Summation formulae and zeta functions

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Johan Andersson

Summation formulae and zeta functions

Department of Mathematics Stockholm University

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Johan Andersson

Summation formulae and zeta functions

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4 Doctoral Dissertation 2006 Department of mathematics Stockholm University SE-106 91 Stockholm Typeset by LA_TEX2e c 2006 by Johan Andersson e-mail: johana@math.su.se ISBN 91-7155-284-7

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ABSTRACT

In this thesis we develop the summation formula X ad−bc=1 c>0 fa b c d = “main terms”+ X m,n6=0 1 π Z ∞ −∞ σ2ir(|m|)σ2ir(|n|)F (r; m, n)dr |nm|ir|ζ(1 + 2ir)|2 + ∞ X k=1 θ(k) X j=1 X m,n6=0 ρj,k(m)ρj,k(n)F 1₂− ki; m, n+ ∞ X j=1 X m,n6=0 ρj(m)ρj(n)F (κj; m, n),

where F (r; m, n) is a certain integral transform of f , ρj(n) denote the Fourier

coefficients for the Maass wave forms, and ρj,k(n) denote Fourier coefficients of

holomorphic cusp forms of weight k. We then give some generalisations and ap-plications. We see how the Selberg trace formula and the Eichler-Selberg trace formula can be deduced. We prove some results on moments of the Hurwitz and Lerch zeta-function, such as

Z 1 0 ζ∗ 1₂+ it, x 2 dx = log t + γ − log 2π + Ot−5/6, and Z 1 0 Z 1 0 φ∗ x, y,1₂ + it 4

dxdy = 2 log2t + O(log t)5/3,

where ζ∗(s, x) and φ∗(x, y, s) are modified versions of the Hurwitz and Lerch zeta functions that make the integrals convergent. We also prove some power sum results. An example of an inequality we prove is that

√ n ≤ inf |zi|≥1 max v=1,...,n2 n X k=1 zv_k ≤√n + 1

if n + 1 is prime. We solve a problem posed by P. Erd˝os completely, and disprove some conjectures of P. Tur´an and K. Ramachandra.

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PREFACE

I have been interested in number theory ever since high school when I read clas-sics such as Hardy-Wright. My special interest in the zeta function started in the summer of 1990 when I studied Aleksandar Ivi´c’s book on the Riemann zeta function and read selected excerpts from “Reviews in number theory”. I remember long hours from that summer trying to prove the Lindel¨of hypothesis, and working on the zero density estimates. My first paper on the Hurwitz zeta function came from that interest (as well as some ideas I had about Bernoulli polynomials in high school). The spectral theory of automorphic forms, and its relationship to the Rie-mann zeta function has been an interest of mine since 1994, when I first visited Matti Jutila in Turku, and he showed a remarkable paper of Y¯oichi Motohashi to me; “The Riemann zeta function and the non-euclidean Laplacian”. Not only did it contain some beautiful formulae. It also had interesting and daring speculations on what the situation should be like for higher power moments. The next year I visited a conference in Cardiff, where Motohashi gave a highly enjoyable talk on the sixth power moment of the Riemann zeta function. One thing I remember -“All the spectral theory required is in Bump’s Springer lecture notes”. Anyway the sixth power moment might have turned out more difficult than originally thought, but it further sparked my interest, and I daringly dived into Motohashi’s inter-esting, but highly technical Acta paper. My first real break-through in this area came in 1999, when I discovered a new summation formula for the full modular group. Even if I have decided not to include much related to Motohashi’s original theory (the fourth power moment), the papers 7-13 are highly influenced by his work. Although somewhat delayed, I am pleased to finally present my results in this thesis.

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ACKNOWLEDGMENTS

First I would like to thank the zeta function troika Aleksandar Ivi´c, Y¯oichi Mo-tohashi and Matti Jutila. Aleksandar Ivi´c for his valuable comments on early versions of this thesis, as well as his text book which introduced me to the zeta function; Y¯oichi Motohashi for his work on the Riemann zeta function which has been a lot of inspiration, and for inviting me to conferences in Kyoto and Ober-wolfach; Matti Jutila for first introducing me to Motohashi’s work, and inviting me to conferences in Turku.

Further I would like to thank Kohji Matsumoto and Masanori Katsurada, for being the first to express interest in my first work, and showing that someone out there cares about zeta functions; Dennis Hejhal for his support when I visited him in Minnesota; Jan-Erik Roos for his enthusiasm during my first years as a graduate student; My friends at the math department (Anders Olofsson, Jan Snellman + others) for mathematical discussions and a lot of good times; My family: my parents, my sister and especially Winnie and Kevin for giving meaning to life outside of the math department.

And finally I would like to thank my advisor Mikael Passare, for his never ending patience, and for always believing in me.

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INTRODUCTION

This thesis in analytic number theory consists of three parts and altogether thirteen papers. Each of the parts has a separate introduction. This thesis is by no means complete and is a result of a number compromises. There are additional papers I would have liked to include, some of which I refer to as forthcoming papers. Even if they are mostly complete they would still need some more work to be included in this thesis. However, even if they would have painted a fuller picture and provided further motivation of my main result, the current version does not depend upon them and it is entirely self contained. Part I and parts of part II of this thesis have also been included in my licentiate thesis [3], and three of the papers have been published in mathematical journals ([1], [2] and [4]).

The Riemann zeta-function

The Riemann zeta-function1

ζ(s) =

∞

X

n=1

n−s (Re(s) > 1) (0.1)

is perhaps the deepest function in all of mathematics. First of all it is arithmetic ζ(s) = Y

p prime

1

1 − p−s, (Re(s) > 1) (0.2)

and has an Euler product. It is symmetric

ζ(s) = χ(s)ζ(1 − s) χ(s) = 2sπs−1sinπ 2s

Γ(1 − s) (0.3) and has a functional equation. Furthermore it is random. A theorem illustrating this is the Voronin universality theorem (see e.g [8]). If f is a non-vanishing analytic function on D for a compact subset D ⊂ {s ∈ C : 1/2 < Re(s) < 1}, there exists a sequence Tk= Tk(D, f ) so that

lim

k→∞maxs∈D |ζ(s + iTk) − f (s)| = 0. (0.4) 1_{Introduced by Euler [6] and studied by Riemann [10].} _{General references: Ivi´}_{c [7] and}

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Introduction 14

The depth of the function comes from its simple definition and these three prop-erties. I particular The Euler product implies that the function contains all infor-mation about the primes, and therefore its study belongs in the realms of number theory. The most fundamental open problem about the Riemann zeta-function is the Riemann hypothesis2_{that says that ζ(s) = 0 implies that either Re(s) = 1/2 or}

s is an even negative integer. Of near equal importance are the weaker conjectures 1. The Lindel¨of hypothesis: |ζ(1/2 + it)| = O(|t|), ∀ > 0.

2. The density hypothesis: N (σ, T ) = O(T2(1−σ)+), ∀ > 0,

where N (σ, T ) denote the number of zeroes ρ with real part Re(ρ) ≥ σ, and imagi-nary part −T ≤ Im(ρ) ≤ T . It is obvious that the Riemann hypothesis implies the density hypothesis and with some complex analysis, it is possible to show that the Riemann hypothesis implies the Lindel¨of hypothesis, and the Lindel¨of hypothesis implies the density hypothesis. Many consequences of the Riemann hypothesis are in fact implied already by the density hypothesis.

Part I - The power sum method

In an attempt to prove the density hypothesis we were lead to study the Tur´an power sum method. The method allows us to obtain lower bounds for

max v=M (n),...,N (n) 1 + n X k=1 z_kv ,

and in the first papers on his method Tur´an stated some conjectures on these bounds that would indeed imply the density hypothesis. However we prove that Tur´an’s conjectures were in fact false. We also prove some other results in the field, in particular solving a problem of Erd˝os. Although it does not strictly use the power sum method, we have also included a disproof of a certain conjecture of Ramachandra. This conjecture would also have had implications on the Riemann zeta function if true. The Voronin universality theorem (0.4) is a crucial part of the disproof. All of the above together constitutes part I of our thesis.

Part II - Moments of the Hurwitz and Lerch zeta functions

A problem closely related to the Lindel¨of hypothesis is the moment problem, which consists in estimating Zk(T ) = Z T 0 ζ 1₂+ it 2k dt. (0.5)

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Introduction 15

The general conjecture is that

Zk(T ) ∼ ckT logk

2

T (0.6)

for all integers k ≥ 1. It should here be mentioned that by the method of Bala-subramanian and Ramachandra (see [9]), the lower bound

Zk(T ) ≥ dkT logk

2

T

for some dk > 0 has been proved. It is also known that the Lindel¨of conjecture is

equivalent to a weak form of the conjecture

Zk(T ) T1+ (0.7)

for all integers k ≥ 1, and not even the Riemann hypothesis seems to imply the strong form of the conjecture (0.6), which has so far only been proved for k = 1, 2. A way of trying to understand the Riemann zeta-function is to see how it relates to other zeta functions, such as the closely related Hurwitz zeta-function

ζ(s, x) =

∞

X

k=0

(k + x)−s, (Re(s) > 1)

and the Lerch zeta-function φ(x, y, s) =

∞

X

k=0

e(kx)(k + y)−s. (Re(s) > 1)

The Lindel¨of hypothesis is expected to hold also for the Hurwitz and Lerch zeta-functions. However the Riemann hypothesis is false. This is due to the fact that while the Hurwitz and Lerch zeta-functions admit a functional equation, they do not have an Euler product. In more generality the Riemann hypothesis is assumed for a class of zeta-functions with Euler product and functional equation (the Selberg class - see Selberg [11]). It is thus interesting to study what differs between the Riemann zeta function and the Hurwitz and Lerch zeta functions, in order to better understand the arithmetic nature of the Riemann zeta function. In particular, for the Hurwitz zeta function it is of interest to study the corresponding moments Z 1 0 Z T 0 ζ∗ 1₂+ it, x 2k dtdx, (0.8) Z 1 0 ζ∗ 1₂+ it, x 2k dx, (0.9)

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Introduction 16 and Z T 0 ζ∗ 1₂+ it, x 2k dt, (0.10)

where ζ∗(s, x) = ζ(s, x) − x−s. In Part II we show some new results on these problems.

Part III - A new summation formula on the full modular group and

Kloosterman sums

In the final part of our thesis we consider the following question. What is the appropriate generalization of the classical Poisson summation formula

∞ X n=−∞ f (n) = ∞ X n=−∞ ˆ f (n)

to the case of the full modular group? There are several generalizations in the literature. The pre-trace formula, or even the Selberg trace formula has been suggested. The pre-trace formula allows us to get an expansion of SO(2) bi-invariant functions f on SL(2, R) in terms of modular forms and Eisenstein series. The central result in Part III of the thesis is a proof of such a formula without assuming SO(2) bi-invariance. We will obtain a formula of the following type:

X ad−bc=1 c>0 fa b c d = “main terms”+ X m,n6=0 1 π Z ∞ −∞ σ2ir(|m|)σ2ir(|n|)F (r; m, n)dr |nm|ir|ζ(1 + 2ir)|2 + ∞ X k=1 θ(k) X j=1 X m,n6=0 ρj,k(m)ρj,k(n)F 1₂− ki; m, n+ ∞ X j=1 X m,n6=0 ρj(m)ρj(n)F (κj; m, n),

where F (r; m, n) is a certain integral transform of f , the ρj(n) denote the Fourier

coefficients for the Maass wave forms, and the ρj,k(n) denote Fourier coefficients

of holomorphic cusp forms of weight k. For an exact version of this formula we refer to Theorem 1 in “A summation formula on the full modular group”. Our method of proof will use the Kloosterman sums

S(m, n; c) = X h¯h≡1 (mod c) 0≤h,¯h<c e mh + n¯h c ,

the Bruhat decomposition, the classical Poisson summation formula and the Kuznetsov summation formula. We also consider the case of integer matrices of determinant

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Introduction 17

D, or in other words, the image of SL(2, Z) under Hecke operators TD. We prove

some related identities about Kloosterman sums. We show that the Kuznetsov summation formula as well as the Selberg trace formula and the Eichler-Selberg trace formula can be proved as consequences. We also indicate how these results are closely connected to the explicit formula given by Motohashi for the fourth power moment of the Riemann zeta function.

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BIBLIOGRAPHY

[1] J. Andersson, Mean value properties of the Hurwitz zeta-function, Math. Scand. 71 (1992), no. 2, 295–300.

[2] , On some power sum problems of Tur´an and Erd˝os, Acta Math. Hun-gar. 70 (1996), no. 4, 305–316.

[3] , Power sum methods and zeta-functions, Licentiate thesis, Stockholm University, 1998.

[4] , Disproof of some conjectures of K. Ramachandra, Hardy-Ramanujan J. 22 (1999), 2–7.

[5] E. Bombieri, Problems of the millenium: The Riemann hypothesis, http://www.maths.ex.ac.uk/˜mwatkins/zeta/riemann.pdf.

[6] L. Euler, Introductio in analysin infinitorum, Bousquet, Lausanne, 1748. [7] A. Ivi´c, The Riemann zeta-function, A Wiley-Interscience Publication, John

Wiley & Sons Inc., New York, 1985, The theory of the Riemann zeta-function with applications.

[8] A. Laurinˇcikas, Limit theorems for the Riemann zeta-function, Kluwer Aca-demic Publishers Group, Dordrecht, 1996.

[9] K. Ramachandra, On the mean-value and omega-theorems for the Riemann zeta-function, Tata Institute of Fundamental Research Lectures on Mathe-matics and Physics, vol. 85, Published for the Tata Institute of Fundamental Research, Bombay, 1995.

[10] B. Riemann, ¨Uer die Anzahl der Primzahlen unter einer gegebenen Gr¨osse, Monatsberichte der Berliner Akademie (November 1859), 671–680, also avail-able at http://www.emis.de/classics/Riemann/.

[11] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) (Salerno), Univ. Salerno, 1992, pp. 367–385.

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Bibliography 19

[12] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.

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INTRODUCTION

The density hypothesis and the power sum method

From the explicit formula

∞ X n=1 Λ(n)f (n) = Z ∞ 0 f (x)dx − X ζ(ρ)=0 Z ∞ 0 xρ−1f (x)dx, (I.11)

where Λ(n) denotes the Van-Mangholt function, it is easy to see that Z H 0 ∞ X n=1 Φ(n/T )Λ(n)n−1/2−it 2 dt Z 1 0 T2σ−1+(N (σ, 2H) + 1)dσ (I.12)

for a positive test-function Φ ∈ C₀∞(]0, ∞[) and log T log H, where N (σ, T ) denotes the number of zeroes of the Riemann zeta function with real part greater or equal to σ, and with imaginary part between −T and T . (Note: the Riemann hypothesis states that N (σ, T ) = 0, for all σ > 1/2.) Can we invert this formula? More precisely, can we find for some H = H(T ) so that log H ∼ log T and such that Z 1 1/2 T2σ−1N (σ, T /2)dσ T Z T 0 ∞ X n=1 Φ(n/H)Λ(n)n−1/2−it 2 dt. (I.13) If this were true we could use a standard mean value formula for Dirichlet-polynomials

Z T 0 X k≤T akkit−1/2 2 dt T X k<T |ak| 2 /k +X k<T |ak| 2 (I.14)

to obtain the estimate

N (σ, T ) = OT2(1−σ)+. (I.15) This result is called the density hypothesis and can be can be used as a substitute for the Riemann zeta function in a lot of applications, such as local prime number

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Introduction 24

theorems. Both the density hypothesis and the Riemann hypothesis imply π(x + h) − π(x) ∼ h

log x, (x

θ_{< h < x)} _(I.16)

for each θ > 1/2.

Turán introduced a new method in 1941 (see [10]) in an attempt to prove the density hypothesis with this idea. However he was never able to prove the density hypothesis. Instead he stated two conjectures that would imply the density hypothesis. The problems with inverting (I.12) comes from the fact that there is no way of knowing that locally the sum over the zeroes in the critical strip do not cancel each other. If we knew that the zeroes in the critical strip (but not on the critical line) were “isolated”, i.e. N (σ, T + 1) − N (σ, T ) = o(log T ), for σ > 1/2, we could have used Turán’s method. However this has been shown to be equivalent to the even stronger Lindelöf hypothesis (see Titchmarsh [9], Chapter 13). In our paper “Disproof of some power sum problems of P. Turán” we show that the two conjectures of Turán are false3_{. Furthermore it can be shown that it is a futile}

attempt to try to prove the density hypothesis just by these methods. One would need at least something like

inf λ∈Cn_,λ 1=0 max x∈[1,C] n X k=1 eλkx > e−o(n), (C > 1)

to be true, and we manage to prove that there is no such estimate.

Even though the density hypothesis was not proved by these methods, the Tur´an power sum theory allowed us to obtain non-trivial estimates (and the best in its time) for the density of the zeroes (and a θ < 1) in (I.16) before more modern zero-detecting methods were found. This could be done because (I.13) can be proved for positive constants c1, c2 and for each T and some H so that

c1log T ≤ log H ≤ c2log T . Even though the Tur´an theory currently does not give

as good estimates as the state-of the art zero-detecting methods, we suspect that it will eventually lead to the same-strength estimates (neither better nor worse).

The power sum method

Turán power sum theory has over the years found a number of other applications. In his book [11] P. Turán has 15 chapters for the theory and 40(!) chapters of applications. Turán himself thought of his method as his most important contri-bution to mathematics. His book also contains two chapters of open problems. In

3_{This result was presented at the Halberstam conference in analytic theory in Urbana}

Cham-paign, 1995 and a slightly different version of this paper also appeared in our Licentiate thesis [2].

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Introduction 25

the paper “On some power sum problem of Tur´an and Erd˝os” we solve some of them4_{. One of the problems we solve completely is the following one of P. Erd˝}_os.

Problem. Does there exist for each integer 1 ≤ m ≤ n − 1 a c = c(m) so that max 1≤v≤c(m)n,v integer |zv 1+ · · · + znv| mink=1,...,n|zk|v ≥ m? (I.17)

We show that in fact one can use c(m) = max(1021m, [(2π2m)m/2]). Another problem of P. Tur´an concerns that of estimating

inf |zi|≥1 max v=1,...,n2 n X k=1 zv_k . (I.18)

We also show that if n + 1 is prime then √n ≤ (I.18) ≤√n + 1, and that we have√n ≤ (I.18) for all integers n.

We would like to state here a similar open problem of Erd˝os-Newman (not power sum though) that Erd˝os did send to us in 1995.

Problem. Is it true that there is an absolute constant c > 0 such that for every choice of ak = ±1, 0 ≤ k ≤ n one has

max |z|=1 n X k=0 akzk > (1 + c)√n? (I.19)

For many similar open problems, see Montgomery [6]. None of the results in this paper have any applications on zeta-functions. (However we use some results from that theory, e.g. the distribution of consecutive primes.) It should be noted that the constants 23/84 and 65/42 in Proposition 2 can be improved to 0.2675 and 1.535 thanks to improvements on the difference between consecutive primes (Harman-Baker [4]).

Omega results for the Riemann zeta-function in short intervals

As well as considering order estimates of the Riemann zeta-function there is in-terest in finding omega-estimates (that is estimates from below) in short intervals. The method used here is the one by Ramachandra-Balasubramian (see Ramachan-dra’s book [7]), which gives omega-estimates for the 2k-th moment:

Z T +H T ζ 1₂+ it 2k dt ≥ ck(log H)k 2

, for H log log T. (I.20)

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Introduction 26

In a series of papers Ramachandra and Balasubramian generalized these results to a wide class of Dirichlet series, called the Titchmarsh series. In a paper in Asterisque [8], Ramachandra states some conjectures which essentially would al-low us to localize these results even further. Ramachandra-Balasubramian wrote another paper [5] which contained further conjectures. In our paper “Disproof of some conjectures of Ramachandra” we disprove these conjectures (2 different disproofs) as well as an even weaker conjecture given by Ramachandra in a private communication5. An excerpt from this paper which essentially disprove all these conjectures, is the following result.

Result. Suppose that H, > 0, 0 < δ < 1/2. Then there exist an N ,a1 = 1,

|ak| ≤ kδ−1 so that max t∈[0,H] N X n=1 annit <

We finally remark that this introduction is also taken in a somewhat modified form from our Licentiate thesis [2].

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BIBLIOGRAPHY

[1] J. Andersson, On some power sum problems of Tur´an and Erd˝os, Acta Math. Hungar. 70 (1996), no. 4, 305–316.

[2] , Power sum methods and zeta-functions, Licentiate thesis, Stockholm University, 1998.

[3] , Disproof of some conjectures of K. Ramachandra, Hardy-Ramanujan J. 22 (1999), 2–7.

[4] R. C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. (3) 72 (1996), no. 2, 261–280.

[5] R. Balasubramanian and K. Ramachandra, On Riemann zeta-function and allied questions. II, Hardy-Ramanujan J. 18 (1995), 10–22.

[6] H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Math-ematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994.

[7] K. Ramachandra, On Riemann zeta-function and allied questions, Astérisque (1992), no. 209, 57–72, Journées Arithmétiques, 1991 (Geneva).

[8] , On the mean-value and omega-theorems for the Riemann zeta-function, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 85, Published for the Tata Institute of Fundamental Re-search, Bombay, 1995.

[9] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.

[10] P. Tur´an, ¨Uber die Verteilung der Primzahlen. I, Acta Univ. Szeged. Sect. Sci. Math. 10 (1941), 81–104.

[11] , On a new method of analysis and its applications, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1984, With the assistance of

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Bibliography 28

G. Hal´asz and J. Pintz, With a foreword by Vera T. S´os, A Wiley-Interscience Publication.

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On some Power Sum problems of Tur´

an and

Erd˝

os

Johan Andersson

∗

1 Introduction

In this paper I will show some elementary inequalities, and e.g. solve a problem

given by Paul Erd˝

os. I will also consider some problems of Paul Tur´

an. Lemma

1 was given by Cassels [1], and its proof will follow his approach, although I will

use it in a more general setting. In this paper e(x) will denote e

2πix

_{, bxc will}

denote the integral part of x, and {x} will denote the distance between x and its

nearest integer. Furthermore v will always be integer valued.

2 Main theorem

Lemma 1. (Cassels) If z

k

are complex numbers we have

n k=1

z

v k

+ z

vk

, Clearly 2 Re(

P

n k=1

z

v

k

) = σ

v

. By the construction the a

0k

s

are real. Now suppose σ

v

< 0 for v = 1, . . . , 2n + 1. We now apply the Newton

identities

σ

v

+ a

1

σ

v−1

+ · · · + a

v−1

σ

1

+ va

v

= 0,

v = 1, 2, . . . , 2n

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∗_{This paper has been published in Acta Math. Hungar. 70 (1996), no. 4, 305–316.}

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and successively get a

1

> 0, a

2

> 0, . . . , a

2n

> 0 and with the relation σ

2n+1

+

a

1

σ

2n

+ · · · + a

2n

σ

1

= 0 we get σ

2n+1

> 0, and a contradiction, hence

max

v=1,...,2n+1

σ

v

≥ 0.

Theorem 1. If z

k

are complex numbers and 1 ≤ m ≤ n we have

max

v=1,...,2nm−m(m+1)+1

|

P

n k=1

z

v k

|

q

P

m k=1

|z

k

|

2v

≥ 1.

Proof. Put s

v

=

P

n_k=1

z

vk

, and s

a,b,v

=

P

b_k=a

z

kv

=

m

X

k=1

|z

k

|

2v

+ |s

m+1,n,v

|

2

+ 2 Re

s

1,m,v

s

m+1,n,v

+

X

1≤j<k≤m

z

vj

z

kv

!

=

m

X

k=1

|z

k

|

2v

+ |s

m+1,n,v

|

2

+ 2 Re B(v),

where B(v) is a pure power sum with nm −

m(m+1)₂

elements. i. e.

B(v) =

nm−m(m+1)₂

X

k=1

w

v k

.

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From Lemma 1 we have an integer v such that 1 ≤ v ≤ 2nm − m(m + 1) + 1,

such that Re(B(v)) ≥ 0, and for that v we have |s

≥

√

m.

Proof. This follows from the theorem and the inequality

v

u

t

m

X

k=1

|z

k

|

2v

≥

√

m min

k=1,...,n

|z

k

|

v

.

From the two extreme cases in corollary 1 (m = 1 and m = n) we get

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Corollary 2. If |z

k

| ≥ 1 then

max

v=1,...,2n−1

n

X

k=1

z

v k

≥ 1

which is the classic result by Cassels (see [1] or [8] section 3). We also get

Corollary 3. If |z

k

| ≥ 1 then

max

v=1,...,n2_−n+1

n

X

k=1

z

kv

≥

√

n

which is a rather interesting new result, which we will use in the following

sections. Note that the only condition is that |z

k

| ≥ 1. Under the assumption

|z

k

| = 1 similar results have been proved earlier by e.g. Cassels, Newman and

Szalay who even proved

max

1≤v≤bcnc

n

X

k=1

b

k

z

kv

≥

v

u

t

1 −

n − 1

bcnc

n

X

k=1

b

k

,

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when b

k

> 0 and c ≥ 1. (See Tur´

an [8] page 80, or Leenmann-Tijdeman [5] for a

weaker result.)

3 Problem 10 of Tur´

an

In his classic book [8] Tur´

an states a number of problems. One of them (Problem

10, page 190) is the following:

If min

j

|z

j

| = 1 then what is

inf

zj

max

1≤v≤n2_{,v integer}

n

X

k=1

z

kv

?

(4)

We see that Corollary 3 clearly gives a lower estimate for this expression. In

the following we will show that this estimate is rather strong, in the sense that

for infinitely many integers, e.g. the primes the upper estimate is very close. We

begin with two lemmas which give upper estimates for (4).

Lemma 2. (H. Montgomery) If p is prime then there exists numbers z

1

, . . . , z

p−1

such that |z

k

| = 1 and |

P

p−1 k=1

z

v k

| ≤

√

p for all integers 1 ≤ v ≤ p

2

_{− p − 1.}

Proof. Take z

k

= χ(k)e(k/p), χ a character mod p of order p − 1, see [8] page 83.

See Queffelec [6] for a fuller account.

(32)

Lemma 3. If q is a prime power then there exists numbers z

1

, . . . , z

q+1

such that

|z

k

| = 1 and |

P

q+1 k=1

z

v k

| ≤

√

q for all integers 1 ≤ v ≤ q

2

_{+ q.}

Proof. (Here I essentially follow Fabrykowski [3].) Suppose q is a prime power.

By the Singer theorem (see [7]), we can find numbers a

1

, . . . , a

q+1

, such that the

q(q + 1) numbers a

i

− a

j

, i 6= j form all nonzero residues modulo q

2

+ q + 1.

Consider z

k

= e(a

k

/(q

2

+ q + 1)). We get |

P

q+1 k=1

z

v k

|

2

= q + 1 +

P

q+1 k6=j

e((a

k

−

a

j

)v/(q

2

+ q + 1)) = q +

P

q2_+q k=0

e(kv/(q

2

_{+ q + 1)) = q +}

1−e(v)

1−e(v/(q2_+q+1))

= q for all

integers v such that 1 ≤ v ≤ q

2

_{+ q, and thus we even have |}

P

q+1

k=1

z

v k

| =

√

q for

those v.

It is now an easy task to state and prove the following proposition.

Proposition 1.

For n+1 prime we have:

√

n ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n+1

n

X

k=1

z

kv

≤

(i)

≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_+n−1

n

X

k=1

z

kv

≤

√

n + 1.

For n-1 prime power we have:

√

n − 2 ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n

n

X

k=1

z

kv

≤

(ii)

≤

√

n − 1 ≤

√

n ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n+1

n

X

k=1

z

vk

.

Proof. The inequality

√

n − 2 ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n

n

X

k=1

z

kv

in (ii) follows from Corollary 1 with m = n − 2. The inequality

√

n ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n+1

n

X

k=1

z

kv

in (i) and (ii) is exactly Corollary 3. The inequality

inf

zk∈C,|zk|≥1

max

v=1,...,n2_+n−1

n

X

k=1

z

kv

≤

√

n + 1

32

(33)

in (i) follows from Lemma 2. The inequality

inf

zk∈C,|zk|≥1

max

v=1,...,n2_−n

n

X

k=1

z

_kv

≤

√

n − 1

in (ii) follows from Lemma 3.

From Proposition 1 (i) we immediately obtain the best interval estimate so

far for (4) and infinitely many integers.

Corollary 4. If n + 1 is a prime we have

√

n ≤

inf

zk∈C,|zk|≥1

max

v=1,...,n2

n

X

k=1

z

v k

≤

√

n + 1.

For arbitrary n we can’t use the methods of Corollary 4 which requires

pri-mality. But since we know that the primes are rather densely distributed we can

use local prime density results from Iwaniec and Pintz (see [4]). First, however

we need a lemma.

Lemma 4. (Erd˝

os and R´

enyi) There exists numbers z

k

for k = 1, . . . , n such

that |z

k

| = 1 and max

v=1,...,m

|

P

n k=1

z

v

k

| ≤

p6n log(m + 1).

Proof. See Erd˝

os and R´

enyi [2].

Similar results with explicit z

k

’s have been proved by Leenmann-Tijdemann

(see [5] or [8] page 82).

Proposition 2.

inf

zk∈C,|zk|≥1

max

v=1,...,bn2_−n65/42_c

n

X

k=1

z

kv

=

√

n + O

n

23/84

p

log n

Proof. We know that there exists a prime p between n−1 and n−

1 2

n

23/42

_{, ∀n ≥ N}

0

by Iwaniec-Pintz’s theorem (see [4]). Choose z

1

, . . . , z

p+1

fulfilling the condition

in Lemma 3, and z

p+2

, . . . , z

n

k=p+2

z

v k

≤

√

p +

p

6(n − p − 1) log(p

2

_{+ p + 1) =}

=

√

_{+ p.}

In the other direction we put m = n − 2bn

65/84

_{c for n ≥ N}

0

, and use Corollary

1. max

v=1,...,2nm−m(m+1)+1

n

X

k=1

z

vk

≥

√

m =

q

n − 2bn

65/84

_{c =}

√

_{n + O(n}

23/84

)

We also get 2nm − m(m + 1) + 1 = 2n(n − 2bn

65/84

_{c) − (n − 2bn}

65/84

_{c)(n −}

2bn

65/84

_{c + 1) + 1 = n}

2

_{− 4bn}

65/84

_c

2

_{− n + 2bn}

65/84

_{c + 1 ≤ n}

2

_{− bn}

65/42

_c.

Queffelec [6] has showed the ≤ part of the proposition with essentially the

same method. See also Queffelec [6] for an application of these estimates in

functional analysis. Note that we still have not considered (4) for all n’s. In the

next section we will do that (and obtain much worse results than e.g. Corollary

4 or Proposition 2), but we will also generalize results in the previous sections.

4 A further result

With Lemma 4 of Erd˝

os and R´

enyi in mind we might consider the following

generalization of (4)

If min

j

|z

j

| = 1 then what is

inf

zj

max

v=1,...,n2m_{,v integer}

n

X

k=1

z

v_k

?

(5)

For general integers m first I prove an analogue of Corollary 3:

Theorem 2. If |z

k

| ≥ 1 then

max

v=1,...,n2m

n

X

k=1

z

kv

≥

n

m

m!

2

_2m1

.

Proof.

n

X

k=1

z

kv

2m

=

n

X

k=1

z

vk

!

m _n

X

k=1

z

v k

!

m

= A(v) + B(v) + B(v)

Where A(v) is a power sum consisting of only positive real elements.

Since

A(v) + B(v) + B(v) consists of exactly n

2m

_{elements and A(v) is non-empty}

we must have that B(v) have less than

1 2

(n

2m

_{− 1) elements. We now find a}

lower estimate of the number of terms of A(v). It is clear that for each m-tuple

(35)

k

1

, . . . , k

m

of distinct integers k

i

we have m!

2

number of terms contributing to

A(v), since each permutation on both sides occur. By elementary combinatorics

we know that we can pick

_mn

such sets of integers from 1, . . . , n. and since

|z

k

| ≥ 1 we have A(v) ≥

_mn

m!

2

, ∀v. We now apply Lemma 1 on B(v) and we

obtain the theorem.

We now consider a well known version of the Stirling formula, which we will

use

n! ≥

√

2πn

n+12

e

−n

(6)

see e.g. Abramowitz-Stegun, Handbook of mathematical functions, formula 6.1.38.

Proposition 3. If |z

k

| ≥ 1, and 1 ≤ m ≤ n then

max

v=1,...,n2m

n

X

k=1

z

v k

≥

p

(n − m)me

−1

Proof. By Theorem 2 we have

max

v=1,...,n2m

n

X

k=1

z

kv

≥

n

m

m!

2

_2m1

= (n(n − 1) · · · (n − m + 1)m!)

2m1

_≥

≥ ((n − m)

m

m!)

2m1

₌

√

_{n − m(m!)}

2m1

≥

≥ (According to (6)) ≥

p

(n − m)me

−1

₍

√

_2πm)

1 2m

≥

p

(n − m)me

−1

_.

We now have the means to obtain an interval estimate for (5)

Corollary 5. We have for 1 ≤ m ≤ n

r

e

−1

_{(1 −}

m

n

) ≤

1 √

mn

zk∈C,|z

inf

k|≥1

max

v=1,...,n2m

n

X

k=1

z

_kv

≤

p

12 log(n)

1 +

1 n

Proof. The first inequality is exactly Proposition 3. The second inequality is seen

by Lemma 4 and the fact that

p

log(n

2m

_{+ 1) =}

r

2m log n + log(1 +

1 n

2m

) ≤

r

2m log n +

1 n

2m

≤

p

2m log n(1 +

1 n

)

35

(36)

We note that the quotient between the upper and lower estimate is essentially

independent of m for small m’s.

5 A problem of Erd˝

os

In this section I will use Corollary 1, Proposition 3 and a classic result by Dirichlet

in Diophantine approximation to solve the following problem of Erd˝

os, given in

Tur´

an [8] as problem 47 on page 196.

Does there exist for each integer 1 ≤ m ≤ n − 1 a c = c(m) so that

max

1≤v≤c(m)n,v integer

|z

v 1

+ · · · + z

v n

|

min

k=1,...,n

|z

k

|

v

≥ m?

(7)

I will start by proving an essential lemma.

Lemma 5. If |z

k

| ≥ 1, and n ≥ m + 1 then

max

v=1,...,b“2π2n n−m ”n/2 c

n

X

k=1

z

kv

≥ m

Proof. We use the following version of Dirichlet’s theorem:

Given real numbers a

1

, a

2

, . . . , a

n

, and an A ≥ 2 then we can find

an integer k in the range 1 ≤ k ≤ A

n

such that {ka

i

} ≤

1 A

. (8)

See e.g. Tur´

an [8] section 15.

Now put A =

q

z

vk

!

= Re

n

X

k=1

e(vθ

k

)|z

k

|

v

!

=

n

X

k=1

cos(2π{vθ

k

})|z

k

|

v

≥ (By the relation 1 −

x

2

2 ≤ cos x) ≥

≥

n

X

k=1

1 −

1

2 (2π{vθ

k

})

2

|z

k

|

v

≥

n

X

k=1

1 −

1

2 2π

A

2

!

|z

k

|

v

≥ min

k=1,...,n

|z

k

|

v

n

1 −

n − m

n

≥ m.

36

(37)

We now state the main result in this section.

Proposition 4. If |z

k

| ≥ 1, we have for n ≥ m + 1 that

max

v=1,...,bmax

(

1021m_,(2π2_m)m/2

₎

_nc

n

X

k=1

z

_kv

≥ m

Proof. We will show this result in three cases.

Case 1: m + 1 ≤ n ≤ 32m

Proof. First we put f (x) = (

2π2_x

x−m

)

x/2

_{. From Lemma 5 we obtain that h(m) =}

1

m

max

n=m+1,...,32m

f (n) clearly is sufficient as c(m) in case 1. We get h(m) ≤

1

m

max

x∈[m+1,32m]

f (x). We now notice that f (x)’s only critical point for x >

m is a minimum point, hence the maximum must be obtained in the interval

end-points, and we get h(m) ≤

_m1

≤

≤ (2π

2

m)

m/2

, for e.g. m > 1000

Since clearly 10

21m

_{is the dominating term for m < 1000 this is enough. We}

have now proved the proposition in Case 1.

Case 2: 32m ≤ n ≤ m

2

Proof. We see that it only makes sense for m ≥ 32. Then we have by Proposition

3 that

max

v=1,...,n2bm/8c

n

X

k=1

z

kv

≥

p

(n − bm/8c)bm/8ce

−1

_≥

≥

r

31m

3

4

1

8 me

−1

_{≥ m}

r

3

31

32 e

−1

_{≥ m.}

Since n

2bm/8c

_{≤ m}

m₂

_{≤ (2π}

2

_m)

m/2

_{when n is in the above interval, we have proved}

the proposition in case 2.

Case 3: m

2

_{≤ n}

Proof. By Corollary 1 we get

max

v=1,...,2nm2

n

X

k=1

z

v k

≥ m

Which clearly is sufficient.

that will be asymptotically significant. It seems that this is essentially the

asymp-totically best possible estimate obtainable with these methods, although for small

m it is easy to obtain better estimates.

6 On Tur´

an’s problem 17 and 18

I will finally present a short solution to two further power sum problems from

Tur´

an’s book [8]. Problem 17 page 191 is the following:

Show that for arbitrarily small ε > 0, there is an m

0

(ε, n), such that for every

integer m > m

0

(ε, n) there exists a system (z

1∗

, . . . , z

n∗

) with max

j

|z

j∗

| = 1 for

which the inequality

max

m+1≤v≤m+n,v integer

n

X

k=1

z

_k∗v

≤ ε

n

(9)

holds. In problem 18 he asks if problem 17 also holds if max

j

|z

j∗

| = 1 is replaced

by min

j

|z

j∗

| = 1. I will now show a proposition which implies that there in fact

exists an m

0

(ε, n) such that the statements in both problems are true.

Proposition 5. For each m ≥ 4πn

2

_ε

−n

_{there exists complex numbers z}

k

, with

|z

k

| = 1 such that

max

₎

k n

≤

≤ (By the relation |e

ix

_{− 1| ≤ 2x) ≤ n}

4π

m + n

m

− 1

≤

4πn

2

m

≤ ε

n

References

[1] J. W. S. Cassels, On the sums of powers of complex numbers, Acta Math.

Acad. Sci. Hungar. 7 (1956), 283–289.

[2] P. Erd¨

os and A. R´

enyi, A probabilistic approach to problems of Diophantine

approximation, Illinois J. Math. 1 (1957), 303–315.

[3] J. Fabrykowski, A note on sums of powers of complex numbers, Acta Math.

Hungar. 62 (1993), no. 3-4, 209–210.

[4] H. Iwaniec and J. Pintz, Primes in short intervals, Monatsh. Math. 98 (1984),

no. 2, 115–143.

[5] H. Leenman and R. Tijdeman, Bounds for the maximum modulus of the first

k power sums, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36

(1974), 387–391.

[6] H. Queff´

elec, Sur un th´

eor`

eme de Gluskin-Meyer-Pajor, C. R. Acad. Sci. Paris

S´

er. I Math. 317 (1993), no. 2, 155–158.

[7] J. Singer, A theorem in finite projective geometry and some applications to

number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.

[8] P. Tur´

an, On a new method of analysis and its applications, Pure and Applied

Mathematics, John Wiley & Sons Inc., New York, 1984, With the assistance

of G. Hal´

asz and J. Pintz.

(40)

(41)

Disproof of some conjectures of P. Tur´

an

Johan Andersson

∗

1 Introduction

In his first paper on power sum theory [4], P. Tur´

an stated a power sum conjecture

which implies the density hypothesis of the Riemann zeta-function. In this paper

we will show that a considerably weaker statement is still false. That is if

Claim. For some C > 1 one has that

inf

λ∈Cn_,λ₁₌₀_x∈[1,C]

max

n

X

k=1

e

λkx

> e

−o(n)

.

(1)

Then

Theorem. Claim is false.

2 Some conjectures

We first state the two original conjectures of Tur`

an [4]. It should be noted that

Hal´

asz in a note in [6], (page 2083) wrote that the conjectures were probably

false, but that no disproof had been found.

Conjecture 1. If ω(n) is a positive increasing function with lim

n→∞

ω(n) = ∞

and for some n ≥ C

1

there exists a M = M (n) such that the inequality

nω(n) ≤ M (n) ≤

n

2

ω(n)

2

,

then

max

M

₍

1− 1 ω(n)

)

≤v≤M v integer

|z

v 1

+ · · · + z

v n

| > e

− n2 M ω(n)

∗_{This result was first presented at the Halberstam conference in analytic number theory in}

Urbana-Champaign, 1995.

(42)

for all systems (z

1

, . . . , z

n

) satisfying the conditions

+ · · · + z

v k

| > e

−n0.09

.

In [4] Tur`

an shows that Conjecture 2 implies the density hypothesis for the

Riemann zeta-function. If we use a modification of his method it is possible

to show that the conjectures can be weakened. However we would still need

something like the existence of a C > 1 so that Claim is true to prove the density

hypothesis by this approach. We have

Proposition 1. Conjecture 1 and Conjecture 2 are false.

Proof. The falsity of the conjectures follows from Theorem and the fact that both

conjectures implies Claim. To prove the first two implications, pick a λ ∈ C

n

_{. We}

can assume that max

k

Re(λ

k

) = 0, since otherwise if j is such that max

k

Re(λ

k

) =

Re(λ

j

), then consider ˆ

λ

k

= λ

k

− λ

j

, with ˆ

λ

1

and ˆ

λ

j

interchanged.

1. Conjecture 1 =⇒ Claim.

Choose z

k

= e

λk/(M (1−ω(n)1 ))

_{. First divide {z}

k

}

nk=1

into two disjoint sets

i=1

is exactly the set that fulfills (6).

Now consider ˆ

z ∈ C

n

_{, where ˆ}

_z

))

M

,

max

M

₍

1− 1 ω(n)

)

≤v≤M v integer

|ˆ

z

v1

+ · · · + ˆ

x∈[1,1/(1−1/ω(n))]

n

X

k=1

e

λkx

,

max

x∈[1,C]

n

X

k=1

e

λkx

,

for some C > 0.

42

(43)

2. Conjecture 2 =⇒ Claim.

Choose z

k

= e

λk/(n 3/2_(1−n−0.42₎₎

. We obtain

e

−o(n)

e

−n0.09

,

max

n3/2_(1−n−0.42_)≤v≤n3/2

|z

v 1

+ · · · + z

v k

|,

≤

max

x∈[1,1/(1−n−0.42_)]

n

X

k=1

e

λkx

,

max

x∈[1,C]

n

X

k=1

e

λkx

,

for some C > 0.

3 Proof of theorem

First we remark that in the next section we will only use the principal part of

the logarithm, and α

β

_{will mean e}

β log α

_{. We have the following lemma}

Lemma. One has for α and β > 1 real numbers that

∞

X

k=−∞

(1 + αki)

−β

=

2π

αΓ(β)

∞

X

k=1

2πk

α

β−1

e

−2πk/α

.

Proof. By using the Poisson summation formula

∞

X

k=−∞

Z

∞ −∞

f (x)e

−ikx

dx = 2π

∞

X

k=−∞

f (2πk)

on

f (x) =

(

x

β−1

_e

−x/α

_,

_{when x > 0,}

0,

when x ≤ 0;

we get for β > 1 the identity

∞

X

k=−∞

Γ(β)(1/α + ik)

−β

= 2π

∞

X

k=1

(2πk)

β−1

e

−2πk/α

.

By dividing both sides with Γ(β)α

β

_{we obtain the lemma.}

(44)

Note that the Lemma is a special case of the functional equation for the

X

k=−n

1 +

ki

n

−nx

= A + B,

(3)

where

A =

X

|k|>n

1 +

ki

n

−nx

,

and

B =

2πn

Γ(nx)

∞

X

k=0

(2πkn)

nx−1

e

−2πkn

.

It is easy to see that

A n2

−na/2

(4)

for x ≥ a. The dominating term will be the first, which will have absolute value

2

−x/2

_{. To estimate B we use the Stirling formula}

p

n/x

−nb

,

(5)

44

(45)

for x ≤ b, since the terms tends to zero fast enough. From equations (3), (4) and

(5), we see that we can choose

C(a, b) = min

b

log

b

2π

+ 1 −

2π

b

,

a log 2

2 − ,

(6)

which is a constant > 0 for some > 0, since the function

f (x) = log

n

|. This result, which is a refined version of Tur´

an’s

second main theorem (see Tur´

an [5](with constant 4e instead of 8e), or

Kolesnik-Straus [1]), seems more powerful than ever when we consider the much more

special case when b

k

= 1 and the maximum with respect to v is considered for

v in the interval [m, n + m] instead of just the integers m, . . . , n + m, for e.g.

n = m and we still obtain results of decreasing exponential order). It should also

be noted that already Makai [2] showed that 4e is in general the best possible

constant in (7). Even though the situation might seem disappointing, note that

conjectures 1 to 2 and the Claim deal with conditions on the maximum norm,

max

k

|z

k

| = 1. Claim should be true for the minimum norm (Re(λ

k

) > 0) and it

is quite likely that we have

(46)

Conjecture 3. If C > 1 one has that

inf

λ∈Cn_,Re(λ_k_)≥0 _x∈[1,C]

max

n

X

k=1

e

λkx

> e

−o(n)

.

(8)

It is possible to show that the corresponding non-pure power sum problem

(i.e. when b

k

= 1 do not need to be true), then the analogue of Conjecture 3 is

false. However in the pure power sum case this seems likely (to us). It would not

(to our knowledge) have any consequences to the theory of the Riemann

zeta-function. Nevertheless it would be of interest to investigate it further. Another

problem is to see to what extent Claim fails. That is

Problem. Given C > 1. Find the smallest constant B = B(C) so that

inf

λ∈Cn_,λ₁₌₀ _x∈[1,C]

max

n

X

k=1

e

λkx

e

−(B+)n

,

(9)

for all > 0.

From (7) it is easy to see that B ≤ log(4eC/(C − 1)). We also see that

equation (6) can give us a lower bound for B.

References

[1] G. Kolesnik and E. G. Straus, On the sum of powers of complex numbers,

Studies in pure mathematics - To the memory of Paul Tur´

an, Birkh¨

auser,

Basel-Boston, Mass., 1983, pp. 427–442.

[2] E. Makai, On a minimum problem II, Acta. Math. Hung. 15 (1964), no. 1–2,

63–66.

[3] F. Oberhettinger, Note on the Lerch zeta-function, Pacific J. Math. 6 (1956),

117–120.

[4] P. Tur´

an, ¨

Uber die Verteilung der Primzahlen (1), Acta. Sci. Math. Szeged

10 (1941), 81–104.

[5]

, On a new method of analysis and its applications, John Wiley &

Sons, New York, 1984.

[6]

, Collected papers of Paul Tur´

an, Akad´

emiai Kiad´

o (Publishing House

of the Hungarian Academy of Sciences), Budapest, 1990.

(47)

Disproof of some conjectures of K. Ramachandra

Johan Andersson

∗

1 Introduction

In a recent paper [9] K. Ramachandra states some conjectures, and gives

conse-quences in the theory of the Riemann zeta function. In this paper we will will

present two different disproofs of them. The first will be an elementary

appli-cation of the Sz´

asz-M¨

untz theorem. The second will depend on a version of the

Voronin universality theorem, and is also slightly stronger in the sense that it

disproves a weaker conjecture. An elementary (but more complicated) disproof

has been given by Rusza-Lazkovich [11].

2 Disproof of some conjectures

2.1 The conjectures

We will first state the three conjectures as given by Ramachandra [9], and

Ramachandra-Balasubramian [5]:

Conjecture 1. For all N ≥ H ≥ 1000 and all N −tuples a

1

= 1, a

2

, . . . , a

N

of

complex numbers we have

1 H

Z

H 0

N

X

n=1

a

n

it

dt ≥ 10

−1000

.

Conjecture 2. For all N ≥ H ≥ 1000 and all N −tuples a

1

= 1, a

2

, . . . , a

N

of

complex numbers we have when M = H(log H)

−2

that

1 H

Z

H 0

N

X

n=1

a

n

it

2

dt ≥ (log H)

−1000 M

X

n=1

|a

n

|

2

.

∗_{This paper has been published in Hardy-Ramanujan J. 22 (1999), 2–7.}

(48)

Conjecture 3. There exist a constant c > 0 such that

Z

T 0

N

X

n=1

a

n

it

2

dt ≥ c

X

n≤cT

|a

n

|

2

Summation formulae and zeta functions

Johan Andersson

Summation formulae and zeta functions

Johan Andersson

Summation formulae and zeta functions

ABSTRACT

PREFACE

ACKNOWLEDGMENTS

CONTENTS

INTRODUCTION

The Riemann zeta-function

Part I - The power sum method

Part II - Moments of the Hurwitz and Lerch zeta functions

Part III - A new summation formula on the full modular group and

Kloosterman sums

BIBLIOGRAPHY

INTRODUCTION

The density hypothesis and the power sum method

The power sum method

BIBLIOGRAPHY

On some Power Sum problems of Tur´

an and

Erd˝

os

Johan Andersson

1

Introduction

In this paper I will show some elementary inequalities, and e.g. solve a problem

given by Paul Erd˝

os. I will also consider some problems of Paul Tur´

an. Lemma

1 was given by Cassels [1], and its proof will follow his approach, although I will

use it in a more general setting. In this paper e(x) will denote e

, bxc will

denote the integral part of x, and {x} will denote the distance between x and its

nearest integer. Furthermore v will always be integer valued.

2

Main theorem

Lemma 1. (Cassels) If z

are complex numbers we have

max

Re

X

z

!

≥ 0.

Proof. Let a

, k = 1, . . . , 2n be defined by

Y

(z − z

)(z − z

) =

X

a

z

and σ

=

P

z

+ z

, Clearly 2 Re(

P

z

) = σ

. By the construction the a

s

are real. Now suppose σ

< 0 for v = 1, . . . , 2n + 1. We now apply the Newton

identities

σ

+ a

σ

+ · · · + a

σ

+ va

= 0,

v = 1, 2, . . . , 2n

(1)

and successively get a

> 0, a

_{, bxc will}