The effect of repeated differentiation on L-functions

Jos Gunns; Chris Hughes

doi:10.1016/j.jnt.2018.07.008

The effect of repeated differentiation on L-functions

Jos Gunns
, Chris Hughes

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

287 Downloads (Pure)

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.

Original language	English
Pages (from-to)	30-43
Number of pages	14
Journal	Journal of Number Theory
Volume	194
Early online date	22 Aug 2018
DOIs	https://doi.org/10.1016/j.jnt.2018.07.008
Publication status	Published - 1 Jan 2019

Keywords

Selberg class of L-functions

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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.",

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AB - 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.

KW - Selberg class of L-functions

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DO - 10.1016/j.jnt.2018.07.008

M3 - Article (Academic Journal)

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SN - 0022-314X

VL - 194

SP - 30

EP - 43

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

The effect of repeated differentiation on L-functions

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Keywords

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