On large values of the divisor function (Q1265269)

Denote by \(d(n)\) the number of positive divisors of a positive integer \(n\) and let \(D(x)=\max\{d(n):n\leq x\}\). In order to investigate properties of \(D(x)\), an important role is played by the highly composite numbers (h.c. numbers): An integer \(n\) is h.c. if \(d(m)<d(n)\) for all \(m\in \mathbb{N}\), \(m<n\). Hence \(D(x)=d(n_{k(x)})\) if \(n_{k(x)}\) is the greatest h.c. number not exceeding \(x(x>1)\). In the present paper both upper and lower bounds are derived for the number of integers \(n\leq x\) with \(xD(x)\leq d(n)\) \((0<z\leq 1)\).