values of the Riemann zeta function in terms of Bernoulli numbers
Theorem.
Let k be an even integer and let Bk be the kth Bernoulli number. Let ζ(s) be the Riemann zeta function
. Then:
ζ(k)=2k-1|Bk|πkk! |
Moreover, by using the functional equation (http://planetmath.org/RiemannZetaFunction) , one calculates for all n≥1:
ζ(1-n)=(-1)n+1Bnn |
which shows that ζ(1-n)=0 for n≥3 odd. For k≥2 even, one has:
ζ(1-k)=-Bkk. |
Remark.
The zeroes of the zeta function shown above, ζ(1-n)=0 for n≥3 odd, are usually called the trivial zeroes of the Riemann zeta function, while the non-trivial zeroes are those in the critical strip
.
Title | values of the Riemann zeta function in terms of Bernoulli numbers |
---|---|
Canonical name | ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers |
Date of creation | 2013-03-22 15:12:07 |
Last modified on | 2013-03-22 15:12:07 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 7 |
Author | Mathprof (13753) |
Entry type | Theorem |
Classification | msc 11M99 |
Related topic | BernoulliNumber |
Related topic | ValueOfTheRiemannZetaFunctionAtS2 |