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

Abstract

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

8 pages, revised

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