{"id":"palomar/PALOMAR-2026-11-03-000001","url":"https://github.com/jkfarland-bot/QuadraticFloorII/tree/86f0b48f6a9384da671762c5da8b0eaf9cf0c807","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-03-000001-v1.json","repo":"https://github.com/jkfarland-bot/QuadraticFloorII","tags":["math.NT","msc:11A15","msc:11B68","msc:11R29","msc:11R16","msc:68V20"],"score":0,"title":"QuadraticFloorII: fibre sums and quartic traces at conductor 8lp","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"Sequel to QuadraticFloor. For an odd prime p, the alternating floor sums S_{A,B,C}(p) = sum over 0 <= n < p of (-1)^floor((A n^2 + B n + C)/p) with admissible (B, C) are averages of first Bernoulli numbers of the odd characters vartheta_{p,psi} = eta_p chi_{-4}^delta psi modulo 8Ap. The development proves that, weighted by psi_0(b)^2 and summed over the units b modulo A, these sums give 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; that for a prime l = 1 (mod 4), a quartic character xi modulo l and an odd prime p != l the real part of…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-03","reactions":{},"formalized":true,"formalizes":[{"title":"Quadratic floor sums II: fibre sums and quartic traces at conductor 8lp, formalized in Lean","identifier":"","relationship":"formalizes"},{"title":"Quadratic Floor Sums, Class Numbers, and the Trace of a Quartic Bernoulli Number","identifier":"","relationship":"formalizes"}]}