Logarithmic derivatives of L-functions and small prime quadratic nonresidues

Genheng Zhao

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arXiv · 2509.03157

Logarithmic derivatives of L-functions and small prime quadratic nonresidues

Genheng Zhao

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Abstract

Let $\chi$ be a real non-principal character modulo a prime $q$ and $L(s,\chi)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,\chi )}{L(s,\chi)}\leq c \log q,$$ where $c$ can be taken arbitrarily close to $1/4$ if we assume $q$ is sufficiently large depending upon it. As a consequence, for all large $q$, there are at least $q^{3/50}$ primes $p$ smaller than $q$ such that $\chi(p)=-1$.

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Genheng Zhao. 2025-09-03. Logarithmic derivatives of L-functions and small prime quadratic nonresidues. https://arxiv.org/abs/2509.03157

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