Euler Products, the Reflection s ↔ 1−s, and Zeta Values at Integers: A Self-Contained Toolkit for the Zeta Anchors of Classical Physics

Abstract

Many dimensionless prefactors of classical physics are values of the Riemann zeta function at integers: π⁴/15 (Stefan–Boltzmann), π²/720 (Casimir), 2ζ(3) (photon number density), π²/3 (Lorenz number) and ζ(3/2) (Bose–Einstein condensation). This paper collects the mathematics needed to treat these prefactors, with complete proofs or precise references. It is the toolkit for a planned series of companion papers. Its aim is an arithmetic explanation of known constants, not new forces or new predictions, and it makes no physical claim. The toolkit covers the Euler product with an explicit truncation bound, the functional equation via the Jacobi theta function, Bernoulli numbers and the von Staudt–Clausen theorem. It also covers Bernoulli polynomials, Faulhaber sums and the Euler–Maclaurin formula, with a proof that the finite part of a cut-off power sum is ζ(−p) for every rapidly decreasing cut-off. Further topics are Bose and Fermi integrals, the Abel–Plana formula with the one-dimensional and parallel-plate Casimir constants, Poisson summation, the Epstein zeta function and the Chowla–Selberg expansion. The number-theoretic part treats Dirichlet L-functions and the Dedekind zeta functions of Q(i) and Q(√−3), including ζ_K(−1/2) = −|d| ζ_K(3/2)/(16π²), together with the Hurwitz zeta function, polylogarithms, the exact Sommerfeld expansion and a fast evaluation of ζ(3/2). The Wien constants are treated via the Lambert W function; they are transcendental and, as far as tested, unrelated to zeta values. The paper also covers Eisenstein series and the theta series of E8 and of the Leech lattice, where 240 = 2/ζ(−3) and 65520/691 = 2/ζ(−11). Almost everything here is classical. What is new:- the bookkeeping of the whole toolkit;- the exact domain of convergence of the two-branch product ∏_p((1−p^{−s})^{−1} + (1−p^{s−1})^{−1}), which is symmetric under s ↔ 1−s;- its decomposition into pure and mixed terms, which shows that the reflection needs the archimedean factor;- the universal exponent m(4) = 240 = 2/ζ(−3), whose squarefree kernel is 30;- Gentile statistics as the change of a single Euler factor, in contrast to the capped primon gas ζ(s)/ζ(qs);- the identification of the mirror orbits of the prime machine modulo 30 with Frobenius classes;- a working conjecture that separates moments (zeta values) from positions such as median and mode (no zeta values);- a negative result: at the physically relevant arguments the eight residue classes modulo 30 carry less than one percent of log ζ(4). A section on the literature records, item by item, what is classical, what was found in prior work and what was not located. Every result carries exactly one status label: proved, proved (cited), classical physics, numerical, negative, open or conjecture. Negative results and documented test errors are reported. Files: the paper (PDF); its complete LaTeX source (a single file that compiles with pdfLaTeX); and a companion archive with eight Python scripts that reproduce every numerical statement. The archive also contains a runner that executes them all in about two minutes, the logs, a README that maps each script to the statements it checks, and SHA-256 checksums. Tested with Python 3.14.3, mpmath and sympy (numpy for one script). Run `python3 run_all_checks.py` inside the folder `code`; it runs all eight scripts, rewrites their logs and writes `run_all_checks.log`. A full run takes about two minutes on two cores and less than 1 GB of memory. Intended large residuals (convergence demonstrations) and documented test errors are listed in README.md.