The Mean Totient Entropy: A New Number-Theoretic Constant and its Dirichlet-Type Representation over the Primes

Abstract

The Mean Totient Entropy: A New Number-Theoretic Constant and its Dirichlet-Type Representation over the Primes For a positive integer n, the divisors carry a canonical probability vector (φ(d)/n) over d | n (Gauss: Σ_{d|n} φ(d) = n). Its Shannon entropy H(n) = −Σ_{d|n} (φ(d)/n)·log(φ(d)/n) is the totient entropy. This work introduces the mean totient entropy constant C = Σ_p (p−1)/p · Σ_{k≥1} H(p^k)/p^k, the prime-density-weighted average of H over prime powers. Results are organized in three rigor tiers. (i) Theorems: a closed form for H(p^k) and a Dirichlet-type series in the prime zeta P(s) and logarithmic prime zeta S_log(s) = −P′(s), symbolically verified. (ii) Computation: C = 0.98876258583724770489978…, to 120+ digits with 40 rigorously certified (error < 10^{−46}); OEIS A396759. (iii) Conjecture: PSLQ searches (130/200/400 digits) detect no linear relation over {1, log 2, log 3, γ, ζ(3), P(2), P(3), P′(2), P′(3)}. Version history. v2.0: companion constant D = A397097 − C, certified to 600 digits, with a Glaisher–Kinkelin decomposition and PSLQ independence tests. v3.0: the mean value over the integers, C′ = C + Δ (C′ = 1.33542510464…). v4.0 (this version): the Rényi totient entropy constants C_α (resolving Open Problem 3) — closed form, convergence for all α > 0, confluence to C as α → 1, and a two-scale Dirichlet representation; C_{1/2}, C_2, C_3 certified to 30+ digits with a rigorous far-tail majorant and interval arithmetic, plus a height-bounded PSLQ non-relation gate with a machine-validated precision guard. A complete reproducibility archive (mpmath, SymPy) with batch orchestrators is included. Acknowledgements: The author thanks Vaclav Kotesovec for extending OEIS A396759 and for the observation leading to the companion constant OEIS A397097 (joint work). Author: Vladimir Khomyakov (Independent Researcher), ORCID 0009-0006-3074-9145. License: MIT. Version 4.0. DOI: 10.5281/zenodo.20533024 (concept, always latest)