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 gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.     

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.     

On universal relatives of the Riemann zeta-function

The Riemann zeta-function ζ has the following well-known properties (M) It is meromorphic in ℂ with a simple pole at z = 1 with residue 1.     

Zeros of derivatives of Riemann's xi-function of the critical line. II

Explicit lower bounds for the proportion of zeros of the derivatives of Riemann's xi-function which are on the critical line and simple are given. These lead to upper bounds for the proportion of zeros of the Riemann zeta-function with given multiplicity.     

On the instability of the Riemann hypothesis for curves over finite fields

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required to understand the theorems and their proofs, for it is known that the zeta-functions of curves over finite fields are very explicit meromorphic functions. We study the approximation properties of these meromorphic functions.     

A discrete version of the Mishou theorem

H. Mishou proved that the Riemann zeta-function and Hurwitz zeta-function with transcendental parameter are jointly universal, i.e., their shifts (continuous) approximate any pair of analytic functions. In the paper, a discrete version of the Mishou theorem is presented. In this case, the parameter of the Hurwitz zeta-function and the step of discrete shifts are connected by a certain independence relation.     

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.     

On the Mishou Theorem with an Algebraic Parameter

Siberian Mathematical Journal - The Riemann zeta-function and the Hurwitz zeta-function with transcendental or rational parameter are universal in the sense of Voronin: their shifts approximate...     

ON RITZ METHOD OF ANINTEGRO-DIFFERENTIAL EQUAITON

This paper deals with the Ritz method of an integro-differential equation related with Riemann zeta-function.     

Partitioning Infinite-Dimensional Spaces for Generalized Riemann Integration

To form Riemann sums for generalized Riemann integrals, thedomain of integration must be partitioned in a suitable manner.The existence of the required partitions is usually proved bya simple method of repeated bisection of the domain of integration.However, when the domain is the Cartesian product of infinitelymany copies of the set of real numbers, this simple method ofproof has frequently failed. A proof which works for infinite-dimensionalspaces is provided here. 2000 Mathematics Subject Classification28C20.     

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.     

On some averages at the zeros of the derivatives of the Riemann zeta-function

In this article we study two problems raised by a work of Conrey and Ghosh from 1989. Let ζ(k)(s) be the k-th derivative of the Riemann zeta-function, and χ(s) be factor in the functional equation of the Riemann zeta-function. We calculate the average values of ζ(j) and χ at the nontrivial zeros of ζ(k).     

Extreme values of zeta and <Emphasis Type="Italic">L</Emphasis>-functions

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions. The author is partially supported by the National Science Foundation (DMS 0500711) and the American Institute of Mathematics (AIM).     

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.     

Disproof of some conjectures of P. Turán

We disprove some power sum conjectures of Turán that would have implied the density hypothesis of the Riemann zeta-function if true. This result was first presented at the Halberstam conference in analytic number theory in Urbana-Champaign, 1995.     

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     

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.     

A central limit theorem for coefficients of the modified Borwein method for the calculation of the Riemann zeta-function

Lithuanian Mathematical Journal - The paper extends the study of the modified Borwein method for the calculation of the Riemann zeta-function. We present an alternative perspective on the proof of...     

Application of bounds for trigonometric sums to the problem of divisors and the mean value of the Riemann zeta-function

In this paper we give a new bound for the Riemann zeta-function in the neighborhood of the straight line =1 and indicate its application to the problem of divisors and the mean value of the Riemann zeta-function.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 539–548, November, 1972.The author is grateful to A. F. Lavrik for formulating the problem and for indicating the work of Richert.