The effect of repeated differentiation on L-functions

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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.
Original languageEnglish
Pages (from-to)30-43
JournalJournal of Number Theory
Volume194
Early online date22 Aug 2018
DOIs
Publication statusPublished - 1 Jan 2019

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