BandParity: the band-parity sum of a quartic character, two laws modulo 4
For a prime p = 5 (mod 8) let chi be the quartic Dirichlet character with chi(2) = i, with values in the Gaussian integers, and let C(p) = f + i e be the sum of chi(n) (-1)^floor(n^2/p) over 0 , the coset form f = sum over the fourth powers H of s(u) s(u^2) and e = the same sum over 2H, the lag identity sum over 2H of s(u) s(2u) = kappa exactly, the exact coset parities sum_H s = k + 2t and sum_{2H} s = t - 2k where B_{1,chi} = k + i t, and the Coordinate Congruence kappa = (p-1)/4 + k + t (mod 4). It also proves the odd-subgroup congruence: for an odd modulus N >= 3, a subgroup H of (Z/N)^x…
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