How quadratic are the natural numbers? (Q1125391)

For a positive integer \(n\) denote by \(a_n\) and \(b_n\) the two divisors of \(n\) closest to \(\sqrt{n}\) which satisfy \(a_n\leq b_n\). Put \(\kappa(n):=a_n/b_n\) and \(K(x):=\sum_{n\leq x}\kappa(n)\). In the present paper the author proves asymptotic upper and lower bounds for \(K(x)\). Further he investigates \(\kappa^*(n):=\sum_{ab=n, a\leq b}a/b\) and its summatory function \(K^*(x)\). It is shown that \(K^*(x)=x/2+O(\sqrt{x}\log x)\) as \(x\to\infty\), and that \(\kappa^*(n)>D(n)^{1/2-\epsilon}\) holds for infinitely many integers \(n\), where \(D(n):=\max_{m\leq n}\tau(m)\). Reviewer's remark: For improvements of the quoted results see Tenenbaum's review MR 2000h:11104.