zeta_2n.sf

#!/usr/bin/ruby

#
## Closed-form for Zeta(2n)
#

# See also: https://en.wikipedia.org/wiki/Riemann_zeta_function

func bernoulli_number(n) is cached {

    n.is_one && return 1/2
    n.is_odd && return 0

    ^n -> reduce({|a, k| a - (__FUNC__(k) * binomial(n, k) / (n - k + 1))}, 1)
}

func zeta_2n(Number n {.is_even}) {
    ((-1)**(n/2 + 1) * 2**(n - 1) * Num.pi**n * bernoulli_number(n)) / n!
}

for n in (1 .. 10) {
    say "zeta(#{2*n}) = #{zeta_2n(2*n)}"
}