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

TL;DR

This paper investigates the asymptotic behavior of the divisor function's maximum within small neighborhoods of natural numbers, aiming to understand its oscillation patterns relative to the divisor count of the original number.

Contribution

It introduces a new sequence analyzing the ratio of maximum divisor counts in neighborhoods to the divisor count, providing insights into the divisor function's local oscillations.

Findings

The sequence $T_{n}()$ exhibits specific asymptotic behaviors.

The divisor function shows rapid oscillations in small neighborhoods.

Results suggest potential patterns in divisor distribution near natural numbers.

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.