Universality of Various Zeta-Functions

This is a survey of results on joint universality in Voroninʼs sense of various zeta-functions, when in the collection of these functions some of them have the Euler product and the others have not.     

On Zeta-Functions of Finite Abelian Groups

In the paper, the zeta-functions of finite Abelian groups of rank 2 and 3 are considered. One- and two-dimensional limit theorems for these zeta-functions on the complex plane and in the space of meromorphic functions are obtained.     

Twists of Lerch Zeta-Functions

We study twists of the Lerch zeta-functions with shifted Dirichlet characters. This generalizes Dirichlet L-functions. These twists turn out to be meromorphic with at most one simple pole and satisfy a functional equation (similar to the functional equation of the Lerch zeta-function) if the Dirichlet character is primitive. We also investigate the distributions of their zeros. In particular, we disprove the analogue of Riemann's hypothesis for some twists.     

Value Distribution of General Dirichlet Series. V

We prove a joint universality theorem for general Dirichlet series. The conditions of this theorem are weaker than those in [7].     

On joint universality of periodic Hurwitz zeta-functions

We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.     

Joint value-distribution theorems on Lerch zeta-functions. II

We give corrected statements of some theorems from [5] and [6] on joint value-distribution of Lerch zeta-functions (limit theorems, universality, functional independence). We also present a new direct proof of a joint limit theorem in the space of analytic functions and an extension of a joint universality theorem. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 332–350, July–September, 2006.     

On the effectivization of the universality theorem for the Lerch zeta-function

In the paper one way of the effectivization of the universality theorem for the Lerch zeta-function is discussed. It is based on the estimate of the rate of convergence in a limit theorem in the space of analytic functions. Partially supported by the Lithuanian State Science and Studies Foundation. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 172–178, April–June, 2000. Translated by A. Laurinčikas     

On a certain class of dirichlet series

For 0 1, let L(s,a) and L?(s,a) be the Dirichlet series L(s,a) = ∑ : cos (2πna)n^-s and L? [001](s,a) = ∑ sin (2πna)n^-s. We show that L(s,a) and L?(s,a[001]) have holomorphic extension in the whole complex plane. Values of L(s,a)andL?(s,a) at the negative integers are given. Moreover values of L?(s,a) at the intergers 0,2,4,... and values of L^?(s,a)at the integers 1,3,5,... are obtained. An exponential sums of certain recursion formulas are obtained by means of bernoulli numbers and Bernoulli polynomials     

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 Joint Limit Theorem for the Riemann Zeta-Function in the Space of Analytic Functions

In this paper, we prove a joint limit theorem for the Riemann zeta-function in the space of analytic functions in the sense of weak convergence of probability measures.     

On Statistical Properties of the Lerch Zeta-Function. II

A discrete limit theorem for the Lerch zeta-function with an integer parameter in the space of meromorphic functions is proved.     

A Joint Limit Theorem for General Dirichlet Series

In this paper, a joint limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for general Dirichlet series is proved. The explicit form of the limit measure is given.     

On a Joint Limit Distribution of the Riemann Zeta-function in the Space of Analytic Functions

In the paper, the explicit form of the limit measure in a joint limit theorem for the Riemann zeta-function in the space of analytic functions is given.     

A Discrete Limit Theorem for the Matsumoto Zeta-Function on the Complex Plane

In the paper, a discrete limit theorem in the sense of the weak convergence of probability measures for the Matsumoto zeta-function on the complex plane is proved.     

Fractional Moments and Zeros of ζ(s) on the Critical Line

A connection is established between the fractional moments of the Riemann zeta-function and the number of its zeros on the critical line.     

Joint discrete limit theorems in the space of analytic functions for general Dirichlet series

We prove joint discrete limit theorems in the sense of weak convergence of probability measures in the space of analytic functions for general Dirichlet series.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 104–116, January–March, 2005.Translated by R. Macaitien     

Joint Discrete Limit Theorems on the Complex Plane for General Dirichlet Series

We prove a joint discrete limit theorem in the sense of weak convergence on the complex plane for general Dirichlet series.     

A Joint Limit Theorem for General Dirichlet Series

We obtain a joint limit theorem in the sense of weak convergence of probability measures for general Dirichlet series in the space of meromorphic functions.     

A Joint Limit Theorem on the Complex Plane for General Dirichlet Series

We prove a joint two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for general Dirichlet series.     

Discrete Limit Theorems for General Dirichlet Series. II

We prove a discrete limit theorem for general Dirichlet series in the sense of weak convergence of probability measures in the space of analytic funtions. Examples are presented.