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arXiv · 2407.14656

On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges

Di Liu·Clayton Williams·Alexandru Zaharescu
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Abstract

We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.

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Di Liu, Clayton Williams, Alexandru Zaharescu. 2024-07-19. On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges. https://arxiv.org/abs/2407.14656

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