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arXiv · 2406.01519

Large values of quadratic Dirichlet $L$-functions

Pranendu Darbar·Gopal Maiti
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Abstract

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\Omega$-results for the family of quadratic Dirichlet $L$-functions $L(\sigma, \chi_d)$, where $d$ runs over all fundamental discriminants with $|d| \leq X$ and $\sigma\in [1/2, 1]$ is fixed. This study advances understanding of the maximum size of $L(\sigma, \chi_d)$ within the segment $\sigma\in [1/2, 1]$. In particular, we improve upon Soundararajan's results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $\sigma \in (1/2, 1]$.

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Pranendu Darbar, Gopal Maiti. 2024-06-03. Large values of quadratic Dirichlet $L$-functions. https://arxiv.org/abs/2406.01519

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