krzysztof_reformulated_zeta_function.sf

#!/usr/bin/ruby

# Formula due to Krzysztof Maslanka

func Ak(k) {
    sum(0..k, {|j|
        (-1)**j * binomial(k, j) * (2*j + 1) * zeta(2*j + 2)
    })
}

func krzysztof_zeta(s, r=100) {
    sum(0..r, {|k|
        (gamma(k + 1 - s/2) / gamma(1 - s/2)) * (Ak(k) / k!)
    }) / (s - 1)
}

say krzysztof_zeta(3)       #=> 1.2020569022705898699637409727804911671219286162
say krzysztof_zeta(5)       #=> 1.0369277551507417486201175962997774385468699839