{"id":"palomar/PALOMAR-2026-11-02-000019","url":"https://github.com/jkfarland-bot/MellinDuality/tree/1a5228293f894c9779404d39808b250705e30137","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-02-000019-v1.json","repo":"https://github.com/jkfarland-bot/MellinDuality","tags":["math.NT","msc:11L40","msc:11A15","msc:11B68","msc:68V20"],"score":0,"title":"MellinDuality: Mellin duality for the band-parity companion","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For an odd prime p, an odd character chi modulo p and m >= 0, the multiplier family C(a) = sum over 0 ) has Fourier transform over the characters 2 sum_a conj(psi)(a) C(a) = G(conj psi) G(chi psi^(2m)), with the multiplier constant G(psi) = sum_m psi(m) (-1)^m = 2 (1 - 2 psi(2)) B_(1,psi). The development proves this Mellin duality over any commutative ring, its inversion over the odd complex characters and its form bilinear in first Bernoulli numbers, the half-sum identity, the exponent family and its two collapses. At p = 5 (mod 8), for the quartic character, it proves the coset…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-02","reactions":{},"formalized":true,"formalizes":[{"title":"Mellin duality for the band-parity companion, formalized in Lean","identifier":"","relationship":"formalizes"},{"title":"What the band-parity companion is: Mellin duality, Weil sums, and the Gauss-sum axis","identifier":"","relationship":"formalizes"}]}