On the distribution of $\log |L(σ, χ)|$ and $\log L(σ, χ_D)$ in the modulus aspect
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915458413756416 |
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| author | Hosoi, Manami Umegaki, Yumiko |
| author_facet | Hosoi, Manami Umegaki, Yumiko |
| contents | Let $χ$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, χ)|$ in $q$-aspect at a fixed $s=σ>1/2$, under Generalized Riemann Hypothesis (GRH), we explain it can be written as integrals involving the same density function ($M$-function) for the average of values of the difference between the logarithms of two symmetric power $L$-functions in the level aspect.
For the distribution of values of $\log L(s, χ_D)$ and $L'/L(s, χ_D)$ in the $D$-aspect at a fixed $s=σ>1/2$ which $L(σ', χ)\neq 0$ in $σ\leq σ' \leq 1$, where $χ_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the distribution of $\log |L(σ, χ)|$ and $\log L(σ, χ_D)$ in the modulus aspect Hosoi, Manami Umegaki, Yumiko Number Theory 11M06 Let $χ$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, χ)|$ in $q$-aspect at a fixed $s=σ>1/2$, under Generalized Riemann Hypothesis (GRH), we explain it can be written as integrals involving the same density function ($M$-function) for the average of values of the difference between the logarithms of two symmetric power $L$-functions in the level aspect. For the distribution of values of $\log L(s, χ_D)$ and $L'/L(s, χ_D)$ in the $D$-aspect at a fixed $s=σ>1/2$ which $L(σ', χ)\neq 0$ in $σ\leq σ' \leq 1$, where $χ_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally. |
| title | On the distribution of $\log |L(σ, χ)|$ and $\log L(σ, χ_D)$ in the modulus aspect |
| topic | Number Theory 11M06 |
| url | https://arxiv.org/abs/2410.20341 |