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

2010/08/04 by Ilgar Shikar Jabbarov, Jabbarov, Ilgar Shikar
Mathematics · #11M26 #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT) #math.GM #math.NT #msc:11M26

paper · pdf · doi:10.48550/arxiv.1008.0852

The method of the paper based on a new measure introduced in infinite dimensional unit cube

openalex publication_date 2010/08/04 · arxiv created 2013/08/04 · arxiv updated 2013/08/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Cited by

• The Riemann Hypothesis

Related

• Partial Euler products as a new approach to Riemann hypothesis
• Family of Approximations for Dirichlet L-functions
• Essential singularities of Euler products
• Euler Product Sieve
• On the validity of the Euler product inside the critical strip