QuadraticFloor: quadratic floor sums and the trace of a quartic Bernoulli number
For an odd prime p and integers A, B, C, the alternating floor sum S_{A,B,C}(p) is the sum over 0 <= n < p of (-1)^floor((A n^2 + B n + C)/p). The development proves the root-count reduction of S_{A,B,C}(p) to a sum of Legendre symbols over an arithmetic progression when A + B is even, and, for a positive odd A, an odd prime p not dividing A and an admissible pair (B odd, prime to A, 1 <= B^2 - 4AC < 4A), the square-class decomposition: S_{A,B,C}(p) equals (-1)^C minus the average over the Dirichlet characters psi modulo A of conj(psi(B))^2 times the first Bernoulli number, at modulus 8Ap,…
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