# New upper bounds for the number of divisors function

## TL;DR

This paper establishes new upper bounds for the divisor counting function $ au(n)$ based on the logarithm of $n$ and its prime factorization, advancing understanding of divisor distribution.

## Contribution

It introduces novel upper bounds for $ au(n)$ that depend on $
$ and its prime factors, improving previous estimates.

## Key findings

- Derived tighter bounds for $ au(n)$ in terms of $
$ and prime factors
- Enhanced understanding of divisor function growth rates
- Provides tools for analyzing divisor distribution in number theory

## Abstract

Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $\tau(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.