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?



Noninteger moments of the divisor function
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.


