An explicit formula for the square of the Riemann zeta-function in the critical strip

In this article, we prove an explicit formula for |ζ(σ + iT)|², where ζ(s) is the Riemann zeta-function and 1/2 < σ < 1, which is an analogue of Jutila’s formula. Our proof differs from that of Jutila. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 381–398, July–September, 2007.     

Short series over simple zeros of the Riemann zeta-function

Recently, Garaev showed that the series Σ_?∥?ζ¹(?)∥⁻¹ diverges, where the sum is taken over the simple zeros ? = β + iγ of the Riemann zeta-function ζ(s). More precisely, he proved . Using a mean-value estimate due to Ramachandra and some result on the distribution of simple zeros in short intervals on the critical line, we prove for T^0.552 ≤ H ≤ T. This leads to a slight improvement of Garaev's result in replacing his lower bound by .     

关于Smarandache函数与Riemann zeta-函数

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即S(n)=min{m∶m ∈N,n|m!).本文的主要目的是利用初等方法研究一类包含S(n)的Dirichlet级数与Riemann zeta-函数之间的关系,并得到了一个有趣的恒等式.     

On simple zeros of the Reimann zeta-function in short intervals on the critical line

We calculate in a new way (following old ideas of Atkinson and new ideas of Jutila and Motohashi) the mean square of the product of a function F(s), involving the Riemann zeta-function ζ(s), and a certain Dirichlet polynomial A(s) of length M=T^θ in short intervals on σ=a near the critical line: if θ<3/8, then

Large gaps between consecutive zeros of the Riemann zeta-function

Combining the amplifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.     

Formulas for the Riemann zeta-function and certain Dirichlet series

Using known theta identities and formulas of S. Ramanujan and G. Hardy among others we prove several formulas for the Riemann zeta-function and two Dirichlet series.     

A remark on negative moments of the Riemann zeta-function

With the use of the boundedness condition, the asymptotics of negative moments of the normalized Riemann zeta-function is obtained. Research supported by the Lithuanian State Science and Studies Foundation. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 28–35, January–March, 2000. Translated by A. Laurinčikas     

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).     

Gaps between consecutive zeros of the Riemann zeta-function on the critical line

LetR denote the number of gaps of length at leastV between consecutive zeros of the function ζ(1/2+i t) in the interval [0,T]. It is proved that $$R<< TV^{ - 2} \min (\log T, V^{ - 1} \log ^5 T).$$ The same problem is also discussed for Dirichlet series associated with cusp forms.     

Limit theorem for the Riemann zeta-function on the critical line. III

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 83–89, January–March, 1989.     

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,

On the mean square formula for the Riemann zeta-function on the critical line

Ifu _n denotes thenth zero of the function ,Ivi has shown thatu _n+1 –u _n u _n ^1/2 for alln andu _n+1 –u _n u _n ^1/2 (log u_n)^–5for infinitely manyn. We sharpen his lower estimate for the gapu _n+1 –u _n o the best possible, namely,u _n+1 –u _n u _n ^1/2 for infinitely manyn.The author wishes to thank Dr. Kai-Man Tsang for his continual guidance.     

A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function

The main purpose of this paper is using the mean value theorem of Dirichlet L-function and the estimates for character sums to study the asymptotic properties of a hybrid mean value of Kloosterman sums with the weight of Hurwitz zeta-function and the Cochrane sums, and give an interesting mean value formula for it.     

On the divisor function and the Riemann zeta-function in short intervals

We obtain, for T ^ε ≤U=U(T)≤T ^1/2−ε , asymptotic formulas for

On the twelfth power moment of the Riemann zeta-function near the critical line

This research was partially supported by the International Science Foundation.