Zeros of derivatives of $L$-functions in the Selberg class on $\Re(s)<1/2$

TL;DR

This paper explores the implications of the Riemann hypothesis for $L$-functions in the Selberg class, showing that derivatives of such functions have finitely many zeros to the left of the critical line.

Contribution

It extends Levinson and Montgomery's 1974 result for the Riemann zeta function to a broader class of $L$-functions within the Selberg class.

Findings

Finiteness of zeros of derivatives on the left of the critical line under the Riemann hypothesis

Generalization of previous results from the Riemann zeta function to the Selberg class

Conditional results based on the Riemann hypothesis for $L$-functions

Abstract

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974.