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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.15311 |
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| _version_ | 1866912400230318080 |
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| author | de Faveri, Alexandre |
| author_facet | de Faveri, Alexandre |
| contents | Let $f \in S_k(Γ_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $Ω\left(T^δ\right)$ simple zeros with imaginary part in $\left[-T, T\right]$, for any $δ< \frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $Ω(\log\log\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_15311 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Simple zeros of $\mathrm{GL}(2)$ $L$-functions de Faveri, Alexandre Number Theory Let $f \in S_k(Γ_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $Ω\left(T^δ\right)$ simple zeros with imaginary part in $\left[-T, T\right]$, for any $δ< \frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $Ω(\log\log\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros. |
| title | Simple zeros of $\mathrm{GL}(2)$ $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2109.15311 |