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Mixed character sums modulo prime powers

Todd Cochrane, Andrew Granville

Abstract

We obtain explicit estimates for the mixed character sum $S= S(χ,g,f,p^m) = \sum_{x=1}^{p^m} χ(g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $χ$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over $\mathbb Q$. Let $f=f_+/f_-$, $g=g_+/g_-$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_-g_+g_-$, and $Δ= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $Δ=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$ if $\text{deg}_p(f) \ge 1$, and $|S| \le 3^{4/3}\, p^{m(1-\frac 1Δ)}$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums.

Mixed character sums modulo prime powers

Abstract

We obtain explicit estimates for the mixed character sum , where is a prime power, is a multiplicative character mod and are rational functions over . Let , in reduced form, and set where is the number of distinct complex zeros of , and for polynomial , otherwise. We show for example that for odd , any non-degenerate sum has if , and if . Analogous bounds are given for degenerate sums.

Paper Structure

This paper contains 27 sections, 29 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. notation and background
  3. Degenerate sums
  4. Background on Prime moduli
  5. Main results on Prime power moduli
  6. Upper bound for $t(\chi,g,f,p^m)$
  7. Proof of Proposition \ref{['tprop']}
  8. Conversion of a mixed sum to a pure exponential sum
  9. Pure exponential sum bounds
  10. Relations between parameters
  11. Proof of Theorem \ref{['maintheorem']}
  12. Proof of Corollary \ref{['maincor1']}
  13. Proof of Corollary \ref{['m2']}
  14. Proof of Theorem \ref{['maintheorem0']} and Corollary \ref{['maincor']} for odd $p$
  15. Degenerate Sums, Proof of Theorem \ref{['degentheorem']}
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that $f(x),g(x)$ are rational functions over $\mathbb Q$, $p$ is a prime, $m$ a positive integer, and $\chi$ is a multiplicative character mod ${p^m}$ such that $S(\chi,g,f,p^m)$ is non-degenerate. i) If $\deg_p(f) >0$, then for odd $p$, and the same for $p=2$, with $3$ replaced by $2^{\frac{5}{3}}$. ii) If $\chi$ is primitive and $\deg_p(g)>0$, then for odd $p$, and for $p=2$, $|S(\chi,

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 3.1
  • Example 3.1
  • Example 3.2
  • Theorem 3.2
  • Remark 3.1
  • Theorem 5.1
  • Corollary 5.1
  • Corollary 5.2
  • ...and 44 more