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The $r$th moment of the divisor function: an elementary approach
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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, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right). $$
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Moments of the shifted prime divisor function
For every k≥2, the k-th moment of the shifted-prime divisor function is asymptotically of order x(log x)^(2^k-k-1), confirming the Fan-Pomerance conjecture.
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