A Note on Large Sums of Divisor-Bounded Multiplicative Functions

Abstract

Given a multiplicative function $f$, we let $S(x,f)={\sum }_{n\le x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products. More precisely, we let $f_j$, $j=1,2$ be such that $|f_j(n)|\le \ddot{I}„(n{)}^{\hat{I}}º$ for some $\hat{I}º\in \mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\ge \hat{I}\cdot x_j(\log x_j{)}^{2^{\hat{I}}º-1}$ for some $x_1,x_2\gg 1$ and $\hat{I}\cdot >{\max }_j\{(\log x_j{)}^{-1/100}\}$. We then show that there exists $x\ge \min \{x_1,x_2{\}}^{\hat{I}{¾}^2}$ such that $\left|S(x,f_1{f}_2)\right|\ge \hat{I}¾x(\log x{)}^{2^{2\hat{I}º}-1}$, where $\hat{I}¾=C\hat{I}{\cdot }^{1+2^{\hat{I}º+3}}$ for some absolute constant $C>0$.

16 pages, project begun at Women in Numbers 6