Logarithmic derivatives of L-functions and small prime quadratic nonresidues
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914028870172672 |
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| author | Zhao, Genheng |
| author_facet | Zhao, Genheng |
| contents | Let $χ$ be a real non-principal character modulo a prime $q$ and $L(s,χ)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,χ)}{L(s,χ)}\leq c \log q,$$ 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 $χ(p)=-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic derivatives of L-functions and small prime quadratic nonresidues Zhao, Genheng Number Theory Let $χ$ be a real non-principal character modulo a prime $q$ and $L(s,χ)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,χ)}{L(s,χ)}\leq c \log q,$$ 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 $χ(p)=-1$. |
| title | Logarithmic derivatives of L-functions and small prime quadratic nonresidues |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.03157 |