New Trends in Probability and Statistics. Vol. 4 Analytic and Probabilistic Methods in Number Theory: Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23–27 September 1996 [Reprint 2012 ed.] 9783110944648

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New Trends in Probability and Statistics - Volume 4 Analytic and Probabilistic Methods in Number Theory

NEW TRENDS IN PROBABILITY AND STATISTICS Volume 4 Analytic and Probabilistic Methods in Number Theory Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996

Editors A. Laurincikas, E. Manstavicius and V. Stakenas

>jEy VILNIUS, LITHUANIA

///VSP/// UTRECHT, THE NETHERLANDS TOKYO, JAPAN

VSP BV P.O. Box 346 370 AH Zeist The Netherlands

TEV Ltd. Akademijos 4 2600 Vilnius Lithuania

© VSP BV & TEV Ltd. 1997

First published 1997 ISBN 90-6764-255-X (VSP) ISBN 9986-546-23-0 (TEV)

All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted in any f o r m or by any means, electronic, nuvluinical, photocopying, recording or otherwise, without the prior permission of the copyright o\\ nor.

Typeset in Lithuania by TEV Ltd., Vilnius, SL 1185 Printed in Lithuania by Spindulys, Kaunas

CONTENTS

Preface

ix

I. ALGEBRAIC NUMBER THEORY Gel'fond's transcendence method by elementary means P. Bundschuh

3

Algebraic conjugates outside the unit circle A. Dubickas

11

Applications of algebraic units C. Kliorys

23

On some diophantine equations connected with Pellian equation D. A. Mitkin

27

II. QUADRATIC FORMS On the general unimprovable estimates of the singular series of positive quadratic forms G. Gogishvili

35

On the representation of numbers by certain quadratic forms in ten variables T. Vepkhvadze and N. Tsalugelashvili

45

III. ZETA AND L-FUNCTIONS The universality theorem with weight for the Lerch zeta-function R. Garunkstis

59

The general additive divisor problem and moments of the zeta-function A. Ivic

69

The correspondence between a Dirichlet series and its coefficients M. Jutila

91

A note on the mean square of ("(.s) in the critical strip A. Kacenas

107

Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions. II M. Katsurada and K. Matsumoto

119

vi

Contents

On limit distribution of the Lerch zeta-function A. Laurincikas

135

The fourth moments of Dirichlet series A. I: Vinogradov

149

IV. MULTIPLICATIVE NUMBER THEORY Estimation of a certain function related to the Dirichlet divisor problem J. Furuya and. Y. Tanigawa

171

On some pairs of multiplicative functions correlated by an equation I. Kätai and Β. M. Phong

191

A characterization of some additive arithmetical functions J.-L. Mauclaire

205

On pseudosquares A. Schinzel

213

A remark on some special arithmetical functions W. Schwarz and J. Spilker

221

Modified zeta functions and the number of ^-integers E. Stankus

247

V. VALUE DISTRIBUTION OF ARITHMETIC FUNCTIONS On a conjecture by Erdös and its extension to additive functions on the set of pairs of integers G. J. Babu

261

Multiplicative functions and stochastic processes G. Bareikis

271

Convolutions of the Poisson laws in number theory D. Bekelis

283

Shifted and Kubilius models P. D.T. primes A. Elliott The distribution of the number of prime divisors of numbers of form ab+ 1

297

P. D.T. A. Elliott and A. Särközy

313

Multiplicative functions of Farey fractions K.-H. Indlekofer and V. Stakinas

323

On some inequalities in the probabilistic number theory J. Kubilius

345

Contents

vii

The asymptotical expansion in the mean value theorem for multiplicative functions A. Maciulis

357

Local distributions of arithmetic functions on semigroups R. Skrabutenas

363

The mean values of multiplicative functions. Ill G. Stepanauskas

371

On the convergence to the Poisson law J. Siaulys

389

VI. PROBABILISTIC THEORY OF NUMBER SYSTEMS AND SERIES Renormalization of algorithms in the probabilistic sense P. Hubert and Y. Lacroix

401

Probabilistic theory of additive functions related to systems of numeration E. Manstavicius

413

On limit theorems for endomorphisms of A. Renyi type G. Misevicius

431

Densities for sums of independent random variables and their applications to the value distributions of A(x), P(x) and C(.s) M. Nakajima

441

VII. MISCELLANEOUS Note on the transcendence of a generating function J.-P. Allouche

461

Solution of the equations of dynamical chaos F. Ivanauskas, T. MeSkauskas and B. Kaulakys

467

Logistic differential equation of neutral type D. Svitra

475

A theorem of Müntz type for Chebyshev polynomials R. Wallisser

485

Complete systems of holomorphic functions R. Wallisser

489

Programme of the Conference

499

PREFACE The Second International Conference "Analytic and Probabilistic Methods in Number Theory" was held in honour of Professor Jonas Kubilius on the occasion of his 75th birthday, September 23-27, 1996 in Palanga, Lithuania. Twentyseven distinguished mathematicians from Belorussia, Estonia, Finland, France, Georgia, Germany, Hungary, Japan, Russia, Sweden, and the USA joined 19 Lithuanian colleagues and their celebrating leader. Five years have passed since the first meeting at the same site. During this period of five years professor Kubilius has spent a lot of time and energy on the preparation of the second edition of the Textbook on Probability and Mathematical Statistics (1996), and the Collection (1996) of his public lectures, several studies and articles. He was also one of the editors of the Proceedings of the Sixth Vilnius Conference on Probability Theory and Mathematical Statistics (1994). Moreover, he headed the team which created and issued the academic Lithuanian-English-Russian Dictionary of Mathematical Terms (1994). The contributions of the first Palanga meeting were published in New Trends in Probability and Statistics, Vol. 2, edited by F. Schweiger and E. Manstavicius. The proceedings of the second Palanga Conference appears in the same series and contains most of the 46 talks delivered at the conference. They cover a broad range of areas within the contemporary theory of numbers, especially its analytic and probabilistic branches. The organizers have included some invited papers by mathematicians who were unable to attend. Some participants have submitted modified versions of their original lectures, which fall within the general scope of the meeting. The contributors to this volume intend to congratulate Professor J. Kubilius by dedicating their achievements. All contributed papers were accepted for publication after thorough refereeig and we would like to express our deep gratitude to all referees. The conference was sponsored by the Lithuanian Studies and Science Foundation, the Open Society Fund-Lithuania, and Vilnius University. We gratefully acknowledge their sponsorship. A. Laurincikas, E. Manstavicius and V. Stakenas, editors

Part One

ALGEBRAIC NUMBER THEORY

New Trends in Prob, and Stat., Vol. 4, pp. 3 - 9 A. Laurincikas et al. (Eds) © 1997 VSP/TEV

GEL'FOND'S TRANSCENDENCE METHOD BY ELEMENTARY MEANS PETER BUNDSCHUH Mathematisches Institut der Universität, Weyertal 86-90, D-50931 Köln, Germany

ABSTRACT An axiomatic description of Gel'fond's transcendence method from 1934 is given in such a way that the analytical parts of the proof can be performed using real analysis only. Some classical applications are added.

1. INTRODUCTION AND RESULTS In the last chapter of their book Gel'fond and Linnik (1966) gave an elementary proof of the theorems of Hermite-Lindemann and of Gel'fond-Schneider, respectively. Here elementary means that the authors avoided any use of complex function theory, but relied only on facts from real analysis in the analytical part of their proofs. In principle, Gel'fond and Linnik used in their two proofs the transcendence method of Gel'fond which this one had developed to solve Hilbert's 7th problem in 1934. It is the main aim of this note to present an elementary version of Gel'fond's method, and we will do this at once in an axiomatic form which should be compared with Waldschmidt's axiomatisation (Waldschmidt, 1974, Theorem 3.3.1) using classical, i.e. function theoretical tools. Our main result is the following THEOREM. Let Κ be a real algebraic number field. Suppose f i , • • • , fs € C°° (E) to be s ^ 2 real-valued functions satisfying the three conditions (i) / ι , f2 are algebraically independent over Q. (ii) For any σ € {1,..., .s} there exists α Ρσ £ K[X\,..., Xs] such that fa = Pa(fu---Js) holds. (iii) For any compact interval I CM. there exists a real constant 7 > 1 {which may depend on I ) such that \f(„k] (x)\ ^ 7 f c + 1 holds for all χ € I, k G No, and σ = 1,2.

P. Bundschuh

4

Then the following inequality

holds

card {a: e Μ | f x ( x ) , f , { x ) € Κ} ^ 2 [K : Q], Obviously the above-mentioned classical results which were proved in (Gel'fond and Linnik, 1966), i.e. the "real" Gel'fond-Schneider theorem and the "real" Hermite-Lindemann theorem are special cases of our theorem: COROLLARY 1 . Let a a, b, ab are algebraic.

€

R +

\

{ 1 }

and b

e

R

\

.

Then not all three numbers

Proof. Suppose to the contrary that, under the remaining conditions, a, b, ab are algebraic. Define Κ := Q(a,b,ab), and take fi(x) := e x , fi{x) : = e b x . Evidently, f\, fa satisfy all hypotheses of our theorem where, for (i), the irrationality of b is essential. But, on the other hand, we have / σ ( η log a) G Κ for all η € Ζ and σ = 1,2. Since log α φ 0 the last fact contradicts the final conclusion of our theorem. COROLLARY 2. Let a e K x . Then not both numbers a and ea are algebraic. Proof. Here we can argue with / i ( x ) : = x, fi{x) left to the reader.

e x , but the details are

In the complex theory the transcendence of sin a for any complex algebraic number α φ 0 (and thus the transcendence of π) is an immediate consequence of the general Hermite-Lindemann theorem. Nevertheless, in our "real" theory, too, we can deduce from our theorem the following corollary. COROLLARY

Let a € R . Then not both numbers a and sin a are algebraic. In particular, π is transcendental. x

3.

Proof. Assume that for some β Ε l x both numbers a, sin α are algebraic. Then cos α is algebraic, too. Take Κ := Q(a, sin a, coso), and put s := 3, f\{x) := x, fi(x) '•= sina;, f^(x) : = cosic. Conditions (ii) and (iii) are obvious whereas (i) has an easy "real" proof. The two equations

i/=0

and

ί

\ 2 i / +" l

(η = 0 , 1 , . . . )

Gel'fond's Transcendence Method by Elementary Means

5

having again easy "real" proofs, by induction, show that we have fa (na) e Κ for any η € Ζ and σ = 1,2,3. Since αφ 0, this contradicts the final conclusion of our theorem.

2. SOME LEMMAS For the proof of our theorem we need three lemmas which will be collected here. The first one is the following Siegel type LEMMA 1. Let Κ be an algebraic number field, Οχ its ring of integers, and D := [Κ: Q]. Let τη, η Ε Ν with η > Dm, and let αμν £ Οκ u — 1,...,

(β = 1 , . . . , τη;

η) be given; suppose |αμ„| ^ A for all possible pairs {ν, μ). Then

there exists a constant c £ M+ which is independent of τη, τι the αμυ, and of A such that the τη equations

are satisfied by some {x\,... 0
iVlogTV ^ DJN, compare (3), we can apply Lemma 1. Taking (3) and (8) into account, it shows that there exist L2 numbers ρ ν (λ) € Ζ, not all zero, with Μ

A) I < exp(c 4 jV)

(9)

such that all J Ν conditions (5) are satisfied. Here, and in the sequel, c\, c 2 , . . . are positive real numbers which are independent on k, ν, N, and later also on M . Step 2. Let now I be an arbitrary, but fixed compact interval in K. We have to estimate on I . From (7), (9) and condition (iii) of our theorem we find

Ι

^

ω

ΐ

^

ν

= e*V

^ + 2 i

Σ Σ(

Σ λ ι

+

fc,

.

^ ^

and therefore \F^\x)\^^nY+2L{2LY

(10)

for all χ 6 I and u € No, where 7 = 7(7) > 1 is the constant from condition (iii) of the theorem. From (10) it follows, in particular, that F χ has an everywhere converging Taylor series expansion about any xq € M. Since /1, / 2 are algebraically independent over Q, by condition (i) of our theorem, the last fact implies that for each xo Ε Κ there exists a ν = v(xq) e No such that F^(xq) Φ 0. Therefore we can choose the largest Μ € Ν such that F^\xj)=0 By (5), we have Μ Ff^\x0)

for ^

Ν,

j = 1,...,

J;

v = 0,...,M-l.

and there exists an xq €

{xi, • • •, x j } with

φ 0. Indeed, from (7) and Lemma 2 we deduce F{NM)(xo)eK\

Step 3. firstly

(11)

(12)

To estimate the absolute value of this number from below we remark

den F(nm) (X0) Ζ dM8stM+2L

< exp(c 6 M)

(13)

Gel'fond's Transcendence Method by Elementary Means

9

where we used (3) and the discussion on denominators after (7). The considerations on houses after (7) and the estimate (9) lead to Κ Λ ^ Μ Ι - ( D - l ) M l o g M - C9M.

(15)

Step 4. Suppose now the interval I to be the convex hull of the points x \ , x j. Then, by (11), the function Fn has at least J Μ zeros in I counting with multiplicities. Therefore Lemma 3, applied with q JM, u : = M , and combined with (10) gives ljl(J-l)M

(16)

\J\(J-\)M

< — eC5N7(/)JM+2L(2L)JM ((J-l)M)! for any χ ς. I. Of course, we may suppose J > 2, since otherwise our theorem is proved. Taking χ = xo we find from (16) log|4M)(*0)| ^ - ( f

- l ^ M l o g M + doM.

(17)

Combining (15) and (17) we get J/2 < D, and this proves our theorem. REFERENCES

Bundschuh, P. (1996). Einführung in die Zahlentheorie. 3rd edn. Springer, Berlin. Gel'fond, A. O. and Linnik, Yu. V. (1966). Elementary Methods in the Analytic Theory of Numbers. MIT Press, Cambridge, MA. Waldschmidt, Μ. (1974). Nombres Transcendants. Springer, Berlin.

2·

New Trends in Prob, and Stat., Vol. 4, pp. 11-21 A. Laurincikas et al. (Eds) © 1997 VSP/TEV

ALGEBRAIC CONJUGATES OUTSIDE THE UNIT CIRCLE ARTÜRAS DUBICKAS Department of Mathematics, Vilnius University, Naugarduko 24, Vilnius 2006, Lithuania. E-mail: [email protected]

ABSTRACT Let a be an algebraic number of degree η and let q be the leading coefficient of its minimal polynomial. We find lower bounds for the product of s conjugates outside the unit circle in terms of n, q and ,s. For a non-reciprocal algebraic number of norm ± 1 this lower bound is given in terms of q.

1. INTRODUCTION Let α be an algebraic number of degree η ^ 2 with conjugates a = a \ , 012,···, OLn. If P(x) = q{x - ai)(x - a2) • • • (x - an) — qxn + qn-\xn~l

Η

+ q0

is the minimal polynomial of a, then we say that q is the denominator of a . Denote the absolute norm of a by λγ( ϊ Ν (a) = a \ a 2 • · · « „ = ( ~ l ) " 9 o . 9

Let η A(a) = J ] m a x ( l , | a j | ) j=ι be the product of conjugates of a outside the unit circle. Lehmer's conjecture on the lower bound for Λ (a) of an algebraic integer a (i.e. g = l) was stated in (Lehmer, 1933): there exists an absolute constant 0 such that Λ(α) ^ 1 + 6 whenever a is not a root of unity. The best unconditional result so far follows from the work of Dobrowolski, Cantor and Straus (Louboutin,

12

A. Dubickas

1983). For a non-reciprocal algebraic integer Smyth (1971) obtained the best possible result Λ ( α ) > θγ, where θ\ is the real root of χ 3 — χ — 1 = 0. In this paper we will consider the case q > 2. If |JV(a)| > 1, then