values of the Riemann zeta function in terms of Bernoulli numbers

Theorem.

Let $k$ be an even integer and let $B_{k}$ be the $k$th Bernoulli number     . Let $\zeta(s)$ be the Riemann zeta function    . Then:

 $\zeta(k)=\frac{2^{k-1}|B_{k}|\pi^{k}}{k!}$

Moreover, by using the functional equation (http://planetmath.org/RiemannZetaFunction) , one calculates for all $n\geq 1$:

 $\zeta(1-n)=\frac{(-1)^{n+1}B_{n}}{n}$

which shows that $\zeta(1-n)=0$ for $n\geq 3$ odd. For $k\geq 2$ even, one has:

 $\zeta(1-k)=-\frac{B_{k}}{k}.$
Remark.

The zeroes of the zeta function  shown above, $\zeta(1-n)=0$ for $n\geq 3$ 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 ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers 2013-03-22 15:12:07 2013-03-22 15:12:07 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Theorem msc 11M99 BernoulliNumber ValueOfTheRiemannZetaFunctionAtS2