Large values of $L(\sigma ,\chi )$ over sets of characters with small product set

Abstract

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi )$ in the critical strip $1/2\le \Re s\le 1$ when the character $\chi$ runs through the ratio set $\mathcal{A}/\mathcal{A}$ of a thin set $\mathcal{A}$ of characters with small product set modulo a prime $q$. Some of these bounds are based on new zero-density estimates on average over sets of characters with a small product set. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots.