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The $r^{th}$ moment of the divisor function, for $r\geq 1,$ is well known to obey

$$ \sum_{n\leq x} \tau(n)^r\sim C_r x (\log x)^{2^r-1} $$. where $C_1=1.$ In a paper by Florian Luca and L. Toth, available at https://arxiv.org/abs/1703.08785, the constant $C_r$ is also given.

What about general $r \in (1,\infty)$? Does this expression still provide a good order of magnitude approximation?

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    $\begingroup$ The Dirichlet series is $\prod_p (1 + 2^r / p^s + \cdots)$, which to first order is “$\zeta(s)^{2^r}$”, which has an “order 2^r pole” at 1. Since each extra pole introduces a factor of log, that’s where the $(\log{x})^{2^r-1}$ comes from (just posting this in case you didn’t already know this heuristic). $\endgroup$
    – alpoge
    Sep 6, 2019 at 16:05

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One can derive a similar asymptotic formula (and even an asymptotic expansion with decreasing powers of $\log x$) for any $r\in\mathbb{C}$. See Ch. II.5 (The Selberg-Delange method) and II.6 (Two arithmetic applications) in Tenenbaum: Introduction to analytic and probabilistic number theory.

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