3
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ 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 at 16:05
8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.