ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Laurinčikas, Antanas, Vilnius University, Šiauliai University, Lithuania Value-distribution theorems on Lerch zeta-functions Let s = σ + it be a complex variable. The Lerch zeta-function L(λ, α, s), for σ > 1, is defined by

L(λ, α, s) = ∑_{m=0}^{∞} \frac{e^{2πiλm}}{(m + α)^{s}} .

Here α and λ are fixed real parameters, 0 < α ≤ 1. The case λ ∉ Z is considered, and then L(λ, α, s) is analytically continuable to an entire function. In the report we discuss: a) limit theorems in the sense of the weak convergence of probability neasures for L(λ, α, s) on the complex plane and in the space of analytic functions; b) universality property of L(λ, α, s);

c) functional independence of L(λ, α, s); d) zero-distribution of L(λ, α, s): zero-free regions, estimates of the number of zeros; e) joint value-distribution theorems for Lerch zeta-functions. Part of these results is joint work with R. Garunkštis and Kohji Matsumoto.