The effect of repeated differentiation on $L$-functions

TL;DR

This paper demonstrates that repeated differentiation of the Selberg Xi-function causes its zeros to become more evenly spaced and converge to a cosine function, revealing new insights into zero distribution behavior.

Contribution

It introduces a novel approach connecting high derivatives of the Xi-function to cosine functions through Fourier transform techniques.

Findings

Zeros become more evenly spaced with repeated differentiation

High derivatives converge to cosine functions

Gamma function products expressed as Fourier transforms

Abstract

We show that under repeated differentiation, the zeros of the Selberg $\Xi$-function become more evenly spaced out, but with some scaling towards the origin. We do this by showing the high derivatives of the $\Xi$-function converge to the cosine function, and this is achieved by expressing a product of Gamma functions as a single Fourier transform.