Antanas Laurinčikas

Limit Theorems for the Riemann Zeta-Function

Preface. 1. Elements of the probability theory. 2. Dirichlet series and Dirichlet polynomials. 3. Limit theorems for the modulus of the Riemann Zeta-function. 4. Limit theorems for the Riemann Zeta-function on the complex plane. 5. Limit theorems for the Riemann Zeta-function in the space of analytic functions. 6. Universality theorem for the Riemann Zeta-function. 7. Limit theorem for the Riemann Zeta-function in the space of continuous functions. 8. Limit theorems for Dirichlet L-functions. 9. Limit theorem for the Dirichlet series with multiplicative coefficients. References. Notation. Subject index.

The lerch zeta-function ∗

This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1/2. If λ = 1/2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given.

The Universality of Zeta-Functions

The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Eulers product as well as zeta-functions without Eulers product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank ≤3 is proved.

Universality of the periodic Hurwitz zeta-function

In this article, the universality in the Voronin sense for the Hurwitz zeta-function with periodic coefficients is proved.

The discrete universality of the periodic Hurwitz zeta function

The periodic Hurwitz zeta function , s=σ+it, 01, by and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In this paper, a discrete universality theorem for the function with a transcendental parameter α is proved. Roughly speaking, this means that every analytic function can be approximated uniformly on compact sets by shifts , where m is a non-negative integer and h is a fixed positive number such that is rational.

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.

On the lerch zeta-function

This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1/2. If λ = 1/2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given.

A JOINT UNIVERSALITY THEOREM FOR PERIODIC HURWITZ ZETA-FUNCTIONS

We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.

ON THE UNIVERSALITY OF THE HURWITZ ZETA-FUNCTION

It is known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that its shifts ζ(s + iτ, α), τ ∈ ℝ, approximate with a given accuracy any analytic function uniformly on compact subsets of the strip D = {s ∈ ℂ : ½ < σ < 1}. Let H(D) denote the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In the paper, the classes of functions F : H(D) → H(D) such that F(ζ(s, α)) is universal in the above sense are considered. For example, if F is continuous and, for each polynomial p = p(s), the set F-1{p} is non-empty, then F(ζ(s, α)) with transcendental α is universal.

On the universality of the Riemann zeta-function

Article history: Received 2 March 2010 Revised 28 April 2010 Communicated by David Goss MSC: 11M06