Extreme values of $L$-functions of newforms
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866917657075253248 |
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| author | Gun, Sanoli Lunia, Rashi |
| author_facet | Gun, Sanoli Lunia, Rashi |
| contents | In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$ L(1/2, f ) ~\geq~ \exp\Bigg( (1 + o(1)) \sqrt{\frac{2\log k}{\log\log k} }\Bigg) $$ for sufficiently large $k \equiv 0 \pmod{4}$. In this note, we show that for any $ε>0$ and for all sufficiently large $k \equiv 0 \pmod{4}$, the number of normalized Hecke eigenforms of weight $k$ and level one for which $$ L(1/2, f ) ~\geq~ \exp\left(1.41\sqrt{ \frac{ \log k }{\log\log k} }\right) $$ is $\gg_ε k^{1-ε}$. For an odd fundamental discriminant $D$, let $B_{k}(|D|)$ be the set of all cuspidal normalized Hecke eigenforms of weight $k$ and level dividing $|D|$. When the real primitive Dirichlet character $χ_D$ satisfies $χ_D(-1)= i^k$, we investigate the number of $f \in B_{k}(|D|)$ for which $L(1/2, f \otimes χ_D)$ takes extremal values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extreme values of $L$-functions of newforms Gun, Sanoli Lunia, Rashi Number Theory 11F11, 11F37, 11F72, 11M99 In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$ L(1/2, f ) ~\geq~ \exp\Bigg( (1 + o(1)) \sqrt{\frac{2\log k}{\log\log k} }\Bigg) $$ for sufficiently large $k \equiv 0 \pmod{4}$. In this note, we show that for any $ε>0$ and for all sufficiently large $k \equiv 0 \pmod{4}$, the number of normalized Hecke eigenforms of weight $k$ and level one for which $$ L(1/2, f ) ~\geq~ \exp\left(1.41\sqrt{ \frac{ \log k }{\log\log k} }\right) $$ is $\gg_ε k^{1-ε}$. For an odd fundamental discriminant $D$, let $B_{k}(|D|)$ be the set of all cuspidal normalized Hecke eigenforms of weight $k$ and level dividing $|D|$. When the real primitive Dirichlet character $χ_D$ satisfies $χ_D(-1)= i^k$, we investigate the number of $f \in B_{k}(|D|)$ for which $L(1/2, f \otimes χ_D)$ takes extremal values. |
| title | Extreme values of $L$-functions of newforms |
| topic | Number Theory 11F11, 11F37, 11F72, 11M99 |
| url | https://arxiv.org/abs/2405.02428 |