On an Asymptotic Property of Divisor τ-Function

Abstract

In this paper for μ > 0 we study an asymptotic behavior of the sequence defined as T_n(μ) = (τ(n))⁻¹\({\max _{1 \leqslant t \leqslant \left[ {{n^{1/\mu }}} \right]}}\) {τ(n + t)}, where τ(n) denotes the number of natural divisors of given positive integer n. The motivation of this observation is to explore whether τ-function oscillates rapidly.