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Results tagged with riemann-zeta-function
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user 32389
The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
8
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2
answers
827
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A recurrence relation for $\zeta(2n)$ - reference request
In the textbook https://www.springer.com/gp/book/9783034851688 (Klassische elementare Analysis, by M. Koecher) the following elegant recurrence relation is proved for $\zeta(2n)$ (on p. 157):
$$\left( …
- 16.5k
11
votes
Accepted
A closed form for an integral expressed as a finite series of $\zeta(2k+1)$, $\pi^m$ and a r...
I think the following articles can give a clue:
http://www.tandfonline.com/doi/abs/10.1080/10652460701688125?journalCode=gitr20 (Closed-form evaluation of some families of cotangent and cosecant integ …
- 16.5k
5
votes
Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?
The following results are quoted in http://www.hindawi.com/journals/ijmms/2007/019381/abs/ (Integer Powers of Arcsin, by J.M. Borwein and M. Chamberland):
$$\large I(4,1)=-\frac{3}{2}\mathrm{Li}_5(g^2 …
- 16.5k
17
votes
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationali...
As Frits Beukers writes in http://www.staff.science.uu.nl/~beuke106/caen.pdf "Ironically all generalisations tried so far did not give any new interesting results. Only through a combination of miracl …
- 16.5k