Shifted Character Sums with Multiplicative Coefficients, II

Abstract

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