CONVERGENCE OF DIRICHLET SERIES AND EULER PRODUCTS

Doug S Phillips,Peter Zvengrowski

doi:10.20903/csnmbs.masa.2017.38.2.111

Abstract

The first part of this paper deals with Dirichlet series, and convergence theorems are proved that strengthen the classical convergence theorem as found e.g. in Serre’s “A Course in Arithmetic.” The second part deals with Euler-type products. A convergence theorem is proved giving sufficient conditions for such products to converge in the half-plane having real part greater than 1/2. Numerical evidence is also presented that suggests that the Euler products corresponding to Dirichlet L-functions L(s, χ), where χ is a primitive Dirichlet character, converge in this half-plane.

Highlights

The general theme of this note is convergence
Theorem 2.9 is a refinement of Theorem 2.7, and Theorem 2.12 a further refinement which gives sufficient conditions for convergence for σ >
Since convergent infinite products cannot equal 0, this would imply the Generalized Riemann Hypothesis for all these L-functions, namely each such Lfunction cannot have a zero if σ >

Summary

INTRODUCTION

The general theme of this note is convergence. In Section 2 this is studied for Dirichlet series and in Sections 3-5 for infinite products, in particular for Euler products. Since (as we shall see in Section 4) convergent infinite products cannot equal 0, this would imply the Generalized Riemann Hypothesis for all these L-functions, namely each such Lfunction cannot have a zero if σ > .5 (cf [6], [10]).

DIRICHLET SERIES

THEORY OF EULER PRODUCT CONVERGENCE

Remark

EXAMPLES AND QUESTIONS