Selected Papers Of Wang Yuan

SELECTED PAPERS OF

WANG YUAN

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SELECTED PAPERS G»F

WANG YUAN editor

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Wang Yuan

Chinese Academy of Sciences, China

\ ^ World Scientific NEW JERSEY

• LONDON

• SINGAPORE

• BEIJING • S H A N G H A I

• HONGKONG

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The editor and publisher would like to thank the publishers of the following periodicals for their assistance and permission to reproduce the articles found in this volume: Acta Arith. Acta Math. Sin. Acta Math. Sin. (New Series) Ann. Pol. Math. Chin. Ann. Math. J. Number Theory Kexue Tongbao Sci. Rec. (New Series) Sci. Sin. Sci. Sin. (Series A) Shuxue Jinzhan While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.

SELECTED PAPERS OF WANG YUAN Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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V

PREFACE Earlier in 1998, Professors C. D. Pan and L. Yang, and many of my friends and colleagues urged and encouraged me to publish a volume of my selected papers since most of these papers were published in China and hard to find elsewhere. The Hunan Normal Publisher and in particular, Ms S. H. Meng published a Chinese edition which appeared in 1999. Now World Scientific Publishing Company and Dr K. K. Phua has invited me to publish an English version of my selected papers. This is really my honour and I accepted the offer. Ms R. T. Tan and Ms E. H. Chionh helped me with the editing work. I have been working in Chinese Academy of Sciences since 1952 when I graduated from the Department of Mathematics, Zhe Jiang University. My teacher, Professor L. K. Hua, led me to the field of Number Theory. We also cooperated for a long time on the applications of number theory to numerical analysis. My other longtime collaborator is Professor K. T. Fang. Our joint work is in experimental designs. In order for the readers to understand my life and works, the volume contains a related paper of Professors W. L. Li and X. D. Yuan. The readers can also see that my works are related and influenced by N. C. Ankeny, N. S. Bahvalov, A. Baker, V. Brun, A. A. Buchstab, D. A. Burgess, J. R. Chen, T. Cochrane, H. Davenport, P. Erdos, K. T. Fang, G. H. Hardy, L. K. Hua, N. M. Korobov, P. Kuhn, Yu. V. Linnik, J. E. Littlewood, T. Mitsui, C. D. Pan, W. M. Schmidt, A. Selberg, C. L. Siegel and H. Weyl. Finally, I would like to take this opportunity to express my sincere thanks to those colleagues and institutions for their kind help. Wang Yuan

vii

CONTENTS

Wang Yuan: A Brief Outline of His Life and Works Li Wen-Lin and Yuan Xiang-Dong

xi

1. Number Theory [1] On the Representation of Large Even Integer as a Sum of a Product of at Most 3 Primes and a Product of at Most 4 Primes, Ada Math. Sin. 6:3 (1956) 500-513 [2] On Some Properties of Integral Valued Polynomials, Shuxue Jinzhan 3:3 (1957) 416-423 [3] On Sieve Methods and Some of the Related Problems, Sci. Rec. (New Series) 1:1 (1957) 9-12 [4] On Sieve Methods and Some of Their Applications, Sci. Rec. (New Series) 1:3 (1957) 1-5 [5] On the Representation of Large Even Number as a Sum of Two Almost-primes, Sci. Rec. (New Series) 1:5 (1957) 15-19 [6] (with A. Schinzel) A Note on Some Properties of the Functions (p(n),a(n) and 6(n), Ann. Pol. Math. 4 (1958) 201-213, Corrigendum: 19 (1967) [7] A Note on Some Properties of the Arithmetical Functions p > n x / a } , where a > 2, n is even and p denotes prime number. If we can prove that Pi+i > 0 when I is a positive integer and n is sufficiently large, then we obtain (1,1). Consider further Qa = {q: 1 < q < n, p\(n — q) =>• p > 7i 1//a }, where q denotes prime number. Then (1,/) follows from Qi+i > 0. T. Estermann was the first who proved in 1932 that every large even integer is the sum of a prime and a product of at most 6 primes under the assumption of Generalised Riemann Hypothesis (GRH). We denote this result by (1,6)#. Wang Yuan [2,5,16] improved Estermann's result to (1,4)* (1956),

(1,3)^(1957).

(2)

By the use of Brun's method, theory of prime number distribution and Yu. V. Linnik's large sieve, A. Renyi proved (l,c) in 1948, where c is an absolute constant. The main part of Renyi's proof of his (l,c) is the following mean * The number within the square brackets refers to Sec. I of the list of publications in this volume (pp. 481-485).

xiv value theorem: There exists a constant 0 such that

(Ms) Y ( lfc)=1 max w(x;k,I) - -k ^ - f ^- = O (-^—] fc '

f( ) J2 ln
6, 0, there exists a positive integer n such that ;(;+:> (p(n + i + l)

Q i


Xx. Similar results hold also if Euler function is replaced by some other arithmetic functions.

xvii 5) Small Solutions of System of Diophantine Equations and Inequalities Given a vector with complex components x = (xi,... ,xs), let |x| = maxi>j> s \xi\. Given a form F, let \F\ be the maximum absolute value of its coefficients. With every form F of degree k, a form F ( x i , . . . , Xfc) is associated which is linear in each vector Xj(l < i < k) and such that F(x) = F ( x , . . . , x). W. M. Schmidt proved in 1980 the following result: Given integers h > 1, m > 1 and odd numbers k\,.. .,kh and given a positive number E, however large, there is a constant CQ = co(&i,... ,kh;rn,E) as follows. If M is > 1 and if F\,..., Fh are forms with real coefficients of respective degree k\,..., kh in x = {x\,..., xs), where s > Co then there are m linearly independent points x ( l ) , . . . , x(m) in Z m with |x(i)|<M,

l 0, however small, there is a constant c\ = Ci(fci,..., kh\ m, e) such that if G\,..., Gh are forms of respective degrees k\,... ,kh in x = (x\,..., xs) with integral coefficients, where x > c\, then G\,..., Gh, vanish on an m-dimensional subspace which is spanned by integer points x ( l ) , . . . ,x(m) having |x(i)|«G£,

l 5). They also gave another generalisation of (9) based on the C. Jacobi-O. Perron algorithm. Let i-j = (rij,..., rti), 1 < i < s be real vectors, q = (qi,...,qt) integral vector and (x, y) = X)*=i ZjT/i the inner product of x and y. Wang Yuan [52,54, 55] proved in 1982 the following result: There exist r i , . . . , ra such that

3

qk = — n

1< k< t

*—1

= O(Cn-2t+E). (11) There is also formula for the general E"(C)(a > 1), but the weights are more complicated. When t = 1, the results were obtained by Bahvalov, Haselgrove, Hua and Wang, Korobov and Niederreiter. The generalisation in rational form was obtained by I. H. Sloan later (see I. H. Sloan and S. Joe, Lattice Method for Multiple Integration, Oxford, 1994). 7) Number Theoretic Methods in Statistics The problems in statistics often need the use of small sample of quasi-random numbers. In 1981, Wang Yuan and Fang Kai-Tai first found a set of small samples of quasi-random numbers and gave applications to the problems of experimental design: Suppose that there are s factors and each factor has q levels in an experiment. Then there are qs distinct combinations among levels of s factors. Each combination may correspond to a lattice point in G3. The main idea of number theoretic method is to find out a set U of O(q) points among these qs points which have lower discrepancy. Then we arrange the experiments on each point of U and use the regression analysis to obtain a better combination. This method is called uniform design, where the number of experiments is far less than the number of experiments O(q2) required in orthogonal array. As a result of the applications of uniform design in Chinese industrial departments, the precision of results is similar to that obtained by orthogonal design. To apply number theoretic methods to the problems of statistics, we shall also need sets of points which are uniformly scattered (i.e. set with lower discrepancy) on some domains other than Gs. Wang Yuan and Fang Kai-Tai [67,68] proposed in 1990 an algorithm to treat this problem: Starting from a set of n points which is

xxi uniformly scattered on Gs, we may derive a set of n points with lower discrepancy on any one of the following domains: As = {x : 0 < X\ < x? < • • • < xs < 1}, Bs = {x : x\ + • • • + x2s < 1}, Ss-l

= {x:x21

+ ---+x2s

= 1} and T3_, = {x :

x\t + • • • + xs = 1, Xi > 0, 1 < i < s}, and also a set of approximately n points which is uniformly scattered on the domain T s _i(a, b) = {x : x\ H

\-xs = l.flj < xt < bi: 1 < i < s},

where a, > 0,1 < i < s,X]i=i a i < 1 a n ^ Z)i=i ^* > 1- A set on T s _i(a, b) is, in fact, corresponding to an experimental design with mixture (see [79]). The evaluation of probability and moment is reduced to the numerical evaluation of multiple integral, so we may try to use number theoretic method. Moreover, the number theoretic method can also be applied to the problems of optimization, of finding out a set of representative points for a multivariate distribution, and of statistical inference in statistical sciences. 3. Miscellaneous 8) Orthogonal Latin Squares Let ( 1 , 2 , . . . , s) be arranged in an s x s square in such a way that every number occurs exactly once in every row and once in every column. Such a square is called a Latin square of order s. Two Latin squares are called orthogonal, if one of the squares is superposed on the other, every number of the first square occurs with every number of the second square once and only once. We denote by N(s) the maximal number of mutually orthogonal Latin squares of order s. Euler proposed a conjecture that N(s) = 0 if s > 6 and s = 2 (mod 4). R. C. Bose, S. S. Shrikhande and E. T. Parker made great contribution on Euler conjecture. They proved in 1960 that N(s) > 2 when s > 6. Using their method and Brun's method, S. Chowla, Erdos and E. G. Straus proved that there exists a constant so such that N(s) > |s5T whenever s > s0. Wang Yuan [22, 31] gave in 1964 the following improvement: There exists a number si such that if s > si, we have N(s)>s&.

(12)

9) History of Modern Chinese Mathematics Wang Yuan wrote a book "Hua Loo Keng". Besides the life and mathematical works of the great modern Chinese mathematician Hua Loo-Keng, this book also describes some pieces of the process in which the modern mathematics was drawn into China from America, West Europe and Japan and its developments around Hua Loo-Keng as a centre. For the references of this paper, we refer to the list of Publications by Wang Yuan.

xxii References [1] Li Wen-Lin and Wang Yuan, The Biography of Modern Chinese Scientists, Vol. 1, ed. Lu Jia-Xi (Science Press, 1991), pp. 81-88. [2] Yuan Xiang-Dong, Wang Yuan and Number Theory, The Eminent Contemporary Chinese Scientists in Mathematics and Information Sciences, ed. Lu Jia-Xi (Hei Longjiang Normal Publishing House, 1994), pp. 13-25.

SELECTED PAPERS OF

WANG YUAN

1

ON THE REPRESENTATION OF LARGE EVEN INTEGER AS A SUM OF A PRODUCT OF AT MOST 3 PRIMES AND A PRODUCT OF AT MOST 4 PRIMES* WANG YUAN Institute of Mathematics, Academia Sinica Received 8 July 1955

0. Introduction V. Brun [1] first proved in 1920 the following result: Every large even integer is the sum of two integers each being a product of at most 9 primes. We denote this theorem by (9,9), and we may define (a, b) similarly. Brun's method and his result were improved by several mathematicians, namely: (7, 7) (Rademacher, 1924) [2] (6,6) (Estermann, 1932) [3] (5,7), (4,9), (3,15), (2,366) (Ricci, 1937) [11] (5,5) (Buchstab, 1938) [4] (4,4) (Buchstab, 1940) [5] Professor Hua Loo Keng pointed out that (4,4) may be possibly improved by the combination of the methods of Selberg [6], Brun and Buchstab. The purpose of this paper is to prove (3,4), i.e. the following: Theorem 1. Every large even integer can be represented as a sum of a product of at most 3 primes and a product of at most 4 primes. Theorem 2. There are infinitely many integers n such that n has at most 3 prime factors and n + 2 has at most 4 prime factors. In this paper, we use p,p',p",... ,pi,P2, • • • to denote prime numbers. It seems possible to use the present method to prove (3,3) but it needs some complicated numerical calculations. *Acta Mathematica Sinica, 6:3 (1956) 50O-513.

2

1. Some Computations Lemma 1.

(n,i)

If x > 1 and N > 1, then

= l

^

+ O (log 2AT • log log 3a;TV) + O ((log log 3 z ) 2 ) ,

where fi(n) denotes the Mobius function and Q(n) the number of distinct prime factors of n. We refer [7] for the proof. Lemma 2. Let z > 1, g(l) = 1, g(2) = 1/2, g(p) = 2/p(p > 2) and g(n) = ripin 9(P) for S1uare free number n. Then

£ Hn)\g{n) 1^(1 - PCP))"1 = ^ I I ^ S Proof.

log2 z

+ °( l o g 2 z ' lo S lo g 3z )-

Set rl>(q) = UP\q(P - 2). Then

X>Wls(n)II(l - S(P))-1 2 2 2fn

PI"

2t"

p|n

^

p

n l«^ /

\«

(2)

where the constant in "0" depends only on F(x), and then Ricci [3] gave the following improvement: _ N->oo

,(N;F(x))logN N

29

(3)

4

where fip is a constant depending on F(x) and its definition will be given later. In this paper we shall use Selberg's method to prove the following: Theorem 1.

We have

^;F ( a :)) k and P = rip