Logarithmic derivatives of L-functions and small prime quadratic nonresidues

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 $\chi(p)=-1$.