Pseudo-Totient Entropy Constants: A One-Parameter Family Extending the Mean Totient Entropy Constant

Abstract

Pseudo-Totient Entropy Constants: A One-Parameter Family Extending the Mean Totient Entropy Constant For a positive integer s, let g_s = Id_s * μ denote the arithmetic function obtained by Möbius-inverting the s-th power function, so that g_s(p^j) = p^(j−1)s(p^s−1) on prime powers and Σ_{d|n} g_s(d) = n^s exactly (Gauss's identity of order s); the case s = 1 recovers the Euler totient function φ. Replacing φ by g_s in the totient probability vector (g_s(d)/n^s)_{d|n} and its Shannon entropy H_{g_s}(n), this paper studies the pseudo-totient entropy constant of order s, C^(s) = Σ_p (p−1)/p · Σ_{k≥1} H_{g_s}(p^k)/p^k, the prime-density-weighted average of H_{g_s} over prime powers, with C^(1) = C the mean totient entropy constant of the author's companion paper (Khomyakov, The Mean Totient Entropy, Zenodo, 2026, doi:10.5281/zenodo.21769315). The paper proves a closed form for H_{g_s}(p^k), a single-fraction exact per-prime kernel (p−1)/p · C_p^(s) = κ(p^s)/(p^{s+1}−1) requiring no inner-exponent truncation (κ(P) := P log P − (P−1) log(P−1)), and a Dirichlet-type series representation of C^(s) over prime-zeta values P(r) and logarithmic prime-zeta values S_log(m) at integer arguments only — the logarithmic channel supported exactly on multiples of s+1, with explicit integer-periodic coefficients. Because the exponent shift s is an integer, the natural PSLQ basis for C^(s) is finite and integral, in the same rigor class as the arithmetic-independence conjecture for C itself, in contrast to the fractional two-scale lattice of the Rényi totient entropy constants of the companion paper. Using the exact kernel, the paper certifies C^(2) = 0.5017794161067596928648696598727803088064654424924367749032997…C^(3) = 0.2677326848396394345274211251517620355365725472212831972554014… to 140 decimal digits each, with a rigorous outer-tail bound below 2.7×10⁻³⁰² and survival under precision doubling dps = 140 → 280, cross-checked by an independent direct prime sum with no Möbius machinery, and independently corroborated — unconditionally — by ball-arithmetic (Arb / python-flint) round-off certification. Integer-relation (PSLQ) searches at coefficient height up to 10¹⁵, run over two independent finite bases — a narrow basis with C as anchor at dps = 200/400, and an extended basis at a precision pair derived from a machine-calibrated, environment-validated precision guard — detect no relation for either C^(2) or C^(3) surviving doubling, recorded as a height-bounded arithmetic-independence conjecture. The repository contains the full LaTeX source, all seven companion Python scripts (symbolic verification, numerical cross-checks, certified high-precision evaluation, two independent PSLQ scans, and independent ball-arithmetic round-off certification), the batch orchestrator, the machine-readable calibration record, and the complete verbatim console log of a full reproducible run (7/7 stages OK). Author: Vladimir Khomyakov (Independent Researcher), ORCID 0009-0006-3074-9145. License: MIT. Version 1.0. DOI: 10.5281/zenodo.22074388 (concept, always latest)