values of the Riemann zeta function in terms of Bernoulli numbers
Theorem.
Let be an even integer and let be the th Bernoulli number. Let be the Riemann zeta function
. Then:
Moreover, by using the functional equation (http://planetmath.org/RiemannZetaFunction) , one calculates for all :
which shows that for odd. For even, one has:
Remark.
The zeroes of the zeta function shown above, for 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 |
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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 |