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Pointwise and correlation bounds on Dedekind sums over small subgroups

Bence Borda, Marc Munsch, Igor Shparlinski

Abstract

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(λ,p)$ modulo a prime $p$ with $λ$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of $L(1,χ)$ over subgroups of characters to some correlations of Dedekind sums and use our recent results to study these correlations.

Pointwise and correlation bounds on Dedekind sums over small subgroups

Abstract

We obtain new bounds, pointwisely and on average, for Dedekind sums modulo a prime with of small multiplicative order modulo . Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of over subgroups of characters to some correlations of Dedekind sums and use our recent results to study these correlations.
Paper Structure (16 sections, 10 theorems, 67 equations)

This paper contains 16 sections, 10 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Dedekind sums and moments of $L$-functions
  3. Summary of our results and methods
  4. Notation and conventions
  5. Bounds on Dedekind sums
  6. Pointwise bounds on Dedekind sums
  7. Correlation of Dedekind sums
  8. Preliminaries
  9. Dedekind sums and continued fractions
  10. Small solutions to linear congruences
  11. Proofs of results on Dedekind sums
  12. Proof of Theorem \ref{['thm:CF']}
  13. Proof of Theorem \ref{['thm-formulahigher']}
  14. Proof of Theorem \ref{['thm:CF-2']}
  15. Proof of Theorem \ref{['thm:CorrCorr']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $p$ be a prime, and assume that $\lambda \in {\mathbb F}_p^*$ has multiplicative order $d \ge 3$ in ${\mathbb F}_p^*$. The continued fraction expansion $\{ \lambda /p \}=[0;a_1,\ldots, a_n]$ satisfies

Theorems & Definitions (14)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Remark 2.8
  • Theorem 2.9
  • Lemma 3.1
  • ...and 4 more