Rational zeta series

For m=2, a number of interesting numbers have a simple expression as rational zeta series:

follows by summing the Gauss–Kuzmin distribution. There are also series for π:

being notable because of its fast convergence. This last series follows from the general identity

where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta

Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one has