Elementary Deuring-Heilbronn Phenomenon

TL;DR

This paper provides an elementary proof of the Deuring-Heilbronn phenomenon, linking zeros of the Riemann zeta function off the critical line to lower bounds on Dirichlet L-functions, using classical analytic techniques.

Contribution

It introduces an elementary demonstration of the Deuring phenomenon employing classical methods, simplifying previous complex proofs.

Findings

Zeros of zeta(s) off the critical line imply lower bounds on L(1,hi)

Uses elementary tools like Dirichlet's hyperbola method and Euler summation

Provides a simplified proof of the Deuring-Heilbronn phenomenon

Abstract

Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of \zeta(s) off the critical line gives a lower bound on L(1,\chi). The necessary tools are Dirichlet's 'method of the hyperbola', Euler summation, summation by parts, and the Polya-Vinogradov inequality.