A Note on Large Sums of Divisor-Bounded Multiplicative Functions

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 $\mathrm{\mid }{f}_j(n)\mathrm{\mid }\le \tau (n{)}^{\kappa }$ for some $\kappa \in \mathbb{N}$, and assume their partial sums satisfy $\mid S(x_j,f_j)\mid \ge \eta x_j(\log x_j{)}^{2^{\kappa }-1}$ for some $x_1,x_2\gg 1$ and $\eta >{\max }_j\{(\log x_j{)}^{-1\mathrm{/}100}\}$. We then show that there exists $x\ge \min \{x_1,x_2{\}}^{{\xi }^2}$ such that $\mid S(x,f_1{f}_2)\mid \ge \xi x(\log x{)}^{2^{2\kappa }-1}$, where $\xi =C{\eta }^{1+2^{\kappa +3}}$ for some absolute constant $C>0$.