A Shiu Theorem for Larger and Smoother Functions

TL;DR

This paper extends Shiu's theorem to larger and smoother multiplicative functions, providing new bounds for sums over arithmetic progressions and applications to divisor functions and smooth numbers.

Contribution

It generalizes Shiu's Brun-Titchmarsh theorem to include larger and smooth-supported functions, introducing new bounds involving the Dickman-de Bruijn function.

Findings

Established bounds for sums of larger multiplicative functions in arithmetic progressions.

Derived bounds for smooth-supported functions involving the Dickman-de Bruijn function.

Applied results to divisor functions and smooth numbers in short intervals.

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.