The effect of repeated differentiation on $L$-functions

Abstract

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

Corrected minor typos; added a reference