Timeline for Recovering basic information about perfect numbers from a Dirichlet series
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2021 at 19:57 | comment | added | Steven Clark | I believe I can give you an analytic formula for $\sigma _1(n)-2 n$ (e.g. see the analytic formula for $\sigma_0(n)$ at mathoverflow.net/q/395950 which is based on the answer I posted to one of my own questions at mathoverflow.net/q/395266), but I'm not sure it's going to be useful as I believe the simplification involves the arithmetic function $2\ \phi(n)-n$ (see oeis.org/A083254) which is related to $\sigma_1(n)-2 n$. | |
| Aug 23, 2021 at 18:47 | comment | added | Steven Clark | I don't understand the integral. There are two variables $s$ and $x$ as well as the integration variable $t$. Evaluating the integral using the Dirichlet series for $A(s)$ leads to $-\frac{i}{2 T}\sum\limits_{n=1}^\infty(\sigma_1(n)-2 n)\frac{\left(\frac{x}{n}\right)^{s+i T}-\left(\frac{x}{n}\right)^{s-i T}}{\log\left(\frac{x}{n}\right)}$ and I'm not sure the limit of the term $\underset{T\to\infty}{\text{lim}}\left(-\frac{i}{2 T}\frac{\left(\frac{x}{n}\right)^{s+i T}-\left(\frac{x}{n}\right)^{s-i T}}{\log \left(\frac{x}{n}\right)}\right)$ is determinate. | |
| Aug 23, 2021 at 18:46 | comment | added | JoshuaZ | @StevenClark Possible that I've made a mistake here. It should be valid for any s in the range of convergence, with a result in x terms of x. But I think this sort of formula should be standard. Checking back, there's an essentially identical formula in Apostol's "Introduction to Analytic Number" (Theorem 11.17). Possible I've made some sort of basic error here though. | |
| Aug 23, 2021 at 13:31 | history | asked | JoshuaZ | CC BY-SA 4.0 |