Noninteger moments of the divisor function

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?

• 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). – alpoge Sep 6 '19 at 16:05

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.