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arXiv · 2410.20341

On the distribution of $\log |L(\sigma, \chi)|$ and $\log L(\sigma, \chi_D)$ in the modulus aspect

Manami Hosoi·Yumiko Umegaki
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Abstract

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, \chi)|$ in $q$-aspect at a fixed $s=\sigma>1/2$, 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 $\log L(s, \chi_D)$ and $L'/L(s, \chi_D)$ in the $D$-aspect at a fixed $s=\sigma>1/2$ which $L(\sigma', \chi)\neq 0$ in $\sigma\leq \sigma' \leq 1$, where $\chi_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally.

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Manami Hosoi, Yumiko Umegaki. 2024-10-27. On the distribution of $\log |L(\sigma, \chi)|$ and $\log L(\sigma, \chi_D)$ in the modulus aspect. https://arxiv.org/abs/2410.20341

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