Mean-value of the Riemann zeta-function on the critical line

This is an expository article. It is a collection of some important results on the meanvalue of      

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.     

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

This research was partially supported by the International Science Foundation.     

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.     

On the universality of the Riemann zeta-function

The research has been partially supported by Grant N LAC000 from the International Science Foundation.     

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.     

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.     

Mean absolute value of the Riemann zeta-function in a critical stripe

An asymptotic formula for the mean absolute value of the Riemann zeta-function in a critical stripe is obtained in the paper.     

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.     

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.     

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