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 $s\ge 1$ there holds $$-\frac{L^{\prime }(s,\chi )}{L(s,\chi )}\le c\log q,$$ 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 $\chi (p)=-1$.

c=1/8+εis available elsewhere