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Mixed incomplete character sums of rational functions with smooth moduli

Todd Cochrane, Andrew Granville, Junren Zheng

TL;DR

This work extends Graham–Ringrose bounds for incomplete character sums to moduli that are $N^{1-\varepsilon}$-smooth, removing squarefree restrictions and allowing general rational functions $f$ and $g$ in $\chi(f(n)) e(g(n)/q)$. It develops a comprehensive framework linking incomplete to complete character-exponential sums, via a $q$-analogue of van der Corput differencing, and establishes sharp bounds for prime and prime-power moduli, including $p$-adic divisibility phenomena for difference functions. A key innovation is a detailed treatment of the p-adic valuation of transformed rational-function differences $H^+$ and a method to convert character values into exponential terms using Taylor expansions, enabling strong, explicit bounds on sums over short intervals. The results yield best-possible features in several aspects, including Brun–Titchmarsh-type inequalities and $L$-value bounds, with broad applicability to problems involving congruences and exponential sums with rational-function arguments.

Abstract

Let $χ=χ_q$ be a primitive character mod $q$ and fix $Δ>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} χ(n)$ in intervals of length $N=q^Δ$ whenever $q$ is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be $N^{1-ε}$. We also discuss various generalizations and applications, obtaining best possible results in several aspects.

Mixed incomplete character sums of rational functions with smooth moduli

TL;DR

This work extends Graham–Ringrose bounds for incomplete character sums to moduli that are -smooth, removing squarefree restrictions and allowing general rational functions and in . It develops a comprehensive framework linking incomplete to complete character-exponential sums, via a -analogue of van der Corput differencing, and establishes sharp bounds for prime and prime-power moduli, including -adic divisibility phenomena for difference functions. A key innovation is a detailed treatment of the p-adic valuation of transformed rational-function differences and a method to convert character values into exponential terms using Taylor expansions, enabling strong, explicit bounds on sums over short intervals. The results yield best-possible features in several aspects, including Brun–Titchmarsh-type inequalities and -value bounds, with broad applicability to problems involving congruences and exponential sums with rational-function arguments.

Abstract

Let be a primitive character mod and fix . In 1989 Graham and Ringrose gave strong bounds on character sums in intervals of length whenever is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be . We also discuss various generalizations and applications, obtaining best possible results in several aspects.
Paper Structure (20 sections, 32 theorems, 151 equations)

This paper contains 20 sections, 32 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. New results on incomplete character sums
  3. Linear $g(x)$ and the Brun-Titchmarsh theorem
  4. Counting solutions to congruences mod $p^m$
  5. Complete character-exponential sum estimates
  6. Complete sums modulo primes $p$
  7. Complete sums modulo prime powers
  8. From character values to exponential values
  9. Interpreting Corollary \ref{['cor:chi-expand2']} away from finitely many primes
  10. The application in practice
  11. The $p$-divisibility of the rational function differences
  12. When $f^*$ is an $r$th power mod $p$
  13. The main complete exponential-character sum for difference functions, in context
  14. Basic inequalities for 1-bounded periodic functions
  15. Summing up displacements
  16. ...and 5 more sections

Key Result

Theorem 1

Fix $\delta,\epsilon>0$. Suppose that $f(x), g(x)\in \mathbb Q(x)$ where $f(x)$ and $g(x)$ are not both constants. There exists a constant $\eta=\eta(f,g,\delta,\epsilon)>0$ and an explicitly determinable positive integer $\mathcal{M}=\mathcal{M}(f,g,\delta)$ such that we have unless $g(x)$ is a polynomial of degree $<1/\delta$ and $\chi^{r_f}$ is induced from a primitive character of conductor

Theorems & Definitions (59)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Proposition 1
  • Proposition 2
  • proof
  • proof : Proof of Proposition \ref{['prop: Soln count']}
  • Proposition 3
  • ...and 49 more