2019/10/20 by Zavosh Amir-Khosravi, Amir-Khosravi, Zavosh
Mathematics · #11E45 #11F46 (Secondary) #11L05 (Primary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1910.08932
openalex publication_date 2019/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using a Möbius transform, and show they have a functional equation, Euler product factorization, and roots only on the critical line. In the second part, we prove a general reciprocity law for sums of exponentials of rational quadratic forms in any number of variables, using the transformation formula of the Riemann theta function on the Siegel upper half-space.