The universality of general Dirichlet series

Abstract

We prove a universality theorem for functions given by general Dirichlet series, satisfying several conditions.

On the Universality of Estermann Zeta-Functions

“…Later, it was even proved that the set of translations of C(s) which approximate a given function g{s) has a positive lower (Lebesgue-) density; see [7]. This result was extended, for example, to Joint universality for Dirichlet L-functions by Voronin himself [15], to universality for Dedekind zeta-functions by Reich [10], for Hurwitz zetafunctions by Bagchi [2], for Dirichlet series with multiplicative coefficients by Laurincikas [7], for more general Dirichlet series by Gonek [5], Laurincikas, Schwarz and Steuding [8], and to Brought to you by | Simon Fraser University Authenticated Download Date | 6/13/15 12:55 AM universality theorems with weight by Garunkstis [4]. The Linnik-Ibragimov conjecture states that all functions given by Dirichlet series, analytically continuable to the left of the half plane of absolute convergence, which satisfy some natural growth conditions, are universal.…”

“…Obviously, a choice of ß dose to 1 implies universality for E{s\ j,a) with less restricted a. Of course, the restrictions on £ and a can be easily relaxed but we cannot remove restriction (8) in Theorem 4.…”

Universality Theorems in Physics

“…The has been considered in [17]. We note that in the case of general Dirichlet series many additional conditions in hypotheses of universality theorems are involved.…”

A SURVEY ON THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

“…A more general situation was considered by Laurinčikas, Schwarz and J. Steuding [121]. Let {λ n } ∞ n=1 be an increasing sequence of real numbers, linearly independent over Q, and λ n → ∞ as n → ∞.…”