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

  • #1
murshid_islam
442
17
i know that zeta(2) = pi^2/6 and zeta(4) = pi^4/90

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

Answers and Replies

  • #2
lurflurf
Homework Helper
2,457
155
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
 
  • #3
murshid_islam
442
17
thanks, but what are bernulli numbers?
 
  • #5
shmoe
Science Advisor
Homework Helper
1,994
1
Bernoulli numbers are the coefficients [tex]B_k[/tex] of:

[tex]\frac{t}{e^t-1}=\sum_{k=0}^{\infty}B_k\frac{t^k}{k!}[/tex]

They begin [tex]B_0=1,\ B_1=-1/2,\ B_2=1/6[/tex] 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:

[tex]\zeta(2m)=\frac{-(2\pi i)^{2m}}{(2m)!.2}B_{2m}[/tex]

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

[tex]\zeta(1-2m)=\frac{-B_{2m}}{2m}[/tex]

for m a positive integer.
 

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

Replies
5
Views
440
Replies
2
Views
580
Replies
37
Views
2K
  • Last Post
Replies
5
Views
321
Replies
7
Views
1K
  • Last Post
Replies
5
Views
682
  • Last Post
Replies
5
Views
486
Replies
24
Views
1K
  • Last Post
Replies
2
Views
486
  • Last Post
Replies
7
Views
1K
Top