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.     

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.     

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.     

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.     

Universality of the Riemann zeta-function

In 1975, S.M. Voronin proved the universality of the Riemann zeta-function ζ(s). This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts ζ(s+iτ). In the paper, we consider the functions F(ζ(s)) which are universal in the Voronin sense.     

Simple zeros of the Riemann zeta-function

By estimating the change in argument of a certain function it has been shown that at least 0.3474 of the nonreal zeros of ζ(s) are simple. It is shown here that a more general function containing a real parameter can be used. An optimal choice of which gives a proportion greater than 0.3532.     

On the number of zeros of certain rational harmonic functions

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function , where is a rational function of degree 1$">, has no more than complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an -point gravitational lens.

     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

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

Shifted fourth moment of the Riemann zeta-function

For the Riemann zeta-function we present an asymptotic formula of a shifted fourth moment in an unbounded shift range along the critical line.     

On the limit measure for the Laplace transform of the Riemann zeta-function

We identify the limit measures in limit theorems in the space of analytic functions and on the complex plane for the Laplace transform of the square of the Riemann zeta-function. Received: 22 December 2006