# The $r$th moment of the divisor function: an elementary approach

## TL;DR

This paper provides an elementary proof for the asymptotic behavior of the sum of the r-th powers of the divisor function, extending understanding of divisor sums with a simplified approach.

## Contribution

It introduces an elementary proof for the asymptotic formula of the sum of divisor function powers, avoiding complex analytic methods used previously.

## Key findings

- Derived the asymptotic formula for sum of divisor function powers
- Explicitly computed the constant $C_r$ involving an infinite product
- Established an elementary approach to a classical number theory problem

## Abstract

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right). $$