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

Abstract

For $\mu >0$ we study an asymptotic behavior of the sequence defined as $$T_n(\mu )=\frac{ma{x}_{1\le m\le n^{\frac{1}{\mu }}}\{\tau (n+m)\}}{\tau (n)},n=1,2,...$$ 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.

arXiv comment15 pages