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 ${\hat{I}}^2<1$ such that $f(p^l)\ll (\log \log x{)}^{l{\hat{I}}^2}$ for every prime $p$ and every $l>1$, and if $f(n)\ll \max \{n^{\hat{I}}\mu ,(\log x{)}^{\hat{I}}\mu \}$ for every $\hat{I}\mu >0$, then $$\sum _{\begin{array}{c}x\le n\le x+y\\ n\equiv a(\mathrm{mod}k)\end{array}}f(n)\ll \frac{y}{\ddot{I}•(k)(\log x{)}^{1-\hat{I}{\mu }_0}}\exp \left(\sum _{\begin{array}{c}p\le x\\ p\nmid k\end{array}}\frac{f(p)}{p}\right)$$ for every $\hat{I}{\mu }_0>0$, 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 $$\sum _{\begin{array}{c}x\le n\le x+y\\ n\equiv a(\mathrm{mod}k)\end{array}}f(n)\ll \frac{y}{\ddot{I}•(k)(\log x{)}^{1-\hat{I}{\mu }_0}}\exp \left(\sum _{\begin{array}{c}p\le x\\ p\nmid k\end{array}}\frac{f(p)}{p}\right)\ddot{I}(u{)}^C,$$ where $u=\frac{\log x}{\log Q}$, $\ddot{I}$ is the Dickman-de Bruijn function, and $C$ depends on whether we choose the bound of $f(p^l)\le A_1^l$ or $f(p^l)\ll (\log \log x{)}^{l{\hat{I}}^2}$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.