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The $r$th moment of the divisor function: an elementary approach (1703.08785v2)
Published 26 Mar 2017 in math.NT
Abstract: Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)r =xC_{r} (\log x){2r-1}+O(x(\log x){2r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right){2r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)r}{p{\alpha}}\right)\right). $$