A Shiu Theorem for Larger and Smoother Functions

Abstract

In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $\beta<1$ such that $f(p^l)\ll (\log\log x)^{l\beta}$ for every prime $p$ and every $l>1$, and if $f(n)\ll \max\{n^\epsilon,(\log x)^\epsilon\}$ for every $\epsilon>0$, then $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{y}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)$$ for every $\epsilon_0>0$, where $x$, $y$, and $k$ are as they were in Shiu's original paper and $(a,k)=1$. Moreover, we prove that if $f$ is a $Q$-smooth-supported function then there exists a constant $C$ for which $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{y}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)\rho(u)^C,$$ where $u=\frac{\log x}{\log Q}$, $\rho$ is the Dickman-de Bruijn function, and $C$ depends on whether we choose the bound of $f(p^l)\leq A_1^l$ or $f(p^l)\ll (\log\log x)^{l\beta}$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.