A Note on Large Sums of Divisor-Bounded Multiplicative Functions

Given a multiplicative function f, we let S (x, f) =₍ ₗf (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) | (n) ^ for some, and assume their partial sums satisfy |S (xⱼ, fⱼ) | xⱼ (xⱼ) ^2^-1 for some x₁, x₂ 1 and >ⱼ\ (xⱼ) ^{-1/100\}. We then show that there exists x \x₁, x₂\^² such that |S (x, f₁f₂) | x (x) ^2^{2-1}, where =C^1+2^{+3} for some absolute constant C>0.