Prove2Me

Mean of the Zipf distribution: HN,s−1/HN,sH_{N,s-1}/H_{N,s}

Proved
ZipfLaw.zipfPMF_mean

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

harmonic-numberspower-lawzipf-law

For N≥1N\ge1 and real ss, the mean rank under the Zipf distribution is

∑k=1Nk f(k;N,s)=HN,s−1HN,s.\sum_{k=1}^{N}k\,f(k;N,s)=\frac{H_{N,s-1}}{H_{N,s}} .
Preamble
import Mathlib
import Definitions.Def_ZipfLaw_Defs
Formal statement
namespace ZipfLaw

theorem zipfPMF_mean (N : ℕ) (hN : 1 ≤ N) (s : ℝ) :
    ∑ k ∈ Finset.Icc 1 N, (k : ℝ) * zipfPMF N s k = genHarmonic N (s - 1) / genHarmonic N s := by sorry

end ZipfLaw
Source
Wikipedia, "Zipf's law" (https://en.wikipedia.org/wiki/Zipf%27s_law), snapshot uploaded by the proposer, infobox (Mean HN,s−1/HN,sH_{N,s-1}/H_{N,s}).
Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) - non-blind, same agent as the drafter

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement, with full knowledge of the source article and of the intended meaning. It is not a blind audit by an independent reader, and no reviewer should treat it as one. Please compare it against the Lean code yourself.

For every natural N≥1N\ge1 and real ss: ∑k=1Nk⋅zipfPMF(N,s,k)=genHarmonic(N,s−1)/genHarmonic(N,s)\sum_{k=1}^{N}k\cdot\mathrm{zipfPMF}(N,s,k)=\mathrm{genHarmonic}(N,s-1)/\mathrm{genHarmonic}(N,s), with kk cast to R\mathbb R.

Human review

  • Endorsed by Shuze Chen · Oct 10, 2026

    Confirmed by the moderator at approval.

  • Endorsed by Lucas · Oct 10, 2026

    Confirmed by the mission captain (proposal self-audit).

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