Gamma-Cocycle Richardson Filtering: A Rigorous Weight-Extraction Approach to Apéry's Constant
Authors/Creators
Description
Apéry's constant
is conventionally studied through series, integrals, polylogarithms, modular forms, differential equations, or motivic periods. This paper develops a rigorous coefficient-extraction perspective based on the Gamma function. Let $L$ denote the unique holomorphic logarithm of $\Gamma(1+z)$ on the unit disk $\vert{}z\vert{} < 1$ normalized by $L(0) = 0$. From the local additive coboundary
we take the odd part of the diagonal specialization. This removes all even zeta values and yields a holomorphic normalized generating function
whose higher coefficients involve only $\zeta(5), \zeta(7), \dots$
We then apply the dyadic operators
These operators are standard Richardson extrapolation steps specialized to an even-power error expansion: they fix the constant term and annihilate the monomial $t^{2m}$. Successive application therefore removes the first $M$ higher odd-zeta contributions exactly.
A complete theorem is proved giving the exact surviving expansion, alternating upper and lower bounds, an explicit error estimate, and a finite linear-combination formula in terms of the values $F_0(t), F_0(t/2), \dots, F_0(t/2^M)$. For rational $0 < t < 1/2$, every finite-stage approximant is a finite rational linear combination of real logarithms of Gamma values at positive rational arguments. The proof uses only absolutely convergent series in their exact domains of convergence and finite linear operations; no unproved functional identity, branch-dependent polylogarithmic reduction, divergent symbolic manipulation, or uncontrolled interchange of limits is used.
The resulting statement is a rigorous Gamma-function realization of Richardson coefficient extraction for the $\zeta(3)$ coefficient. The work does not claim that Richardson extrapolation itself is new, nor does it claim a finite closed form for $\zeta(3)$; the object of study is the specific Gamma-derived generating function, its exact sign structure, and the finite rational-Gamma approximants produced by this filtering process.