СМЕШАННАЯ СОВМЕСТНАЯ ФУНКЦИОНАЛЬНАЯ НЕЗАВИСИМОСТЬ ДЛЯ ДЗЕТА-ФУНКЦИИ РИМАНА И ПЕРИОДИЧЕСКИХ ДЗЕТА-ФУНКЦИИЙ ГУРВИЦА

Р Качинскайте,С Рапимбергайте

doi:10.22405/2226-8383-2016-17-4-57-64

Abstract

The functional independence of zeta-functions is an interesting nowadays problem. This problem comes back to D. Hilbert. In 1900, at the International Congress of Mathematicians in Paris, he conjectured that the Riemman zeta-function does not satisfy any algebraicdifferential equation. This conjecture was solved by A. Ostrowski. In 1975, S.M. Voronin proved the functional independence of the Riemann zeta-function. After that many mathematicians obtained the functional independence of certain zeta- and L -functions. In the present paper, the joint functional independence of a collection consisting of the Riemann zeta-function and several periodic Hurwitz zeta-functions with parameters algebraically independent over the field of rational numbers is obtained. Such type of functional independence is called as “mixed functional independence” since the Riemann zeta-function has Euler product expansion over primes while the periodic Hurwitz zeta-functions do not have Euler product.

Highlights

The functional independence of certain functions has a long history and is relevant in nowadays
Hilbert noted [2] that the Riemann zeta-function ζ(s) does not satisfy any algebraic differential equation
Analogue results on the joint mixed functional independence of certain zeta-functions we may obtain in the same way as Theorem 1 if the function ζ(s) will be replaced by certain zeta-functions, namely by the zeta-functions of normalized Hecke cusp forms, the zeta-functions of newforms with a Dicirhlet characters, the L-functions from the Selberg class, and etc

Summary

Introduction

The functional independence of certain functions has a long history and is relevant in nowadays. Hilbert noted [2] that the Riemann zeta-function ζ(s) does not satisfy any algebraic differential equation, He proposed a more general problem, i.e., to prove that the function. The equality (1) gives an analytic continuation of the periodic Hurwitz zeta-function ζ(s, α; a) to the whole s-plane, except, maybe, for a simple pole s = 1 with residue k−1. The joint functional independence of a collection of periodic Hurwitz zeta-functions with parameters algebraically independent over the field of rational numbers Q was obtained by A. Let, for positive integer lj, ajl = {amjl : m ∈ N ∪ {0}} be a periodic sequence of complex numbers amjl with a minimal period kjl ∈ N, and let ζ(s, αj; ajl) denote the corresponding periodic Hurwitz zeta-function, j = 1, ..., r, l = 1, ..., lj.

Auxiliary results

The mixed joint universality theorem

A denseness lemma

Proof of Theorem 1

Findings

Conclusions