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arXiv · 1404.2204

Shifted Character Sums with Multiplicative Coefficients

Ke Gong·Chaohua Jia
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Abstract

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $χ$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)χ(n+a)\ll {N\over q^{1\over 4}}\log\log(6N)+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}. $$ 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 $t\geq 2$, $a_1,\,\cdots,\,a_t$ are pairwise distinct integers modulo $q$.

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Ke Gong, Chaohua Jia. 2014-07-03. Shifted Character Sums with Multiplicative Coefficients. https://arxiv.org/abs/1404.2204

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