A Note on Large Sums of Divisor-Bounded Multiplicative Functions

Abstract Given a multiplicative function f , we let $$S(x,f)=\sum _{n\le x}f(n)$$ S ( x , f ) = ∑ n ≤ 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 $$f_j$$ f j , $$j=1,2$$ j = 1 , 2 be such that $$|f_j(n)|\le \tau (n)^\kappa $$ | f j ( n ) | ≤ τ ( n ) κ for some fixed positive constant $$\kappa $$ κ , and assume their partial sums satisfy $$\left| S(x_j,f_j)\right| \ge \eta x_j (\log x_j)^{2^\kappa -1}$$ S ( x j , f j ) ≥ η x j ( log x j ) 2 κ - 1 for some $$x_1, x_2\gg 1$$ x 1 , x 2 ≫ 1 and $$\eta >\max _j\{(\log x_j)^{-1/100}\}$$ η > max j { ( log x j ) - 1 / 100