A Note on Large Sums of Divisor-Bounded Multiplicative Functions

Given a multiplicative function f, we let S(x,f)=∑n≤ xf(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 fⱼ, $j=1,2$ be such that |fⱼ(n)|≤ τ(n)^κ for some κ, and assume their partial sums satisfy |S(xⱼ,fⱼ)|≥ η xⱼ (log xⱼ)2^κ-1 for some x₁, x₂ 1 and η>maxⱼ\(log xⱼ)-1/100\. We then show that there exists x≥ min₁, x₂\ξ² such that |S(x,f₁f₂)|≥ ξ x (log x)^22κ-1, where ξ=Cη^1+2κ+3 for some absolute constant $C>0$.