A Note on Large Sums of Divisor-Bounded Multiplicative Functions

TL;DR

This paper establishes that lower bounds on the partial sums of divisor-bounded multiplicative functions imply similar bounds for their products, revealing how the behavior of individual functions influences their multiplicative combinations.

Contribution

It provides a new result linking lower bounds of partial sums of multiplicative functions to those of their products, under divisor-bounded conditions, extending understanding of their sum behaviors.

Findings

Lower bounds on individual functions' sums imply bounds on their products.

The bounds depend on the divisor-bounded parameter .

Explicit relation between bounds of functions and their products is established.

Abstract

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ 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 $|f_j(n)|\leq \tau(n)^\kappa$ for some $\kappa\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq \eta x_j (\log x_j)^{2^\kappa-1}$ for some $x_1, x_2\gg 1$ and $\eta>\max_j\{(\log x_j)^{-1/100}\}$. We then show that there exists $x\geq \min\{x_1, x_2\}^{\xi^2}$ such that $\left|S(x,f_1f_2)\right|\geq \xi x (\log x)^{2^{2\kappa}-1}$, where $\xi=C\eta^{1+2^{\kappa+3}}$ for some absolute constant $C>0$.