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.     

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 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.     

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

We obtain a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Lerch zeta-function with algebraic irrational parameter. Partially supported by the Lithuanian Studies and Science Foundation.     

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     

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.     

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.     

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.     

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      

Note on zeros of the derivative of the Selberg zeta-function

Wenzhi Luo studied the distribution of nontrivial zeros of the derivatives of Selberg zeta-functions on cocompact hyperbolic surfaces, and obtained an asymptotic formula for zero density with bounded height. Then he related the distribution of zeros to the multiplicities of Laplacian eigenvalues. Using better bounds for the growth of the Selberg zeta-functions we improve some of the above results. This work was partially supported by a Grant from the Lithuanian Foundation of Studies and Science.     

Value-distribution of twisted automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper, we prove a limit theorem for twisted with character automorphic L-functions with an increasing modulus of the character.     

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.Dedicated to Jean-Louis Nicolas on the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11M06, 11M35, 11Y60     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions. III

In this paper, we consider upper and lower bounds of the same order with explicitly given constants for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 280–295, April–June, 2007.     

An L-Function of Degree 27 for Spin9

This paper studies a Rankin-Selberg integral for a degree 27 L-function on Spin(9). It makes use of an Eisenstein series on the exceptional group F ₄.     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions,II

We obtain upper and lower bounds for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 606–621, October–December, 2006.     

On Fractional Moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-Functions

We obtain upper and lower bounds of the same order for fractional moments of Dirichlet L-functions.Supported by a grant from the Lithuanian Science and Studies Fundation.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 208–228, April–June, 2005.     

On the riemann zeta-function and the divisor problem

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If with , then we obtain

On the index of fractions with square-free denominators in arithmetic progressions

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

K2 of a Fermat quotient and the value of its L-function

Two linearly independent elements of k ₂ of a certain quotient of a Fermat curve is exhibited in an explicit manner. The covolume of the regulator and the value of the L-function of the curve is numerically computed, and their ratio is nearly equal to a simple rational number.