Uniform Asymptotic Moments of Class Numbers for Real Quadratic Fields with Small Regulator — E8 Intelligence Research

FINDING: Asymptotic formula for the k-th moment of class numbers h(d) of indefinite binary quadratic forms, uniformly in k, over discriminants d with fundamental unit ε_d ≤ x — arXiv:1609.01630 (Kim–Sarnak-type moment problem for real quadratic fields). MATH: - Object: h(d) = class number of indefinite BQF of discriminant d > 0; ε_d = fundamental unit of ℚ(√d). - Regulator: R_d = log ε_d; class number formula (real quadratic): h(d)·R_d = √d · L(1, χ_d) (up to 2/√d normalization conventions). - Moment being studied: M_k(x) = Σ_{d>0, ε_d ≤ x} h(d)^k. - Key asymptotic structure: moments grow like (log x)^{k(k+1)/2} · (polynomial in k) — the exponent k(k+1)/2 is the signature result, matching the symmetry of the k-th moment of L-functions on the central line (conjectured by Keating–Snaith / CFKRS). - Uniformity in k: allows k → ∞, connecting to extreme value statistics of h(d) (large deviations). - Extreme values: max h(d) for ε_d ≤ x behaves like exp(c·(log x)/(log log x)) type — sub-exp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com