Timeline for Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 18 at 1:17 | vote | accept | geocalc33 | ||
| Jan 21, 2023 at 23:25 | answer | added | Caleb Briggs | timeline score: 3 | |
| S Jul 31, 2021 at 19:00 | history | bounty ended | CommunityBot | ||
| S Jul 31, 2021 at 19:00 | history | notice removed | CommunityBot | ||
| S Jul 23, 2021 at 17:50 | history | bounty started | geocalc33 | ||
| S Jul 23, 2021 at 17:50 | history | notice added | geocalc33 | Authoritative reference needed | |
| Jul 23, 2021 at 17:33 | comment | added | geocalc33 | Any updates on this? | |
| S Mar 24, 2021 at 23:08 | history | bounty ended | CommunityBot | ||
| S Mar 24, 2021 at 23:08 | history | notice removed | CommunityBot | ||
| S Mar 16, 2021 at 21:28 | history | bounty started | geocalc33 | ||
| S Mar 16, 2021 at 21:28 | history | notice added | geocalc33 | Draw attention | |
| Mar 4, 2021 at 20:58 | comment | added | GH from MO | @fedja: Thanks for the clarification. I wrote my remark in a rush, not thinking about the oscillatory behavior of $\Re(n^{x+iy})=n^x\cos(y\log n)$ for $y\neq 0$. | |
| Mar 4, 2021 at 16:39 | comment | added | fedja | @GHfromMO The function is clearly analytic for $s>0$ as a real variable No, no, and once more no! When $s>1$, it is $C^\infty$ and even in a quasi-analytic class, but not real analytic. | |
| Mar 4, 2021 at 5:51 | history | edited | geocalc33 | CC BY-SA 4.0 |
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| Feb 12, 2021 at 23:29 | comment | added | GH from MO | Note also that, for a complex function, there are usually several maximal analytic continuations. For example, $\log(z)$ on $\Re(z)$>0 can be continued analytically to $\mathbb{C}\setminus i[0,\infty)$, and also to $\mathbb{C}\setminus -i[0,\infty)$, and these are two distinct maximal analytic continuations. | |
| Feb 12, 2021 at 22:49 | comment | added | fedja | @geocalc33 Yep. Basically at this point the main question is whether the line $\Re z=1$ is the natural boundary for metamorphy's analytic continuation. | |
| Feb 12, 2021 at 22:43 | comment | added | geocalc33 | @fedja maximal analytic continuation of $\varphi(s)$. Better? | |
| Feb 12, 2021 at 22:42 | history | edited | geocalc33 | CC BY-SA 4.0 |
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| Feb 12, 2021 at 14:44 | comment | added | M.G. | @fedja: oh, apologies, I was not aware of the MSE thread. | |
| Feb 12, 2021 at 10:01 | comment | added | fedja | @M.G. In this interpretation it has already been established by metamorphy in the MSE thread (the function can be analytically extended to the half-plane $\Re s<1$ from $(0,1)$ with just one pole at $0$) | |
| Feb 12, 2021 at 3:16 | comment | added | M.G. | @fedja: The way I understand it, he or she is asking if this function can be analytically continued to the half-plane $\Re (s) < 0$. | |
| Feb 12, 2021 at 1:02 | comment | added | fedja | Erm... And what are the function values for $s<0$ that you would like to extend? The series, as written, certainly diverges there, so you surely meant something different from what you wrote :-) | |
| Feb 12, 2021 at 0:43 | history | asked | geocalc33 | CC BY-SA 4.0 |