Growth of the Lerch zeta-function

It is believed that the Lindelöf hypothesis is also true for the Lerch zeta-function. Here we present results supporting this conjecture. We first consider the growth of the Lerch zeta-function assuming the generalized Lindelöf hypothesis for Dirichlet L-functions. We next prove that Huxleys exponent 32/205 in the Lindelöf hypothesis for the Riemann zeta-function holds also for the Lerch zeta-function.__________Partially supported by a grant from the Lithuanian State Science and Studies Foundation.Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 45–56, January–March, 2005.     

On the mod-Gaussian convergence of a sum over primes

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates ${\mathrm{Im }}\log \zeta (1/2+it)$ . This Dirichlet polynomial is sufficiently long to deduce Selberg’s central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that ${\mathrm{Im }}\log \zeta (1/2+it)$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.     

Upper bounds for discrete moments of the derivatives of the riemann zeta-function on the critical line*

Assuming the Riemann hypothesis, we establish upper bounds for discrete moments of the Riemann zeta-function and its derivatives on the critical line. Moreover, we express continuous moments of the Riemann zeta-function and its derivatives in terms of these discrete moments. This allows us to give conditional upper bounds for $ {\int_0^T {\left| {{\zeta^{(l)}}\left( {{{1} \left/ {2} \right.} + {\text{i}}t} \right)} \right|}^{2k}}{\text{d}}t $ , where l and k are nonnegative integers.     

A note on S(t) and the zeros of the Riemann zeta-function

Let S(t) denote the argument of the Riemann zeta-function atthe point 1/2 + it. Assuming the Riemann hypothesis, we sharpenthe constant in the best currently known bounds for S(t) andfor the change of S(t) in intervals. We then deduce estimatesfor the largest multiplicity of a zero of the zeta-function,and for the largest gap between the zeros.     

On Balazard,Saias, and Yor’s equivalence to the Riemann Hypothesis

Balazard, Saias, and Yor proved that the Riemann Hypothesis is equivalent to a certain weighted integral of the logarithm of the Riemann zeta-function along the critical line equaling zero. Assuming the Riemann Hypothesis, we investigate the rate at which a truncated version of this integral tends to zero, answering a question of Borwein, Bradley, and Crandall and disproving a conjecture of the same authors. A simple modification of our techniques gives a new proof of a classical Omega theorem for the function S(t)$S\left(t\right)$ in the theory of the Riemann zeta-function.     

A new estimate of the Montgomery constant in the theory of zeros distribution of the Riemann zeta-function in the critical stripe

The new value α = 91/116 is obtained for the constant α such that for all σ ≥ α the “density hypothesis” is valid for the Riemann zeta-function in the critical stripe. The previous value α = 11/14 was obtained by M. Jutila in 1977.     

The density of zeros of the Riemann zeta-function in the critical strip

A new proof of Ingam’s theorem on the density of zeros of the Riemann zeta-function in the critical strip is given basing on an idea of H. Bohr and F. Carlson. Multiplication of segments of the Dirichlet series for the functions ζ(s) and 1/ζ(s) is used, which permits to simplify the proof.     

Zeros of the Riemann zeta-function

A proof that the Riemann zeta-function (+ it) has no zeros in the region where R=9.65 and T=12.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 419–429, October, 1970.     

A generalization of the riemann zeta-function

A generalization of the Riemann zeta-function which has the form

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.     

Convolution of the Fourier coefficients of the Eisenstein-Maass series

The convolution is defined as the sum where for n≠0 and W₀,W₁ are arbitrary smooth functions. Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms. The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function.     

Simple Zeros of the Riemann Zeta-Function

Assuming the Riemann Hypothesis, Montgomery showed by meansof his pair correlation method that at least two-thirds of thezeros of Riemann's zeta-function are simple. Later he and Taylorimproved this to 67.25 percent and, more recently, Cheer andGoldston increased the percentage to 67.2753. Here we proveby a new method that if the Riemann and Generalized LindelöofHypotheses hold, then at least 70.3704 percent of the zerosare simple and at least 84.5679 percent are distinct. Our methoduses mean value estimates for various functions defined by Dirichletseries sampled at the zeros of the Riemann zeta-function. 1991Mathematics Subject Classification: 11M26.     

On the number of partitions into primes

There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in and is readily seen to be monotonic. Research supported by NSA grant, no. MDA904-03-1-0082.     

Equivalents of the Riemann hypothesis

We obtain necessary and sufficient conditions for the Riemann hypothesis for the Riemann zeta-function, in terms of the functional distribution of quadratic Dirichlet L-functions. Received: 29 November 2004     

Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument. The construction of the corresponding Lagrangians L starts with the exact Lagrangian for the effective field of the p-adic tachyon string, which is generalized by replacing p with an arbitrary natural number n and then summing over all n. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself. Dedicated to Vasilii Sergeevich Vladimirov on his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 364–372, December, 2008.     

The Riemann zeta-function and moment conjectures from Random Matrix Theory

On the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line,

关于ζ函数的积分表示

由Riemannζ函数的函数方程得到Hurwitzζ函数的Hermite公式,再从Hermite公式得到Γ(s)的Binet′s第二表达式,从而由ζ函数推得Γ(s)的性质.     

On the zeta-function of systems of nonlinear equations

We introduce the concept of zeta-function for a system of meromorphic functions f = (f ₁,..., f _n) in ?ⁿ. Using residue theory, we give an integral representation for the zeta-function which enables us to construct an analytic continuation of the zeta-function.     

On a conjecture of Shanks

The conjecture in question concerns the function ?n related to the distribution of the zeroes of the Riemann zeta-function, γn, over the Gram points gn. It is the purpose of this article to show that for any α>0 the sum and this was conjectured, on numerical evidence, by Shanks (1961) [7] to be true for .     

On the Pair Correlation of Zeros of the Riemann Zeta-Function

To study the distribution of pairs of zeros of the Riemann zeta-function,Montgomery introduced the function where is real and T 2, and ' denote the imaginary parts ofzeros of the Riemann zeta-function, and w(u) = 4/(4 + u²). Assumingthe Riemann Hypothesis, Montgomery proved an asymptotic formulafor F() when || 1, and made the conjecture that F() = 1 + o(1)as T for any bounded with || 1. In this paper we use anapproximation for the prime indicator function together witha new mean value theorem for long Dirichlet polynomials andtails of Dirichlet series to prove that, assuming the GeneralizedRiemann Hypothesis for all Dirichlet L-functions, then for any > 0 we have uniformlyfor and all T T₀(). 1991Mathematics Subject Classification: primary 11M26; secondary11P32.