Some asymptotic formulas on generalized divisor functions,II

Let A be an infinite sequence of positive integers a₁ < a₂ <… and put $\text{f}{}_{\mathrm{A}}\text{(x) =}\text{Σ}{}_{\mathrm{a\in A,\; a\le x}}\text{(}\text{1}\text{a}\text{)}$, $\text{D}{}_{\mathrm{A}}\text{(x) =}\text{max}{}_{\mathrm{1\le n\le x}}\text{Σ}{}_{\mathrm{<mtext>a\in A,</mtext><mtext>a</mtext><mtext>n</mtext>}}\text{1}$. In Part I, it was proved that $\text{lim}{}_{\mathrm{x\to +\infty }}\text{sup}\text{D}{}_{\mathrm{A}}\text{(x)}\text{f}{}_{\mathrm{A}}\text{(x)}\text{= +∞}$. In this paper, this theorem is sharpened by estimating D_A(x) in terms of f_A(x). It is shown that lim_x→+∞sup D_A(x) exp(?c₁(logf_A(x))²) = +∞ and that this assertion is not true if c₁ is replaced by a large constant c₂.