Timeline for Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 24, 2022 at 12:58 | history | edited | Fred Hucht | CC BY-SA 4.0 |
Added reference to Abel–Plana formula.
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| Aug 10, 2022 at 2:34 | vote | accept | Sidharth Ghoshal | ||
| Aug 10, 2022 at 2:34 | comment | added | Sidharth Ghoshal | Ah I was simply typing this formula in wrong the whole time. Turns out that $a \ne \frac{1}{a}$. Sorry about that | |
| Aug 9, 2022 at 8:12 | comment | added | Claude Leibovici | This is very interesting for sure. Thanks for providing such an answer. Cheers :-) | |
| Aug 9, 2022 at 7:30 | history | edited | Fred Hucht | CC BY-SA 4.0 |
cosmetics
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| Aug 9, 2022 at 7:12 | comment | added | Fred Hucht | @SidharthGhoshal You are right, the sum in (5) starts from $k=0$, I have fixed that error. I also added a Mathematica snippet that checks (3) against (5). | |
| Aug 9, 2022 at 7:11 | history | edited | Fred Hucht | CC BY-SA 4.0 |
Fixed $k=0$ in (5), added Mathematica code
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| Aug 9, 2022 at 0:26 | comment | added | Sidharth Ghoshal | should that last index of summation on line (5) with the zeta function be from $0$ to $\infty$ assuming $\zeta(0) =-\frac{1}{2}$? | |
| Aug 9, 2022 at 0:19 | vote | accept | Sidharth Ghoshal | ||
| Aug 9, 2022 at 0:23 | |||||
| Aug 8, 2022 at 22:14 | history | edited | Fred Hucht | CC BY-SA 4.0 |
Added remark on square-free numbers
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| Aug 8, 2022 at 21:36 | history | edited | Fred Hucht | CC BY-SA 4.0 |
Added Edit 08.08.22,23:20 CEST
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| Aug 8, 2022 at 14:29 | history | edited | LSpice | CC BY-SA 4.0 |
`\eqref`
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| Aug 8, 2022 at 7:15 | comment | added | Fred Hucht | @JoshuaZ One must be careful about the position of the branch cuts. For integer $a=1,2,\ldots$, (2) might be the correct analytic continuation to |x|>1. Furthermore, (1) and (2) seem to be correct for complex $x$ with $|x|<1$, I have not checked this in detail. | |
| Aug 8, 2022 at 2:00 | comment | added | Sidharth Ghoshal | I was trying to expand the function as a power series of $\log(x)$ and also got the constant term was $\frac{1}{2}$. For some reason when I graph the series I have found so far it does not match at all with the graph of the original function so I'm probably missing something more, im gonna explore the integral you have here and see a log-series of that looks like | |
| Aug 8, 2022 at 0:09 | comment | added | JoshuaZ | Hmm, does your formula 2 allow an analytic continuation then? The original series only converges for $|x|<1$, but your left-hand identity might converge for a bigger region. | |
| Aug 7, 2022 at 20:46 | history | edited | Fred Hucht | CC BY-SA 4.0 |
a > 1 -> a >= 1
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| Aug 7, 2022 at 20:18 | history | edited | Fred Hucht | CC BY-SA 4.0 |
added 8 characters in body
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| Aug 7, 2022 at 20:13 | history | answered | Fred Hucht | CC BY-SA 4.0 |