Shifted Character Sums with Multiplicative Coefficients, II

TL;DR

This paper establishes bounds on shifted character sums with multiplicative coefficients, extending previous results and demonstrating that these sums are small relative to N, with bounds involving logarithmic factors of q.

Contribution

It proves new upper bounds for shifted character sums with multiplicative coefficients, generalizing earlier work and covering multiple shifts with explicit logarithmic bounds.

Findings

Bound on sum of f(n)χ(n+a) is O(N(log log q)/log q)

Bound on product of shifted character sums is similar in magnitude

Results hold for large N within specified q-range

Abstract

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{\frac12+\varepsilon}\ll N\ll q^A$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)\chi(n+a)\ll N\frac{\log\log q}{\log q} $$ and that $$ \sum_{n\leq N}f(n)\chi(n+a_1)\cdots\chi(n+a_t)\ll N\frac{\log\log q}{\log q}, $$ where $t\geq 2, a_1, \ldots, a_t$ are distinct integers modulo $q$.