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

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

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