Timeline for Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 18 at 1:17 | vote | accept | geocalc33 | ||
| Jul 28, 2023 at 14:51 | comment | added | Sidharth Ghoshal | The idea of abrupt changes in continuation at least vaguely reminds of the ideas here: en.m.wikipedia.org/wiki/Stokes_phenomenon. Supposedly high powered summation techniques like Ecalle’s resurgence theory become multivalued because of that, and so the multiple inconsistent branches you are finding might be indicating a similar thing here | |
| Feb 1, 2023 at 2:22 | comment | added | geocalc33 | sent you an email | |
| Jan 31, 2023 at 23:33 | comment | added | Caleb Briggs | @geocalc33 Sure-- what is your preferred method of communication? My own email is [email protected] if this is how you prefer to discuss. | |
| Jan 31, 2023 at 22:57 | comment | added | geocalc33 | would you like to discuss this further outside of mathoverflow? | |
| Jan 31, 2023 at 1:58 | comment | added | Caleb Briggs | @geocalc33 You are probably correct-- for instance, I believe this function should be related in some suitable way to the Laplace transform of $\sum_{n=1}^\infty \frac{1}{n^{s}+n^{-s}}$ (see here: math.stackexchange.com/questions/4622671/…) which obviously shares some resemblance to the zeta function. So, I suspect something like finding the zeroes is very difficult. But I still have some hope that it's possible to find the location of the poles. | |
| Jan 30, 2023 at 23:06 | comment | added | geocalc33 | seems like this is an extremely difficult problem that nobody has ever thought about before | |
| Jan 30, 2023 at 0:48 | comment | added | Caleb Briggs | @geocalc33 If my understanding of this function is correct, the branch cut of $\varphi$ comes from the pole of the $\zeta$ function, and the poles of $\varphi$ come from the contour being forced to change directions due to the angle of $s$. Some more consideration would be needed for me to fully understand the behaviour of the analytical continuation. | |
| Jan 30, 2023 at 0:40 | comment | added | Caleb Briggs | @geocalc33 The piecewise nature means that there can't be a single analytic function that captures the behavior on the real line. However, this doesn't mean that it's not possible for each of the branches to be analytically continued everywhere (for instance, log has a branch cut, but the branches can be continued everywhere minus the cut). When that $n^w$ term is present, the function has a natural boundary-- I think there are poles when $k$ makes $n^{sk}+w=0$, $n>0$ However, at $w=0$ the method breaks down and I'm not sure if the poles get pushed to infinity or if they remain somehow. | |
| Jan 30, 2023 at 0:26 | comment | added | geocalc33 | so the piecewise nature of the function is the problem with the analytic continuation? | |
| Jan 26, 2023 at 13:58 | comment | added | geocalc33 | thanks for this insight. did you ever find anything with respect to the integral $\varphi(s)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{2K_1(2\sqrt{z})}{\sqrt{z}}\bigg(\sum_{n=1}^\infty e^{zn^{-s}}\bigg)~dz$? You had said it agreed with what you had obtained in the other post | |
| Jan 21, 2023 at 23:25 | history | answered | Caleb Briggs | CC BY-SA 4.0 |