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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Recursive formula for zeta function of positive even integers

**Physics Forums | Science Articles, Homework Help, Discussion**