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arXiv · 2604.02960

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

Pranendu Darbar·Bryce Kerr·Marc Munsch·Igor Shparlinski
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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.

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Pranendu Darbar, Bryce Kerr, Marc Munsch, Igor Shparlinski. 2026-04-03. Large values of $L(\sigma,\chi)$ for subgroups of characters. https://arxiv.org/abs/2604.02960

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