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

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.