Timeline for An elementary number theoretic infinite series
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2010 at 6:08 | vote | accept | Gil Kalai | ||
| Apr 22, 2010 at 6:08 | history | bounty ended | Gil Kalai | ||
| Mar 18, 2010 at 20:20 | comment | added | Victor Miller | @David: thanks for that. But, in some sense your argument is a Tauberian theorem, just with weaker hypotheses and a simpler proof. | |
| Mar 18, 2010 at 18:42 | comment | added | Dror Speiser | @Hansen: That is a, and is the mentioned, Tauberian theorem. | |
| Mar 18, 2010 at 16:16 | comment | added | David Hansen | ("\alpha" should be "a") | |
| Mar 18, 2010 at 16:15 | comment | added | David Hansen | Suppose $D(s)$ is a Dirichlet series with multiplicative coefficients. If $D(s)=\zeta(s)^{a}F(s)$ where $a$ is rational and $F(s)$ is an Euler product which converges without zeros or poles in $\mathrm{Re}(s)> \delta$ for some $\delta < 1$, then you do NOT need a Tauberian theorem to study partial sums of the coefficients of $D(s)$. It is enough to apply a usual contour integration argument, but instead of shifting the whole contour you deform it around the singularity at $s=1$. The partial sums end up being $~c(\alpha)X(\log{X})^{a-1}$ where I forget $c(a)$ (it is simple). | |
| Mar 17, 2010 at 21:52 | comment | added | Dror Speiser | The Tauberian theorem you mention was actually proved by Raikov: "Generalisation of a theorem of Ikehara–Landau (in Russian). Mat. Sbornik, 45 (1938)". If I am not mistaken, the Selberg-Delange method is more about getting a better error term (dating 1947?). Of course the particular instance of Landau was also proved and can be found in: "Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate." Arch. Math. Phys. 13, 305-312, 1908. | |
| Mar 17, 2010 at 21:40 | comment | added | Victor Miller |
@Dror: Yes, I was reminded of the Landau result you quoted too. I think what Delange did was to give a Tauberian theorem that was strong enough to deduce things like the following: $F(s) = \sum_n \frac{a_n}{n^s}$. and $F(s)^r/\zeta(s)$ has a non-zero limit when $s \rightarrow 1$, for some integer $r$ then $sum_{n \le x} a_n \sim C (\log x)^{1/r}$ for some non-zero $C$. Ramanujan's paper only announces the results. It doesn't have any proofs.
|
|
| Mar 17, 2010 at 20:52 | comment | added | Dror Speiser | I might be wrong, but I believe this idea predates Selberg and Delange by almost half a century. This was done already in 1908 by Landau to find the number of numbers represented by a sum of two squares. But since Selberg-Delange has been mentioned quite a few times in this post, I am doubting myself. | |
| Mar 17, 2010 at 20:23 | history | answered | Victor Miller | CC BY-SA 2.5 |