• Komaravolu Chandrasekharan
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 281)

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.

Keywords

Transformation Formula Quadratic Residue Analytic Number Theory Chapter Versus Algebraic Number Field
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