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
.