Uniform approximation of some Dirichlet series by partial products of Euler type

TL;DR

This paper demonstrates that certain Dirichlet series with Euler products, including the Riemann zeta and L-functions, can be approximated by partial Euler products within the critical strip, supporting an analog of the Riemann Hypothesis.

Contribution

It establishes conditions under which Dirichlet series with Euler products can be uniformly approximated by partial products in the critical strip, extending understanding of their analytic properties.

Findings

Dirichlet series with Euler products can be approximated by partial products in the critical strip.

The approximation supports an analog of the Riemann Hypothesis for these series.

Includes widely used functions like the zeta and Dirichlet L-functions.

Abstract

In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in the critical strip, if the primes over which are taken the products are distributed by a suitable way. The family of such series includes many of widely used Dirichlet series as the zeta-function, Dirichlet L-functions and etc. As a consequence the analog of the Riemann Hypothesis for such series is proven.