- #1

- 410

- 12

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

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

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

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter murshid_islam
- Start date

- #1

- 410

- 12

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

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

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

- #2

lurflurf

Homework Helper

- 2,440

- 138

have simple representations in terms of pi and bernulli numbers

for n=3,5,7,9,...

no such expressions have been found

- #3

- 410

- 12

thanks, but what are bernulli numbers?

- #4

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,784

- 18

- #5

shmoe

Science Advisor

Homework Helper

- 1,992

- 1

[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.

Share: