The Completed Zeta Function Is Smallest Just Beyond the Dimension of Largest Spheres (M2)

The surface area |Sn−1| = 2πn/2/Γ(n/2) of the unit sphere, read as a function of a continuous dimension n, is largest at the classical constant nmax = 7.2569464048… This note shows that the completed Riemann zeta function Λ(s) = π−s/2Γ(s/2)ζ(s) is smallest just beyond it: on (1, ∞) it has exactly one minimum, at smin = 7.3170702166268495123…, and always smin > nmax. The reason fits in one line. Since Λ(s) = 2ζ(s)/|Ss−1|, the minimum of Λ is the maximum of the sphere area, pushed to the right by the decreasing factor ζ(s). In probabilistic language the push is twice the mean of log n under the zeta distribution. MAIN RESULTS 1. Theorem 3 (proved). Λ is strictly log-convex on (1, ∞), has a unique minimum smin, characterised by ψ(s/2) − log π = −2ζ′(s)/ζ(s), and smin > nmax. By the functional equation there is a mirror minimum at 1 − smin = −6.3170702… 2. Proposition 4 (proved inequality, numerical evaluation). A two-sided bound for the gap δ = smin − nmax that needs only a few function values; iterated, it gives 0.0597 < δ < 0.0632. The exact gap is δ = 0.0601238117662727… 3. Proposition 5 (proved; table numerical). Adding the primes 2, 3, 5, … to the Euler product one at a time moves the minimum strictly to the right, from nmax towards smin. The prime 2 alone accounts for 92.5% of the shift. 4. Theorem 6 (proved). For every c > 0 the minimum of c−sΛ(s) lies just beyond the dimension in which the sphere of radius c has largest area. For c = 2 (the Casimir plate normalisation) the gap is 4.76·10−7, for c = 3 it is 3.73·10−16. The note includes three figures, key takeaways, implications for mathematicians (archimedean versus finite Euler factors; the same argument applies to completed Dedekind zeta functions) and for physicists (Casimir coefficients on a circle and between plates as functions of the dimension), and a table separating proved, numerical and open statements. The real minimum of Λ and the inequality smin > nmax were first obtained in Krause, P1 v2 (doi:10.5281/zenodo.23930220); this note gives a self-contained account with complete proofs and new quantitative results. FILES M2_Completed_Zeta_Minimum_EN.pdf: the paper (9 pages, 3 figures).M2_overleaf_source.zip: LaTeX source and figures, compiles with pdfLaTeX on Overleaf.M2_companion_code.zip: check_m2.py (36 checks), make_figures.py, reference logs at 50 digits (seed 1) and 100 digits (seed 2), README, SHA-256 checksums. Requires Python 3 and mpmath.