Timeline for Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and related integrals
Current License: CC BY-SA 4.0
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| Sep 8, 2020 at 20:42 | comment | added | Iosif Pinelis | @VincentGranville : I have now simplified the expression for $G_1$, using expressions for $f(t+k\pi/2)$, where $f$ is a trigonometric function function and $k$ is an integer. | |
| Sep 8, 2020 at 20:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 18:48 | comment | added | Vincent Granville | @ Iosif: wonderful, thank you! You can simplify the $\cos A - \cos B$ expression in parenthesis: it becomes a product of two sin functions. My guess is that one factor becomes +1 or -1. | |
| Sep 8, 2020 at 18:37 | vote | accept | Vincent Granville | ||
| Sep 8, 2020 at 13:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 12:35 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 12:32 | comment | added | Iosif Pinelis | @GerryMyerson : Thank you for your comment. | |
| Sep 8, 2020 at 12:31 | comment | added | Iosif Pinelis | @VincentGranville : I have now written out the expression of for $G_1$ in terms of the logarithmic and trigonometric functions, as you apparently desired. The case of $G_2$ is quite similar, since $\psi(1+a/2)=\psi(a/2)+2/a$. | |
| Sep 8, 2020 at 12:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 11:22 | comment | added | Vincent Granville | @ Gerry: Thank you. I know you can test trillions of combinations of "simple" numbers to see if one matches $\zeta(3)$ (or any other constant). I reached out to David a while back about a different issue, he is a very nice guy. Same with Andrew Granville. At the end, this is not really a calculus problem (however advanced it might look) but a pure number theory problem. | |
| Sep 8, 2020 at 10:41 | comment | added | Gerry Myerson | There are numerical algorithms for determining whether given constants, like $\zeta(3)$, have expressions in terms of simple functions. Might be worth looking at the work of David Bailey, Jonathan Borwein and the like. | |
| Sep 8, 2020 at 10:27 | comment | added | Vincent Granville | Thanks for the link to the Gauss digamma theorem. | |
| Sep 8, 2020 at 8:52 | comment | added | Vincent Granville | @ Iosif: thanks for your insights. I know these sums can be expressed in terms of digamma functions or their derivatives. But I want a formula based on simpler functions - log or trigonometric functions. Essentially, my goal is to get an exact original formula for $\zeta(3)$, as per my last post (see dsc.news/3i9P1kh). | |
| Sep 8, 2020 at 2:41 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 2:25 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 2:08 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 2:01 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 8, 2020 at 0:49 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |