Timeline for Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$
Current License: CC BY-SA 3.0
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| Feb 20, 2017 at 6:38 | comment | added | Lucian | @OlivierOloa: For instance, $~\displaystyle\prod_{n\in\mathbf M}\bigg(1-\frac1{n^4}\bigg)~$ yields $~\dfrac1{\zeta(4)}~$ for $~\mathbf M=\mathbb P,~$ and $~\dfrac1{4\pi\text{ csch }\pi}~$ for $~\mathbf M=\mathbb N_{>1}.$ | |
| Feb 19, 2017 at 21:51 | comment | added | Olivier Oloa | @Lucian Even if I've some difficulty to grasp your definition, I would say keep on working on your intuition. | |
| Feb 18, 2017 at 16:26 | comment | added | Lucian | @OlivierOloa: I've noticed your fondness for various generalizations of $\zeta$ functions, based on infinite series. In your opinion, would a generalization of the same functions, but based on infinite products rather than infinite series, hold any merit ? | |
| Aug 1, 2015 at 0:13 | answer | added | Olivier Oloa | timeline score: 4 | |
| Feb 24, 2015 at 10:49 | vote | accept | Olivier Oloa | ||
| Feb 24, 2015 at 4:12 | answer | added | Antonio Vargas | timeline score: 8 | |
| Feb 23, 2015 at 9:20 | comment | added | Olivier Oloa | @AntonioVargas Yes, I have the same conjecture, I've defined some Laurent-Stieltjes type constants by $$\zeta(s \mid a,b) = \frac{1}{s} + \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\gamma_k(a,b) s^k \tag1$$ near $0$. Then we may extend $ζ(⋅∣a,b)$ to a meromorphic function on $\mathbb{C}$, as is the classic Hurwitz zeta function. The point is to prove $(1)$. Maybe the Euler–Maclaurin formula could be an interesting tool to prove $(1)$. | |
| Feb 22, 2015 at 23:35 | comment | added | Antonio Vargas | Such a series exists, at least. There are coefficients $c_k(a,b)$ such that, for some $\delta > 0$, $$\zeta(s | a,b) = \frac{1}{s} + \sum_{k=0}^{\infty} c_k(a,b) s^k$$ for all $0 < s < \delta$. The series here can be used to analytically continue $\zeta(s|a,b)$ to the annulus $0 < |s| < \delta$. | |
| Feb 22, 2015 at 20:47 | comment | added | Antonio Vargas | Are you looking for a closed form for the coefficients? | |
| Feb 22, 2015 at 15:54 | history | edited | Olivier Oloa | CC BY-SA 3.0 |
A typo corrected
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| Feb 22, 2015 at 12:47 | history | edited | Olivier Oloa |
A tag added
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| Feb 22, 2015 at 2:43 | history | edited | Olivier Oloa | CC BY-SA 3.0 |
edited title
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| Feb 22, 2015 at 2:33 | review | First posts | |||
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| Feb 22, 2015 at 2:29 | history | asked | Olivier Oloa | CC BY-SA 3.0 |