Quadratic floor sums II: fibre sums and quartic traces at conductor 8ℓp, formalized in Lean
Year: 2026
Type: preprint
Abstract: For a positive odd integer A, an odd prime p not dividing A and integers B, C, the alternating floor sum S_{A,B,C}(p) = sum over 0 <= n < p of (-1)^floor((An^2 + Bn + C)/p) is, for admissible (B, C), an average of first Bernoulli numbers of odd characters vartheta_{p,psi} modulo 8Ap weighted by conj(psi(B))^2; this was formalized earlier. Here we prove what the family of these sums determines. Weighted by psi_0(b)^2 and summed over the units b modulo A, it gives minus the sum of B_{1,vartheta_{p,psi}} over the fibre psi^2 = psi_0^2, which is real when psi_0^4 = 1. For a prime l = 1 (mod 4) and a quartic character xi modulo l, this separates the real part of B_{1,Theta_{l,p}}, for a quartic character Theta_{l,p} of conductor 8lp when p != l, from the quadratic Bernoulli number B_{1,chi_{-8lp}}, as signed combinations of floor sums. We also prove the Euler-factor relation that puts the earlier formulas for the coefficients three and five in class-number shape, the quarter-range formula B_{1,vartheta_{p,psi}}^{(8Ap)} = -sum_{0<x<=2Ap} vartheta_{p,psi}(x), and an elementary proof that both coordinates of B_{1,Theta_{l,p}}^{(8lp)} are even integers for every odd prime p. Version 2 qualifies the conductor claims for the included cases p = l. There the modulus 8lp is not the conductor, which is 8p (so Theta_5 has conductor 40, not 200), and the Bernoulli number is the same at both moduli (Remark 5.3). It also removes the half-range form of the triangular sum, which is a symmetry of the case A = 1 rather than a separate result. Nothing else has changed. The paper restates these results with proofs that follow the formal ones and describes a complete verification in Lean 4 with Mathlib (QuadraticFloorII v1.1).
Author: Farland, Joseph S.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224865898