Chebyshev's bias for modular forms
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908768234635264 |
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| author | Koyama, Shin-ya Sheth, Arshay |
| author_facet | Koyama, Shin-ya Sheth, Arshay |
| contents | We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $Γ_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chebyshev's bias for modular forms Koyama, Shin-ya Sheth, Arshay Number Theory We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $Γ_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples. |
| title | Chebyshev's bias for modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.04187 |