Timeline for What is the value of this product (with gamma and zeta function)?
Current License: CC BY-SA 4.0
4 events
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| Jul 27, 2020 at 19:01 | comment | added | Ralph Furman | The functional equation for $\zeta(s)$ gives a formula for $\zeta(n)$, $n=0,-1,-2,\cdots$ in terms of Bernoulli numbers. As you mentioned, this formula just gives 0 for negative even integers. The value you are interested in is equivalent to computing $\zeta(3), \zeta(5), \cdots$, but these are much more mysterious, with no closed formulae. In fact, despite there being a closed formula for even natural numbers, just proving that $\zeta(3)$ is irrational ended up a very subtle argument by Apery, and there are few irrationality results for the others. | |
| Jul 26, 2020 at 6:26 | comment | added | user155294 | @user64494 Thank you. | |
| Jul 26, 2020 at 5:51 | comment | added | user64494 |
The command of Mathematica Limit[1/Zeta[s]/Gamma[s], s -> -2*n, Assumptions -> n \[Element] Integers && n > 0] produces $$ \frac{(2 n)!}{\zeta '(-2 n)}.$$
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| Jul 26, 2020 at 5:32 | history | asked | user155294 | CC BY-SA 4.0 |