Large values of $L(\sigma,\chi)$ for subgroups of characters

TL;DR

This paper investigates large values of L-functions for characters in subgroups, providing new bounds and zero-density estimates, with applications to primitive root gaps and mean value estimates.

Contribution

It introduces novel bounds for L-functions in subgroups, based on new zero-density estimates and mean value techniques, extending previous results unconditionally.

Findings

Established bounds on L-functions in subgroups of characters.

Developed new zero-density estimates on average over subgroups.

Provided an unconditional version of a previous conditional result on primitive root gaps.

Abstract

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. 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.