On the Critical Line Re(s) = 1/2, the Irrationality Measure of {\pi}, and the Automorphic Structure of the Flint Hills Series

TL;DR

This paper develops a comprehensive theory linking the algebraic, analytic, and automorphic properties of special series related to pi, proposing a new perspective on the irrationality measure of pi and the Riemann Hypothesis.

Contribution

It introduces a novel algebraic and automorphic framework connecting the irrationality measure of pi with spectral theory and automorphic forms, leading to new insights into the Flint Hills series.

Findings

Derived explicit formulas for coefficients in Stirling-cosecant decomposition.

Proved the spectral form and functional equations for related zeta functions.

Linked the convergence of specific series to the irrationality measure of pi.

Abstract

We develop, from first principles, a theory connecting the algebra of the Stirling-cosecant decomposition csc^q(z) = sum_k a_{q,k} V_k(z) + E_q(z) with the irrationality measure mu(pi) and the spectral theory of SL(2, Z), leading to an analogue of the Riemann Hypothesis for the Flint Hills auxiliary series. Part I (Algebra) proves the Master Theorem a_{q,k} = (sin z / z)^(-q) * z^(q-k), determines denominators via a von Staudt-Clausen product, identifies boundary coefficients as Wallis ratios a_{2m+1,1} = binomial(2m, m) / 4^m, and establishes recurrence and convolution identities. Part II (Analytic number theory) gives the Hurwitz zeta form H_k(s) = sum_n V_k(n)/n^s, proves sigma(H_k) = k(mu(pi) - 1), and derives F(q,s) converges iff mu(pi) < s/q + 1, recovering the case (2,3). Part III (Automorphic bridges) shows H_k(s) = A_k(s) K_k(1 - s) and K_k(u) = D_k(u) H_k(1 - u), yielding meromorphic continuation with a single pole at s = 1 of residue 2/(pi^k (k - 1)), and induces a functional equation for D(s, rho; pi). Part IV expresses Xi_fl_k(s; pi) = H_k(s) - (2/(pi^k (k - 1))) zeta(s) as a spectral sum over Maass-Hecke forms, implying Xi_fl_k(s; pi) = Xi_fl_k(1 - s; pi) for even k >= 2. The critical line Re(s) = 1/2 arises from SL(2, Z) symmetry. Convergence of F(2,3) is equivalent to Xi_fl_k(3; pi) finite, i.e. mu(pi) < 5/2.