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

Abstract

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

8 pages, revised