Value distribution of the Lerch zeta-function with algebraic irrational parameter. IV

We obtain a joint limit theorem in the space of analytic functions for Lerch zeta-functions with algebraic irrational parameter. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-81/09.     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. II

In this paper, we obtain a joint limit theorem in the sense of weak convergence of probability measures on the complex plane for Lerch zeta-functions with algebraic irrational parameters and give an explicit form of the limit measure. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 482–496, October–December, 2007.     

Value-distribution of the Lerch zeta-function with algebraic irrational parameter. I

We prove a limit theorem in the sense of weak convergence of probability measures on the complex plane for the Lerch zeta-function with algebraic irrational parameter. The limit measure in this theorem is explicitly given. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 163–176, April–June, 2007.     

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.     

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     

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.     

Discrete value distribution of the Hurwitz zeta-function with algebraic irrational parameter. I

In the paper, the statistical properties of discrete values of the Hurwitz zeta-function with algebraic irrational parameter in the space of analytic functions are investigated.     

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.     

Value distribution of general Dirichlet series. VII

We prove a limit theorem in the space of meromorphic functions for a new class of general Dirichlet series. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 193–202, April–June, 2006.     

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.     

Approximation of the Lerch Zeta-Function

We consider uniform in parameters approximations of the Lerch zeta-function by Dirichlet polynomials. It allows us to obtain uniform in parameters bounds in the critical strip.     

A two-dimensional limit theorem for Mellin transforms of the Riemann zeta-function

In this paper, we obtain a two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for Mellin transforms of the square and the fourth power of the Riemann zeta-function. Partially supported by the Lithuanian Foundation of Studies and Science.     

Value Distribution of General Dirichlet Series. IV

In the paper, a limit theorem in the space of meromorphic functions with explicitly given limit measure for general Dirichlet series is proved. The theorem obtained generalizes and simplifies the results of work [7].     

On the Periodic Zeta-Function

We consider the periodic zeta-function introduced by B. C. Berndt and L. Schoenfeld. We obtain the asymptotics of the mean square and prove limit theorems in the space of analytic functions for this function.     

The Mean-Square of the Lerch Zeta-Function Near the Critical Line

In the paper, the asymptotics of the mean-square of the Lerch zeta-function near the critical line is obtained. Three different definitions of the closeness to the critical line are considered.     

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.     

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.     

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.     

A limit theorem for the Hurwitz zeta-function with an algebraic irrational parameter

We prove a limit theorem in the complex plane for the Hurwitz zeta-function with algebraic irrational parameter. Received: 14 June 2004; revised: 12 April 2005     

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