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 $x$ be a large parameter, and let $f_x(n)$ be a nonnegative multiplicative function parametrized by $x$. We prove that if there exists a small constant $\beta$ and a constant $A_1$ such that $f_x(p^l)\le (A_1\log \log x{)}^{l\beta }$ for every prime $p\le x$ and every $l\ge 1$, and if $f_x(n)\ll \max \{n^{\epsilon },(\log x{)}^{\epsilon }\}$ for every $\epsilon >0$ and every $n\le x$, then $$\sum _{\begin{array}{c}x-y\le n\le x\\ n\equiv a(\mathrm{mod}k)\end{array}}{f}_x(n){\ll }_{{\epsilon }_0}\frac{y}{\varphi (k)(\log x{)}^{1-{\epsilon }_0}}\exp \left(\sum _{\begin{array}{c}p\le x\\ p\nmid k\end{array}}\frac{f(p)}{p}\right)$$ for every ${\epsilon }_0>0$, where $x^{\kappa }\le y\le x$, $k<y^{1-\alpha }$, and $(a,k)=1$ for relatively small $\alpha$ and $\kappa$, and the implied constant does not depend upon the parameter $x$. Moreover, we prove that if $f$ is a $Q$-smooth-supported function then there exists a constant $C$ for which $$\sum _{\begin{array}{c}x-y\le n\le x\\ n\equiv a(\mathrm{mod}k)\end{array}}{f}_x(n){\ll }_{{\epsilon }_0}\frac{y}{\varphi (k)(\log x{)}^{1-{\epsilon }_0}}\exp \left(\sum _{\begin{array}{c}p\le x\\ p\nmid k\end{array}}\frac{f(p)}{p}\right)\rho (u{)}^C,$$ where $u=\frac{\log x}{\log Q}$, $\rho$ is the Dickman-de Bruijn function, and $C$ depends on whether $f_x(p)$ is bounded by a constant or by a slowly-growing function of $x$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.