Zipf distribution (Zeta and HarmonicNumber normalizations)
ZipfDistribution[ρ]
ZipfDistribution[n, ρ]
Mean[ZipfDistribution[ρ]]
→ Piecewise[{{Zeta[ρ]/Zeta[1 + ρ], ρ > 1}}, Infinity]PDF[ZipfDistribution[ρ], k]
→ Piecewise[{{k^(-1 - ρ)/Zeta[1 + ρ], k >= 1}}, 0]CDF[ZipfDistribution[ρ], k]
→ Piecewise[{{HarmonicNumber[Floor[k], 1 + ρ]/Zeta[1 + ρ], k >= 1}}, 0]