complex_zeta_function_representations.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 08 August 2017
# https://github.com/trizen

# Simple series representations for the complex zeta function `zeta(r + t*i)`.

var r = 3
var t = 4

var l = 1000

say sum(1..l, {|n|
    1/(n**(r + t.i))
})

say sum(1..l, {|n|
    (cos(log(n) * t) - sin(log(n) * t).i) / n**r
})

say sum(1..l, {|n|
    ((n**(-t.i) + n**(t.i))/2 - ((n**(t.i) - n**(-t.i))/(2i)).i) / n**r
})

say sum(1..l, {|n|
    ((exp(log(n) * (-t.i)) + exp(log(n) * (t.i)))/2 - ((exp(log(n) * (t.i)) - exp(log(n) * (-t.i)))/(2i)).i) / exp(log(n) * r)
})

__END__
0.890555106546179052695024750176771877852377471469-0.00807604576150195980214980067395921761728053036092i
0.890555106546179052695024750176771877852377471469-0.00807604576150195980214980067395921761728053036092i
0.890555106546179052695024750176771877852377471469-0.00807604576150195980214980067395921761728053036092i
0.890555106546179052695024750176771877852377471469-0.00807604576150195980214980067395921761728053036092i