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Large values of $L(σ,χ)$ for subgroups of characters

Pranendu Darbar, Bryce Kerr, Marc Munsch, Igor Shparlinski

Abstract

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,χ)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $χ$ 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.

Large values of $L(σ,χ)$ for subgroups of characters

Abstract

We obtain (conditional and unconditional) results on large values of -functions in the critical strip when the character runs through a thin subgroup of all characters modulo an integer . 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.

Paper Structure

This paper contains 23 sections, 13 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Motivation and set-up
  3. Notation and conventions
  4. Main results
  5. Extreme values
  6. Zero density estimates over subgroups of characters
  7. Mean values of character sum over sets of characters with small multiplicative doubling
  8. Preliminaries
  9. Double sums of character sums
  10. Green--Rusza covering lemma
  11. Approximation and bounds for some finite Euler products and $L$-functions
  12. Proofs of bounds on character sums and zero density results
  13. Proof of Theorem \ref{['thm:MeanVal']}
  14. Proof of Theorem \ref{['thm:mean-value-sub']}
  15. Proofs of results on extreme values
  16. ...and 8 more sections

Key Result

Theorem 2.1

Assume that $q$ is a sufficiently large integer and let $\delta > 0$ be fixed. Let ${\mathcal{H}}$ be a subgroup of $\mathcal{X}_q$ of order $H=\#{\mathcal{H}}$, satisfying There exists a non-principal character $\chi \in {\mathcal{H}}$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 9 more