On simple zeros of the Riemann zeta-function

TL;DR

This paper proves that at least 19/27 of the zeros of the Riemann zeta-function are simple, removing the need for the Generalised Lindel"of Hypothesis by employing the Vaughan identity under the assumption of the Riemann Hypothesis.

Contribution

It improves previous results by establishing simplicity of zeros without assuming the GLH, using advanced analytic techniques.

Findings

At least 19/27 of the zeros are simple under RH.

Removal of the GLH assumption from previous results.

Application of the Vaughan identity in zero analysis.

Abstract

We show that at least 19/27 of the zeros of the Riemann zeta-function are simple, assuming the Riemann Hypothesis (RH). This was previously established by Conrey, Ghosh and Gonek [Proc. London Math. Soc. 76 (1998), 497--522] under the additional assumption of the Generalised Lindel\"of Hypothesis (GLH). We are able to remove this hypothesis by careful use of the generalised Vaughan identity.