The law of quadratic reciprocity

Abstract

As a limiting case of the transformation formula connecting the theta-function θ ₃ (υ, τ) with $% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aa % baWdbiaadAhacaGGSaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aaba % Wdbiabes8a0baaaiaawIcacaGLPaaacaGGSaaaaa!40B9! {\theta _3}\left( {v, - \frac{1}{\tau }} \right),$, we shall prove a transformation formula for exponential sums (Theorem 1), which yields, as a special case, a reciprocity formula for generalized Gaussian sums (Corollary 2) which, in turn, enables us not only to evaluate Gaussian sums but to prove the law of quadratic reciprocity.