Some asymptotic formulas on generalized divisor functions, II

AbstractLet A be an infinite sequence of positive integers a1 < a2 <… and put fA(x) = Σa∈A, a≤x(1a), DA(x) = max1≤n≤xΣa∈A,an1. In Part I, it was proved that limx→+∞supDA(x)fA(x) = +∞. In this paper, this theorem is sharpened by estimating DA(x) in terms of fA(x). It is shown that limx→+∞sup DA(x) exp(−c1(logfA(x))2) = +∞ and that this assertion is not true if c1 is replaced by a large constant c2