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Bounds for moments of quadratic character sums and theta functions

Marc Munsch, Yuichiro Toma

Abstract

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a multivariate tauberian theorem. As a consequence, we prove unconditional lower bounds for all even integer moments of quadratic character sums in a wide range of parameters. Moreover, assuming the Generalised Riemann Hypothesis (GRH), we prove a sharp upper bound on moments of character sums of arbitrary length. In a similar fashion, we obtain unconditional lower bounds on moments of quadratic theta functions and matching conditional upper bounds under GRH. In the case of the second moment of theta functions, we prove an optimal upper bound unconditionally improving the previous results of Louboutin and the first named author.

Bounds for moments of quadratic character sums and theta functions

Abstract

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a multivariate tauberian theorem. As a consequence, we prove unconditional lower bounds for all even integer moments of quadratic character sums in a wide range of parameters. Moreover, assuming the Generalised Riemann Hypothesis (GRH), we prove a sharp upper bound on moments of character sums of arbitrary length. In a similar fashion, we obtain unconditional lower bounds on moments of quadratic theta functions and matching conditional upper bounds under GRH. In the case of the second moment of theta functions, we prove an optimal upper bound unconditionally improving the previous results of Louboutin and the first named author.

Paper Structure

This paper contains 15 sections, 21 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Overview of the proofs
  3. Background results
  4. Proofs of the lower bounds
  5. Proof of Theorem \ref{['lowquad']}
  6. Proof of Theorem \ref{['lowtheta']}
  7. Proof of the asymptotic formula
  8. Proof of Theorem \ref{['asymptotic for S_k(X,Y)']}
  9. Proofs of the upper bounds
  10. Proof of Theorem \ref{['th_smooth']}
  11. Proof of Proposition \ref{['bound_intshift']}
  12. The base case $k=2$
  13. The induction step
  14. Proof of Theorem \ref{['th_theta']}
  15. The second moment of $\theta(1,\chi_d)$: Proof of Theorem \ref{['theta2nd']}

Key Result

Theorem 1

For any fixed even integer $k\geq 2$ and $X^{\varepsilon} \ll Y\ll X^{2/3-\alpha}$ for some $\alpha>0$, we have Moreover, assuming GRH, lowSk remains valid in the range $X^{\varepsilon} \ll Y\ll X^{1-\alpha}$ for some $\alpha>0$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Remark 8
  • Theorem 9
  • Corollary 10
  • ...and 22 more