Zeta(2) = pi^2/6 and zeta(4) = pi^4/90 what is zeta(3)?

[murshid_islam]

i know that zeta(2) = pi^2/6 and zeta(4) = pi^4/90

what is zeta(3)? can i use Fourier series?

 

[lurflurf]

zeta(n) n=2,4,6,8,...
have simple representations in terms of pi and bernulli numbers
for n=3,5,7,9,...
no such expressions have been found

 

[murshid_islam]

thanks, but what are bernulli numbers?

 

[quasar987]

He probably meant Bernoulli numbers.

http://mathworld.wolfram.com/BernoulliNumber.html

 

[shmoe]

Bernoulli numbers are the coefficients $$B_k$$ of:

$$\frac{t}{e^t-1}=\sum_{k=0}^{\infty}B_k\frac{t^k}{k!}$$

They begin $$B_0=1,\ B_1=-1/2,\ B_2=1/6$$ and can also be defined from the Bernoulli polynomials (I can supply their definition as well, but you can also use google for things like this).

They are related to zeta as:

$$\zeta(2m)=\frac{-(2\pi i)^{2m}}{(2m)!.2}B_{2m}$$

for any nonnegative integer m. The functional equation can then give:

$$\zeta(1-2m)=\frac{-B_{2m}}{2m}$$

for m a positive integer.