On an asymptotic behavior of the divisor function $\tau(n)$

Abstract

For $\mu>0$ we study an asymptotic behavior of the sequence defined as $$T_{n}(\mu)=\frac{max_{1\leq m \leq {n^{\frac{1}{\mu}}}}\{\tau (n + m)\}}{\tau(n)},\ n=1,2,...$$ where $\tau(n)$ denotes the number of natural divisors of the given $n\in \mathbb{N}$. The motivation of this observation is to explore whether $\tau$ function oscillates rapidly in small neighborhoods of natural numbers.