Shifted Character Sums with Multiplicative Coefficients

Abstract

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\le 1$, $q$ $(\le N^2)$ be a prime number and $a$ be an integer with $(a,q)=1$, $\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$\sum _{n\le N}f(n)\chi (n+a)\ll \frac{N}{q^{\frac{1}{4}}}\log \log (6N)+q^{\frac{1}{4}}{N}^{\frac{1}{2}}\log (6N)+\frac{N}{\sqrt{\log \log (6N)}}.$$ We shall also prove that \begin{align*} &\sum_{n\leq N}f(n)χ(n+a_1)\cdotsχ(n+a_t)\ll {N\over q^{1\over 4}}\log\log(6N)\\ &\quad+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}, \end{align*} where $t\ge 2$, $a_1,\cdots ,a_t$ are pairwise distinct integers modulo $q$.