Euler Product Asymptotics for Dirichlet $L$-Functions

TL;DR

This paper establishes the asymptotic behavior of partial Euler products for Dirichlet L-functions under GRH, linking the Deep Riemann Hypothesis to the Generalised Riemann Hypothesis through asymptotic analysis.

Contribution

It provides a new asymptotic analysis of Euler products for Dirichlet L-functions assuming GRH, elucidating the connection to the Deep Riemann Hypothesis.

Findings

Asymptotic behavior of Euler products derived under GRH

Relation between GRH and DRH clarified

Extends Ramanujan's work on Euler products

Abstract

Via the work of Ramanujan, we establish the asymptotic behaviour of partial Euler products for Dirichlet $L$-functions under the Generalised Riemann Hypothesis (GRH). Understanding the behaviour of Euler products on the critical line is called the Deep Riemann Hypothesis (DRH). This work manifests the relation between GRH and DRH.