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arXiv · 1803.10001

The effect of repeated differentiation on $L$-functions

Jos Gunns·Christopher Hughes
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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.

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Jos Gunns, Christopher Hughes. 2018-03-27. The effect of repeated differentiation on $L$-functions. https://arxiv.org/abs/1803.10001

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