)sj(modN)

for a l l

Pj(x) = x J - 1 (x-l))

the order of its decency at

Proof.

N

precisely,

k,

value

observation.

a polynomial

for all

is p o l y n o m i a l - l i k e

PROPOSITION

of

integer-valued

of o r d e r

that

this

is f o r m e d

f(p) = P ( p ) . M o r e

such

value

(j=1,2 ..... k) .

then we

(denoted

for w h i c h

exists

possible

of decency of

for t h i s o n e n u m b e r ,

to a p p l y

as t h o s e

function Z

only

functions

we define

a polynomial-like

the order

for a p a r t i c u l a r conditions

of d e c e n t

for w h i c h

primes

and t h e m a x i m a l

be called

an o p p o r t u n i t y

which

PI ( x ) , . . . , P k ( x )

of

WUD these

not have

-like f u n c t i o n s , functions

with

will

Re s ~I.

j ~ I) should be positive.

(r ~ T ,

decent,

be called

infinite)

to c o n s i d e r

shall

for all

will

a(r,j)

p-S=

j , with congruence

any.

Pr(X)

b e given.

~j (mod N)

Then

k

j=1

(j,N) :I

p-S = P p--xj(mad N)

k

log

+ g(s)

7S

with

g(s)

rem.

We m a y

regular

that

congruence,

hence

for

put

Re

s al

a(r,j)

then

the

by the D i r i c h l e t ' s

=k/o(N).

If there

are

Prime

Number

no s o l u t i o n s

Theoof

series

p-S P

Pr (p)-j (mod N) reduces

to a f i n i t e

2. T H E O R E M

sum and

If

5.7.

f

so we m a y

take

of

vanish and

f

Proof. sible,

f

at

is

for

a(r,j) .

r

assume

then it is also

N)

that

the

i=I,2, .... r-1

index

and

all

Q

function,

not exceeding

such that not all numbers

D-WUD(mod

We m a y

thus

N

for

is a decent m u l t i p l i c a t i v e

a given integer and there exists an index of decency

O

a(r,j) WUD(mod

((j,N)=I) N).

r

is c h o s e n

as s m a l l

j

prime

N

to

N a 3

the order

the

as p o s -

series

p-S P

f(pi)---J (mod N) represents

a function

regular

for

Re

s a I, and h e n c e

the

same

applies

to E p-S P (f(pi) ,N)=I

Note

also

that

E

the

series

I/p

P (f (pr) ,N)=I

certainly

diverges,

since

otherwise

the

series

p-S P (f (pr) ,N)=I

would

represent

r. T h i s also

shows

that

for

a function

that

for

f

any c h a r a c t e r

regular

at

the n u m b e r X(mod

N)

s =I,

contrary

m(f,N) and

to the c h o i c e

coincides

Re s > I / r

with

of

r. O b s e r v e

74 X(f(pr))p-rS =

Z X(j) (j ,N] =I

p

X(j) (a(r,j)log E (j ,N) =I =

holds

+gr,j (rs)) =

with

hj (s)

regular

for

+hi (s)

Re s -> 1/r

n-S -~(N)I I ~ X(j) E n [X#X o f (n) -=j (rood N)

Also

~ X(j)a(r,j) (j ,N)=I hy Lemma 5.2 we get

F(S)

=

and

B (X) =

Since

~ n -s = n (f~n) ,N)=I

and thus by Lemma

5.2

hj (s,x)~(X___~+ h(s'X°) ~ (s-I/r) (s-I/r) B (X°) ]

hj (s,X)

regular

for

Re s a I/r,

h ( s , X O) (s-I/r)B (Xo)

by a s s u m p t i o n

l'm s÷~+0

F~ (s)/F(s)

and also = B(X O)

Re

: 1

8 (X) _< 6(X o)

it follows

equality

g o (s)

go(S) ..... gq(S)

f(n)-j(mod

+

regular Proposition

N) } =

possible Fj (s)

only

in case

B (x)

in the form

I -aj

q

(s-I/r) 8 (Xo)

go(l/r) = h ( I / r , X o) A p p l y i n g finally

#{n_<xz

with

that we can write

I

Fj (s) - ~(N)

with

I

I(j ,N)~ =I X(j)a(r,j) }log ~ i

Fj (s)

with

~

~ p-rS p f(pr) __-j(mod N)

~ gj (s) (s - r ) j=1 for

Re s -> I/r,

4.3 to

F(s)

go (I/r> (r (p(N)F(B(Xo))

Re ~j < ~ (Xo) and

+o(1))

Fj (s)

xlog

and

we arrive

B (Xo) -I

x

at

75

and

#{n-<x:

(f(n),N)=1}

rg o(I/r) B (Xo)-1 (~(N) F(S(Xo)) +o(I)) x l o g x

:

which due to the arbitrariness

of

We may thus use T h e o r e m

j

proves that

5.1 for checking

f

is

WUD(mod N).

WUD(mod N)

D

for decent

functions. 3. In the case of polynomial-like a criterion P1,P2,...

for

COROLLARY. order

WUD(mod N)

occuring Let

the m u l t i p l i c a t i v e sisting

of those

RI,R2,...,R E are empty). Then

such

(in case

m =m(f,N)

Let

the function

f

Let

E =~

~D(mod

be

X(mod N)

N)

prime

to

N,

con-

be the sub-

that not all

sets

that not all sets

Rm

index with if and only

trivial

G(N),

of

the congruence

Aj :Aj (f,N) further

we assume

be the smallest

will

character

G(N).

for which

and assume

of exact

the subset

(mod N)

classes

in

Rj

by

and p o l y n o m i a l - l i k e

r ~G(N)

classes

is given by the following

Rj :R~(f,N)

by

of residue

generated are empty

non-principal p

be m u l t i p l i ~ a t i v e

has a solution

G(N)

of

f

group

Such a criterion

denote

it is useful to have

in terms of the polynomials

(5.8).

residue

Pj(x) ~r(mod N) group

expressed

in

j:],2, .... E

E. For

fnnctions

on

Am

is

~D(mod

non-empty.

if for

there

every

exists

a prime

that

X(f(pj))p-j/m

=

0

•

j=O In particular, is an odd

Am(f,N) =G(N)

if

prime

such

Am(f,pk) = G ( p k)

and

holds

however

for

Proof.

with

p =2,

is

WUD(mod

one

has

Note that the group

A m . Indeed,

the series

p-I P

f

f (pro)_--j(rood N)

then

f

Am(f,p 2) = G ( p 2)

that

p )

(with for

to assume

A

occuring

N)

and if

m =m(f,p2))

k ~I.

A similar

then result

Am(f,23) :G(23) in Theorem

Ri

5.1 coincides

76

diverges

if and o n l y

x

to

p r im e

having

N.

It s u f f i c e s

in mind,

k=I,2, . . . .

if the c o n g r u e n c e

cases

one can

Am(f,p2)

cases

PROPOSITION

corollary,

if

Prmof.

has a s o l u t i o n

5.1 and

m[f,p)

its C o r o l l a r y ,

= m ( f , p k)

for

in the last part of this C o r o l l a r y

= G ( p 2)

is a prime P m (x)

follows

by the w e a k e r

larger than

Let

Am(f, P) = G ( p ) .

and notation 2n, where

then from

"

for

condition

replace

in the n e x t P r o p o s i t i o n .

of the preceding

n

denotes

the de-

the equality

A m (f,p) = G ( p )

k=1,2 .....

In v i e w of the p r e v i o u s k =2.

a •G(p2).

Corollary

it s u f f i c e s

By a s s u m p t i o n

to c o n s i d e r

one can find

rl,...,r s •

satisfying

Pm(rl) ...Pm(rs)

Observe all t h o s e

now,

- a ( m o d p).

t h a t if the d e r i v a t i v e

x • G(N)

X P m ( X ) P 'm (x)

for w h i c h

p, and we get it v a n i s h e s

identically

1+n+n-1 = 2 n a p (mod p)

Since

find

ro EG(N)

Pm(ro)P-1

(with

the d e r i v a t i v e p)

(mod p),

p),

then

then

its d e g r e e

by

p

for

we have

p. If

equals

our a s s u m p t i o n .

P~(x) (mod p) V(x)

by

then the p o l y n o m i a l

points divisible

contradicting

polynomial

x = r o cG(p)

of the l e f t - h a n d there

such that

~I (~od p),

Pm(rl)...Pm(rs)Pm(ro)P-2pm(x)

x ~ro(mod

IO(mod

is d i v i s i b l e

at least

If h o w e v e r

vanishes

identi-

Pm(X) = v ( x P ) ,

thus

a contradiction.

H e n c e we m a y

is s o l v a b l e

P'm (x)

at i n t e g e r

identically,

thus w i t h a c e r t a i n

p, again

Pm(x)

has all its v a l u e s

it d o e s not v a n i s h

c G(N).

one has

Under the assumptions

5.8.

p

Am(f, p ) = G ( p k)

n

p

is p r e s e n t e d

gree of the polynomial

cally,

n o w to r e c a l l T h e o r e m

that for p r i m e

the a s s u m p t i o n One of t h Q s e

G(N)

(x) z j(mod N)

D

In c e r t a i n

the case

P

eG(N)

and

P~(r o)

the c o n g r u e n c e

-a

and side

exists a solution

Pm(rl) . . . P m ( r s ) P ~ ( r o ) P - 2 p m ( y )

Pm(ro)

- O ( m o d p)

Pm(x)

~P

(r o) @ O ( m o d

d o e s not v a n i s h of

- a ( m o d p2)

p))

(mod p)

and at

since

77

with

y-ro

/O(/~od p).

Pro(Y) - P m ( r o ) (rood p~ proving

Since implies

Pro(Y) E A re(p21)

I. N O W we g i v e

of d i v i s o r s

some e x a m p l e s .

for a l a r g e c l a s s

a e A re(p2) ,

for c u r v e s

we p r e f e r

to u t i l i z e

only simpler

first example

l e a r n the r e a s o n s

PROPOSITION

Proof. j

converse

however

the n u m b e r

N

d(pJ) = l e j

d(n)

d(n)

of p o s i t i v e

complicated.

root

holds

it s u f f i c e s

we assume

that

~D(mod

Later

divisors we shall

thus

p

X mod N

is

q

N-I ~ X(k) k=1

Aq_ I ~ G ( N )

thus N-I X(i)z i = 0 .

~ zJ = z - I ( I j ~2 j-k (mod N)

denotes

(5.2)

N-I ~ X(i)z i. i=I

q the

c a n n o t be

p. To o b t a i n write

the

if

N). To p r o v e

then

z =p'1(q-~)

-zN) -I

the d i v i -

the first

and we see that %~D(mod

and ~ p r i m e

h o l d s and p u t t i n g

j zl

q-l, w h e r e

Aq-1 = g P { q } d(n)

and

Rj = {1+j} n G(N),

equals

then

(mod N).

0 = k X ( d ( p J ) ) z 0 = ~ X ( 1 + j ) z J = 1 + z -I ~ X(j)zJ = ~=0 ~=0 j=2

i=I

stage

if and only if

N)

root

for p r i m e s

Because

to show that if

with a character (5.2)

N,

(mod N),

is

is a primitive

it is n o n - e m p t y

satisfied

= I + z -I

to

at this

tools.

The function

not d i v i d i n g

is a p r i m i t i v e

leads to an

on a C o r o l l a r y

by A . W E I L ,

looks r a t h e r

is p o l y n o m i a l - l i k e .

for w h i c h

least prime

M*(f~,

proved

an a l g o r i t h m

f

for that.

5.9.

Since

set

concerns M*(f)

the least prime not dividing

sor f u n c t i o n

L a t e r we shall p r e s e n t

based

conjecture

n. Here the set

~-function

of the

determination

Our

and E u l e r

functions

effective

index

we o b t a i n

of p o l y n o m i a l - l i k e

Riemann's

of

(rs) eArn(p21), and

Am(p2) = G ( p 2) .

§ 3. T h e n u m b e r

which

Pm(ro~,Pm(r11) ..... P

this

78

This tainly

shows

not

the

that

z

case,

and

must

b e an a l g e b r a i c

this

contradiction

integer, shows

but

that

this

(5.2)

is c e r -

cannot

hold.

One

sees

dividing list:

easily,

N

that

all n u m b e r s

is a p r i m i t i v e

N =4,

N =2"3 a

root

(a k l ) ,

is a p r i m i t i v e

root

(mod pa))

is a p r i m i t i v e

root

(mod pa)).

2. O u r set

second

M*(f)

PROPOSITION

to

example

looks much

5.10.

N,

for w h i c h

(mod N)

N =pa and

are

(p - an o d d N =2p a

concerns

the

least

included

prime

in the

prime

a ~I,

(p - an o d d

prime,

the E u l e r

function

~(n).

not

following and

2

a ~I, 3

Here

the

simpler:

The set

consists

M*(9)

of all numbers

prime

6. Proof.

values and

If

for

N

2 cG(N)

{r:

are all

the

set

I -O 21k L

from the Corollary

(see exercise

f(pk).

This Proposition

dispose quickly of the possibility

,

to T h e o r e m 5.7.

large without

I), however

that this is not the case under

on the values

0

=

can be arbitrarily

weak uniform d i s t r i b u t i o n implies

=

spoiling the

the next P r o p o s i t i o n

certain additional

assumptions

allows also in many cases to

of having

I,~D(mod N)

without

A(f,N) =G(N) . PROPOSITION

5.11.

(mod N)

of order

propert y

that

is an integer

d

Let

N a3,

and

al,a2,..,

the sequence T

such

- X(aj)(rood N)

If for

k=1,2,..,,d-1

1 +

Xk(aj)p -j/M = 0

~

be integers, a sequence

x(aj) (mod N)

X

a character

of integers

is purely

with

periodic,

the

i.e.

that

X(aj+ T)

j=1

M ~ I

(j=1,2 .... ) .

there

exists

a prime

p =p(k)

such

that

there

81

then

p(k)

=2,

X(aj)

the character

-I

cf

M]j

o

if

Mfj

X

i.e.

d =2

and m o r e o v e r

=

Proof.

Let

T

be a p e r i o d

of

restricting

the

generality

characters

Xk

(k=I,2 .... ,d-l)

with

is real,

that

X(aj) (mod N) T

exceeds

and

M.

and d e n o t e

assume

Let

by

×

dk

without

be one

of the

its order.

We have,

p =p(k),

T

O = 1 +

~

x(aj)p -j/M = I +

j=1

,~

X(ar)

Z

j:1

p-j/M

j_>1 j-r (rood T)

= I +pT/M(pT/M

T

_1)-I

X(ar)P -r/M r=]

Putting

Y

T

for

shortness

T-I ~ r=1

+

x(a r) yT-r

Since

the v a l u e s

have

x(a T) = 0

or

which

it is not.

sible

by

and

to

D

the

is a unit, must we

×(a i]

]

since

Thus

D

dk

Now

K

If

be

also

y

would

is d i v i s i b l e

in the

would

a unit,

of u n i t y

denotes

cannot unit

integer,

of u n i t y

divi-

×(a T)

we

the field g e n e r a t e d

N(x)

a prime

which

we

be an a l g e b r a i c

algebraic

and by

is n o t

and

form

the n o r m power

then

is a n o n s e n s e .

since

N(y)

d

let

is a p o w e r

by

from

K

X(aT)-] Thus of

p

Then

X

t

= pt[Dt]dt

by

be a p r i m e

q,

and

We

show

p(k), divisor

applying

that

D

_ s D --po

say

q

now

if

-I).

power,

or r o o t s

y

of the r o o t

N(y) IN(×(aT)

is of o r d e r shall

order

of u n i t y

a certain

let

zero

is a n o n - z e r o

the

equality

(5.9)

case

d k. M o r e o v e r ,

p(k) I N ( y ) i N ( x ( a T) -I)

hence

either

root

hence

this

.

d-th

be a p r i m e

get w i t h

are

= 0

in that

×(a T) -1 by

divides primitive

Q, t h e n

we can w r i t e

+ × (aT) _i

of

y. D e n o t i n g

see that y

Y =pl/M

q =2

the and

of

last

and

observation

to do

this

X =xd/q. we

get

we a s s u m e ,

q =p(d/q). a contrario,

D

82

that

q =p =p(d/q)

vial Y

powers

of

is an odd

X

are

( = p(d/q) I/M)

y

for

T

T-I ~ r=1

+

also

prime.

Note

of o r d e r

q

also

that

thus w i t h

since the

all

same

non-tri-

value

for

we have

X k(a r)y

T-r

Xk

+

(a T ) - I = O

k = I , 2 , .... q-1. Adding

these

(q-1)yT

equalities

(q-l)

+

~

we

get

yr

_

yr

I ~rO

A

similar

argument

shows

also

WUD(mod

N) . Indeed,

from

Lemma

follows

immediately

that

all

(rood 3), using

T(2)

by 3 for gruent =

thus

RN,

3)

n >-J

to u n i t y

on

cannot

-O(mod

all

(m/3)X o(m)

unity

RN

(mod

(with in c a s e

3) Xo of

5.16

~(3)

~(2 n)

and

of

RN

G(N).

-=O(mod

the

WUD(mod

n.

we c a n n o t

congruent

that

T(3 n)

3)

Since

have

T(p2) = ~ ( p ) 2 _p11 are

from

principal N)

31N

Moreover

is d i v i s i b l e

for e v e n being

in case

(ii)

elements

generate

and

and

that

by 3 for

(5.~3]

character satisfy

we

infer,

is d i v i s i b l e

odd

the c h a r a c t e r

it s h o u l d

it

to u n i t y

n

and c o n X(~n] --

(rood N]) (5.2]

equals

with

93

p =2

or

3 and

m =2,

T(3 n)

show

that

ruling

thus

out

The

N.

residue

classes

nerates

G(q~)

of the

It s u f f i c e s

is c o n s t a n t

the

~O

argument

(mod q~)

on

it.

Lemma

5.15

follows

the

for

for

holds

resp.

further,

shows

same

~(2 n) for

lines

i=1,2,...,s

qk ~2)

and

=O

and

all

n ~I,

(5.2).

that

(if

G(8)

congruences

X(~(3n))

of

to s h o w

resp.

obtain

and

the p o s s i b i l i t y

remainder

of odd

however

X(T(2n))

(mod

that

that

as

the 8)

in the c a s e

set

(if

Rk

qk =2)

to n o n - p r i n c i p a l

it is e n o u g h

of ge-

character

to do this

for

qk e { 2 , 5 , 7 , 6 9 1 } . In case In the hence

Rk

Since

qk = 2

same

contains

-principal (n/5),

Rk

In c a s e

qk = 7

(mod 72 ) that

all

However

~(32)

elements

in

if

provided

does

must

qk = 6 9 1 ,

not

Lemma

Rk to

to 7)

for

and

6

42 < 2 - 2 2

Since

only

5).

non-

character R k.

a(mod

elements,

72 )

residue

7) lie

at l e a s t

and in

this

R k.

3-7 +I = 2 2 the

only

one.

implies

that

is c o n g r u e n t

11~(6912)

3(mod

52 )

generates

every

(mod 7)

so we h a v e

(iii)

which

any

(a 4 +a) 2 - a 1 1 ( m o d

and

5.17

in

contain

be the p r i n c i p a l

(mod 6912 )

I or Rk

so the

3 lie

that will

3,5

has

to

the q u a d r a t i c

1 and

implies

Rk

to

every

contains

I + a 11 +a22(mod 691),

residue

(mod 6912)

thus

6912 ) : x -~1+y+y2(mod

Rk

contains

character

occur.

be

R k =G(8).

~I + p 2 + p 1 1 ( m o d

that

10 e l e m e n t s ,

must

set

G(72)

Rk

(1 +I +I 2 691 ) = -1 ,

this

(i)

~2(mod

if a n o n - p r i n c i p a l

quadratic

shows

this

and b o t h

congruent

since

on

power,

see t h a t that

and

congruent

Rk

the

to get

~(p2)

(mod 52 )

is c o n g r u e n t

643

class

R k = {x(mod

shows

which

691~a(1+a11+a22).

ll-th

and we

5.17

(5.14)

that

25) least

on

(mod 7)

=-113

Rk

residue

is an

Lemma

residues

constant

Finally, every

at

(i) and

qk = 5

(mod

trivial

a £1,2,4

character

5.16

(I/5) =1 ~ - I = (3/5)

satisfying

shows

root

contains

character

but

class

Lemma in case

all r e s i d u e s

3 is a p r i m i t i v e

G(52) . M o r e o v e r

the

we u s e

way we obtain

at

least

character

(n/691),

691(691-3)/2 is c o n s t a n t

however

'I +4__+42.) 691

This

691) , 6 9 1 Z y ( l + y + y 2 ) }

establishes

elements. on

in v i e w

of

Theorem

for

Rk,

This

it m u s t

= +I

the

N

even.

D

be

94

§ 7. N o t e s

I. The n o t i o n J.~LIWA

[76],

Theorem proved

of D i r i c h l e t - W U D ( m o d

where

in W . N A R K I E W I C Z

5.1 can be

(mod N)

KIEWICZ

[66],

was o b t a i n e d the general

of

images

In m a n y

cases

It w o u l d

fill

is due

they

m a n y values

of these

to

happens

p(n)

and

jectured

c(n))

(A.O.L.ATKIN,

that for every

N

was

settled.

D.W.MCLEAN

[80]

functions

for all

m,n)

satisfying

are

31.

are known.

of the m o d u l a r unadapted

values

~67]).

of

of

13

M.NEWMAN

(mod N) p(n),

[59] proved In T . K L ~ V E

solved

[77].

in-

to ful-

N

there class

(both for [60] con-

contains

and proved

it for

N =2

and

F70] the case

by A . O . L . A T K I N

in-

this

N =121

[68].

Cf.

results. (mod N)

[77],

who g a v e

was c o n s i d e r e d necessary

and

for m u l t i p l i c a t i v e sufficient

of c o m p l e t e l y m u l t i p l i c a t i v e

for all primes

the e x i s t e n c e UD(mod N]

coefficients

in e v e r y r e s i d u e

class

strongly multiplicative

f(pk) =f(p)

them he d e d u c e d tions w h i c h

and

and

The

for the p a r t i t i o n

for all powers

function

was also

for n u m e r i c a l

in the case

for certain

O.KOLBERG

N =7

distribution

by H . D E L A N G E

for that

13.

N =7,17,19,29

The case

3. U n i f o r m

tions

N =5

for

WUD c(n)

J.N.O'BRIEN

and

for

and

every r e s i d u e

conjecture

only odd

and m u l t i p l i c a t i v e

representations

functions

of the p a r t i t i o n

[68]

and in

no new problems.

seem c o m p l e t e l y

e.g.

finitely many values

T.KL~VE

case

5.11

by H . P . F . S W I N N E R T O N - D Y E R

UD

that

(mod N)

N. This

[82]

to the F o u r i e r

coefficients

are i n f i n i t e l y prime

A special

who c o n s i d e r e d

also

Z-adic

the known m e t h o d s only

proved in W . N A R -

Proposition

F.RAYNER

are integral

to study

It is known

[75],

applicable

and the Fourier

task.

[45].

case p r e s e n t s

image was d e t e r m i n e d

be i n t e r e s t i n g

5.7 was

theorem)

5.10 appear.

f

class

[83b].

of the c o r r e s p o n d i n g

p(n)

to T h e o r e m tauberian

5.9 and

to J . P . S E R R E

provided

[76]. Delange

function

in a fixed r e s i d u e

in L . G . S A T H E

the general

forms,

this

form in H . D E L ~ q G E

lies

5.5 appears

from it and

integer-valued

in W . N A R K I E W I C Z ,

in W . N A R K I E W I C Z

5.18

j, however

this

case

is in p r i n c i p l e

and the

function

already

in W . N A R K I E W I C Z ,

Proposition

resulting

The C o r o l l a r y

also P r o p o s i t i o n

N, however

same a p p r o a c h

f(n)

first

proved.

also via D e l a n g e ' s

in a special

of other m o d u l a r

variant

for w h i c h

5.9 occurs

case

5.1 was

the c r i t e r i o n

a density.

where

2. T h e o r e m values

n's

appears

in an e q u i v a l e n t

(although

of P r o p o s i t i o n

N)

for a m u l t i p l i c a t i v e

has always way

however

found

that

the set of those

in a n o t h e r

also T h e o r e m

[77],

further

and c o m m e n t s

p

(i.e. m u l t i p l i c a t i v e and

k ~I]

N.

and

functions.

of i n f i n i t e l y m a n y m u l t i p l i c a t i v e

for all

condi-

(if(mn) = f ( m ] f ( n )

Using

func-

95

Exercises

I. S h o w t h a t cative Am(f,N)

f

f

M(f,q)

D-~D(mod

be

=I.

qk)

assumption 4.

is

5. P r o v e set of a l l if a n d o n l y

WUD(mod

N)

mean-value 8. N's

that

those if

for

=I

if

is

such

D-WUD(mod

function

f

is

and

N

and a multipli-

that

the

index of

N). q

D-WUD(mod

excercise

by the mere Let

f

satisfying

n's

N.

Prove

[76]).

an odd

q2)

then

functions

of t h e M o e b i u s [803,

from

f(p) a 2 that

squares,

prime

such

it is a l s o

replace

the

M(f,q). completely

for a l l p r i m e s

p

multi-

and which

f(n) = n . integer-valued has

function

a positive

f

the

density

FN.

from Proposition

4.1

a criterion

for

F N-

function

5.7

to d e d u c e

of p r i m e

r2(n) , c o u n t i n g WUD(mod

the v a n i s h i n g

of the

~(n) .

in t h e c a s e

is

of

(f(n),N) = I

to T h e o r e m

function

cannot

integer-valued

to the class

Deduce

one

existence

be an

for w h i c h

belongs

(O.M.FOMENKO

as a s u m of t w o

f

integer

m(f,N)

for a m u l t i p l i c a t i v e

f

the

find an

k al.

the Corollary

for w h i c h

and

that

for a l l

(H.DELANGE

7. U s e

T

[77]).

function, N)

can

in the p r e c e d i n g

M(f,q)

UD(mod

6.

than

Prove

(H.DELANGE

plicative

one

well-defined

a multiplicative

for a l l

3. S h o w t h a t

T

with

is l a r g e r

2. L e t that

for a n y

function

N) .

N).

Determine

all

the r e p r e s e n t a t i o n s

those of

n

CHAPTER

VI

POLYNOMIAL-LIKE

§ I. G e n e r a t i n g

I. T h e uniform

x

to h a v e

P(x)

with

shall

[48]),

stated

p

P(x)

with

Zet

shows

that

for c h e c k i n g

set of values

(xP(x),N)

=1

in o r d e r

which

P(x) the

is b a s e d

conjecture

it is

for a g i v e n

by

generate

to c h e c k w e a k

function

whether,

attained

does

such a procedure,

(For a p r o o f

X

If

polyno-

at integers

group

G(N).

on a c o r o l l a r y

for a l g e b r a i c

and

let

see e . g . W . S C ~ 4 I D T P(x)

P

[76],

be a p o l y n o m i a l

be a n o n - p r i n c i p a l

the p o l y n o m i a l

curves

We to

(A.WEIL

character

does

not

Ch. II,

over

Z

(mod p)

satisfy

the

th.2C) . of degree

and denote

by

K. d

congruence

- c W d(x) (mod p)

a certain

1

5.7

of a p o l y n o m i a l

below:

6.1.

order.

s e t of v a l u e s

for a p o l y n o m i a l - l i k e

o n the R i e m a n n

be a p r i m e

Further, its

property

theorem

LE~4A Let

Z, t h e

now present

A.Weil's

(mod N)

a procedure

over

the

by the

to T h e o r e m

distribution

important mial

Corollary

G(N)

FUNCTIONS

~

constant

c

and a polynomial

W(x),

then

one

has

X(P(x)) I -< (K - I ) P ½ •

x(mod p)

First ERDOS,

l e t us d e d u c e

posed

He a s k e d , function

on one

whether will

from this

lemma

an answer

of the n u m b e r - t h e o r e t i c a l

a "well-behaved"

be necessarily

(in a c e r t a i n

WUD(mod

p)

to a q u e s t i o n

meetings

for all

sense)

o f P.

at O b e r w o l f a c h . multiplicative

sufficiently

large

p.

97

This

cannot hold

way,

since

this

property.

the answer

However

Let

degree

Let

6.2.

d ~1

is a large

question

f

all sufficiently

large primes p .

We need

If

6.3.

P(x)

reducible factors not divide

Proof.

of

P

into

implies

P

W(x)

of

D

in

a certain P

and by

prime

ideal of

P(x)

splits

is

~D(mod

of degree

with a constant

p)

for

and a poly-

p

P(x)

factors

that

is a prime which does

over

with

i.e.

now that constant

its h i g h e s t

Our

p

cannot as above.

coeffi-

factorization assumption

is a p r i m e w h i c h

c, k a 2

~cWk(x) (mod p) . Let

P(x)

c,k,W(x)

be the

the rationals.

Assume

Then

P(x).

cWk(x)

is monic,

bmt with a c e r t a i n P(x)

c, k ~2

which

and assume

P(x) =V~I (x)...vSn(x)

GF(p),

d a I

of the product of all ir-

P, W

and a p o l y -

be the images

thus

in

ZK

non-zero ZK

constant

the ring

containing

p

K, the p o l y n o m i a l

write

n

P(x) =

~

(x-ai)

ai

i=1

with

of

= cw(x) k

holds with field

func-

P(x)

the d i s c r i m i n a n t

that

Let

we have W

polynomial-like

for a certain constant f

is a p o l y n o m i a l

(al,a2,...,a n) =I.

resp.

~(x)

Then

to a polynomial

irreducible

does not d i v i d e

of

D

P(x)

first

unity.

that

nomial

of

Assume

not have

for w h i c h

with a p o l y n o m i a l cWk(x)

k ~2.

P(x) = c w k ( x )

(mod p)

equals

in a natural does

of f u n c t i o n s

D, nor the leading c o e f f i c i e n t of

be congruent

cient

and

e Z[x]

Denote by

W(x).

d(n)

a lemma.

is not of the form nomial

p

which is not of the form W(x) ~Z[x]

LEM~

class

be a multiplicative,

c, a polynomial

Proof.

occuring

function

is positive.

for all primes

f(p) =P(p)

functions

5.9 the divisor

there

to Erd~s's

PROPOSITION

tion.

for all m u l t i p l i c a t i v e

by P r o p o s i t i o n

a l , . . . , a r ~ZK,

distinct,

and

c. Denote

of integers and let P(x)

by

K

of it. Let

the s p l i t t i n g P

be any

K = Z K / P ~ GF(pf) . must

split

in

K,

Since

thus we m a y

98

n

[(x)

~

=

(x-Ti) al

=

c

W

(x)

k

i=1

with

~i

all

~i's

being

the

were

it e x i s t s

a pair

divisible

by

trary If

~i =~j and

ai

under

then

k

with

i #j.

since

=Ax d +...

the

would

canonical

divide

But

map

Z K + K.

( a l , . . . , a n) =I

then

it is a r a t i o n a l

is not

~i,~2,...,~n

suitable

integer

of

~i - ~ j ~ P

integer

so

w e get

If

hence D

pID,

is con-

assumption.

P(x)

distrinct with

P

to our

image

distinct,

algebraic

q, w h o s e

all

then

monic

for

n ai ~ (x -~i) with i=I we m a y w r i t e ~i =Si/q

P(x)

=A

i=1,2,...,n

integers

prime

and

81,...,8 n

factors

divide

and A

a rational

hence

pXA.

positive If we n o w

put

n

F(X)

then

:

~ i=I

(x - 8i) ai

F(x) = q d A - I P ( x / q ) If n o w

P(x)

for

certain

F(x)

tain

c,

F(x)

k a2

(Z[x].

and

W(x)

we h a v e

eZ[x]

- c W k(x) (rood p)

and we d e f i n e

and

and

A' , q

by

cc'

- qdA'cW(xq')k(mod

since

F(x)

constant

is m o n i c a,

k >_2

P (x) = A q - d F (qx)

contradicting

our

_--AA' - I (mod p) , then

p)

we m u s t

and

have

V(x)

either

eZ[x]

F(x)

= a V k(x)

for a c e r -

hence

= a A q - d v k (qd)

assumptions,

or

the d i s c r i m i n a n t

of the p o l y n o m i a l

n

H (x-Si) i=I P(x) must

is d i v i s i b l e be

TO p r o v e

R =

also

by

divisible

the p r o p o s i t i o n

{P(x) : (xP(x) ,p) = I }

p, but by

in t h a t

p, w h i c h

it s u f f i c e s

case

the d i s c r i m i n a n t

is i m p o s s i b l e .

to

show

that

the

set

of

99

generates

G(p)

all p rime factors

for s u f f i c i e n t l y

factors

of

of

satisfying

and a s s u m e P(x) . If

a non-principal x

of the d i s c r i m i n a n t

P(x)

coefficient

large

R

character pfxP(x)

further

L e t thus

p

p), e q u a l

we h a v e

p

of the p r o d u c t that

does

d o e s not g e n e r a t e X(mod

p-1 ( d - 1 ) / p >_ I Z X(P(x) I = x:O

and thus we h a v e o n l y

p.

be l a r g e r

of all i r r e d u c i b l e

not d i v i d e

G(p),

the

then t h e r e

to u n i t y on

X(P(x)) = I ,

than

R. Thus

and we o b t a i n

leading exists

for all

from Lemma

6.1

l#{x(mod p) : p ~ x P ( x ) } + X ( P ( O ) ) ] - > p - d - 2

finitely many possibilities

for

P(x)

Z

p.

2. N o w we p r o v e

THEOREM

6.4.

not

of the form

and

let

pa

Let

cWk(x)

be a prime

X(mod

pa),

vides

the d i s c r i m i n a n t

or the

constant

leading

max{d 2 +2d,

power.

on the set D

aO

term

3d +2}.

be ~ p o l y n o m i a l (with a constant

exists

X

unity

in

d ~1

,

W(x) cZ[x]) character

then either

of i r r e d u c i b l e

P, or finally equals

of degree and

a non-principal

R ={P(x) : p~xP(x)}

of the product of

If

If there

over c, k ~2

factors

p

does

not exceed

R

then

either

p

di-

of

pIDao

P,

or

p ~ d 2 +2d.

Proof.

Lemma

5.14

We may

our a s s u m p t i o n s Cp

that

Since

imply

cyclic

x(mod

p2) ÷ < x ( m o d p ) , ~ >

x

of

~

elements)

element

of P r o p o s i t i o n

p. If 6.2.

is odd and

of

moreover

at least

n o w that

G(p 2) =G(p)

the i s o m o r p h i s m

G ( p 2)

the c h a r a c t e r

eCp,

of elements, (with

g i v e n by

satisfying X

in v i e w

pa-1(p-l-d)

x z 1(mod p)

and

acts by

= ~ (x)× (x)

is a c h a r a c t e r

order d i v i d i n g

p

contains

Observe

p2) . M o r e o v e r

X ( < x mod p,~>) where

p

is the u n i q u e

~p-1 ~ x P - 1 ( m o d

R

R #~.

being

where

have

assume

a s 2 .

~

(mod p) , and

×

is n o n - p r i n c i p a l

Indeed,

since

X p =]

is a c h a r a c t e r we p r o c e e d and

~P

(mod p2)

of

as in the proof

is n o n - p r i n c i p a l ,

we

100

xP(R)

with

= YP(R)

a constant

Lemma

6.3)

: c

c,

thus

we o b t a i n ,

with

P (d-l) /p _> ] ~ X x=O

thus

p < (d+1) 2. Assume

for

all

~

with

1 ~r ~p-] X(X)

= X(X)

By a s s u m p t i o n ,

utilize

the

of

reduction

in v i e w

R(mod

of

p) ,

_> p - d - 2

we

are r e a d y

y

is the p r i n c i p a l generated

by

character.

1+p(mod

p2)

Since

we can write

satisfying

n =exp{2~i/p}, = X (1+p) t(x)

for e v e r y

x

Q 5t(x] then

for

all

If x

X(1+p)

=n r

with

we h a v e

= n rt(x)

satisfying

C],

) = ~1

so

rt(P(x))

holds

- C2Cmod

implying

p]

in turn

(1+p) rt (P(x))

_ C3(mod

p2)

i.e.

P(x)r

and t h i s assumes

_= C 3 ( m o d

p2)

shows,

that

at m o s t

p-3

for

x

values

subject

to

(mod p2),

p~xP(x)

the polynomial

all d i s t i n c t

P (x] r

tmod p). D e n o t e

101

these values x ~O(mod

by

p)

of

c I, .... c r

and

let

N(c)

be the number

of s o l u t i o n s

the congruence

P(x) r - c ( m o d

p2)

On one h a n d w e h a v e

r

N(cj)

= ~ R > p(p-I

-d)

~=I

a n d on t h e o t h e r

N(c)

_< # { x

+ p #{x

hand

mod

p: p ~ x , ( P ( x ) r] ' ~ O ( m o d

rood p:

p) ,pr(x)

pXx, (Pr(X)) ' -O(rNod p),

Pr(x]

-c(mod

~c(mod

p) } ÷

p) } ,

thus with

S = ~ { x rood p;

p~x,P(x)

/O(mod

p]

(Pr{x))'

~O(mod

p) }

we get

r

p (p-1 -d]

N(C

_
_p(S-d]

(pr(x))' x(mod

vanishes

for

at

S-d