I was working with fourier series and I found the following recursive formula for the zeta function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{p \\ \pi^{2p}}{2p+1} + \sum_{k=1}^{p} \frac{(2p)! (-1)^k \pi^{2(p-k)}}{(2(p-k)+1)!} \zeta(2k) = 0[/tex]

where [itex]\zeta(k)[/tex] is the Riemann zeta function and p is a positive integer. I know this has already been found, but I was wondering who found it and whether it had a name, and whether it is interesting at all. You can use it to calculate the values of the zeta function pretty easily. For example, for p=1, we have:

[tex]\frac{\pi^2}{3} -\frac{2!}{1!}\zeta(2)=0[/tex]

[tex]\zeta(2)=\frac{\pi^2}{6} [/tex]

for p=2:

[tex]\frac{2\pi^4}{5} -\frac{4! \\ \pi^2}{3!}\zeta(2)+\frac{4!}{1!}\zeta(4)=0[/tex]

[tex]\zeta(4)=\frac{1}{24}(4\pi^2 (\frac{\pi^2}{6})-\frac{2\pi^4}{5}) =\frac{\pi^4}{90}[/tex]

etc.

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# Recursive formula for zeta function of positive even integers

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