Timeline for What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
Current License: CC BY-SA 4.0
25 events
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| Apr 18 at 1:16 | vote | accept | geocalc33 | ||
| Nov 2, 2022 at 0:18 | answer | added | Caleb Briggs | timeline score: 3 | |
| Oct 31, 2022 at 20:49 | history | edited | geocalc33 | CC BY-SA 4.0 |
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| S Oct 17, 2022 at 17:03 | history | bounty ended | CommunityBot | ||
| S Oct 17, 2022 at 17:03 | history | notice removed | CommunityBot | ||
| S Oct 9, 2022 at 15:25 | history | bounty started | geocalc33 | ||
| S Oct 9, 2022 at 15:25 | history | notice added | geocalc33 | Authoritative reference needed | |
| Feb 11, 2022 at 2:21 | history | edited | geocalc33 | CC BY-SA 4.0 |
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| Feb 10, 2022 at 15:49 | history | edited | geocalc33 | CC BY-SA 4.0 |
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| Feb 12, 2021 at 0:21 | comment | added | fedja | Sure. If you need to know more, just state clearly what you are now interested in and link to what has been done already to give everybody a clear picture of where the things stand. Follow-ups are pretty common on this site. | |
| Feb 11, 2021 at 21:24 | comment | added | geocalc33 | @fedja Do you think it would be appropriate to ask a follow up question, about a possible further continuation for Re(z)<0? | |
| Jan 9, 2021 at 3:13 | comment | added | fedja | I see nothing wrong with it (though I haven't checked all the statements about fast decay carefully, I'm pretty sure that @metamorphy will supply all details if you request clarification). At least, it is in good agreement with the notes I linked to: the sum of the series is real analytic for $0<s<1$ and extends to the half-plane $\Re z<1$ but it loses the real analyticity property on the line for $s>1$, so no extension from the real line is possible there. My proof promptly breaks down when $s<1$, metamorphy's series promptly diverges for $s>1$, so everything seems to fit together :-) | |
| Jan 6, 2021 at 21:37 | comment | added | fedja | @geocalc33 I discussed the problem with Misha Sodin and you can find the result of this discussion in the set of handwritten notes at drive.google.com/file/d/191PhSQzr5Q-MbfuMzmuiogJZrk2bh9Ko/… .It is supposed to show that the sum of the series is not real analytic at any point $s>1$. I hope there is no mistake, but the argument is a bit involved. Maybe I'll post it as a proper answer later. The possibility still remains that you can go through the boundary of the half-plane $\Re z<1$ somewhere far from the real line but it is another story. | |
| S Oct 16, 2020 at 13:01 | history | bounty ended | CommunityBot | ||
| S Oct 16, 2020 at 13:01 | history | notice removed | CommunityBot | ||
| S Oct 8, 2020 at 11:59 | history | bounty started | geocalc33 | ||
| S Oct 8, 2020 at 11:59 | history | notice added | geocalc33 | Authoritative reference needed | |
| Sep 20, 2020 at 19:01 | history | edited | geocalc33 | CC BY-SA 4.0 |
added the information that there is a partial answer on the original question
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| Aug 13, 2020 at 7:07 | comment | added | fedja | @JohnJiang You are missing the fact that the exponents $n^s$ not only grow in size but also rotate for complex $s$, so the minus sign becomes quite useless and you get arbitrarily huge terms. I suspect that there is no continuation from the real line anywhere though I cannot offer a proof off hand. | |
| Aug 9, 2020 at 6:22 | comment | added | Gerry Myerson | I dunno, I just thought it was an important piece of information that should be on record here. | |
| Aug 9, 2020 at 5:00 | comment | added | John Jiang | @GerryMyerson Re s > 0 or am I missing something obvious? | |
| Aug 8, 2020 at 23:54 | comment | added | Gerry Myerson | To lay the groundwork for this question, in what region of the complex plane does the given series already converge to an analytic function? | |
| Aug 8, 2020 at 16:37 | comment | added | Wojowu | Do you have a reason to believe this function odes have an analytic continuation? If so, what do you mean with "what is it"? I really doubt there is an expression for this analytic continuation in terms of more classical functions. | |
| Aug 8, 2020 at 15:34 | history | edited | Stopple | CC BY-SA 4.0 |
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| Aug 8, 2020 at 15:17 | history | asked | geocalc33 | CC BY-SA 4.0 |