On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909262953840640 |
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| author | Liu, Di Williams, Clayton Zaharescu, Alexandru |
| author_facet | Liu, Di Williams, Clayton Zaharescu, Alexandru |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges Liu, Di Williams, Clayton Zaharescu, Alexandru Number Theory Primary 11M41, Secondary 11F66, 11M26 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. |
| title | On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges |
| topic | Number Theory Primary 11M41, Secondary 11F66, 11M26 |
| url | https://arxiv.org/abs/2407.14656 |