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The effect of repeated differentiation on L-functions

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JournalJournal of Number Theory
DateAccepted/In press - 25 Jul 2018
DateE-pub ahead of print - 22 Aug 2018
DatePublished (current) - 1 Jan 2019
Volume194
Pages (from-to)30-43
Early online date22/08/18
Original languageEnglish

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.

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© 2018 Elsevier Inc. All rights reserved. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.

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