An asymptotic expansion for the generalised quadratic Gauss sum revisited

2014/03/31 by R. B. Paris, R B Paris, Paris, R B · 1 citation
Mathematics · #11L05 #41A30 #41A60 #41A80 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Theory of Mathematics #math.CA #msc:11L05 #msc:41A30 #msc:41A60 #msc:41A80

paper · pdf · doi:10.48550/arxiv.1403.7973

9 pages, no figures

arxiv created 2014/03/31 · openalex publication_date 2014/03/31 · arxiv updated 2014/04/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

An asymptotic expansion for the generalised quadratic Gauss sum S_N(x,θ)=∑_j=1^N exp (πixj²+2πijθ), where x, θ are real and N is a positive integer, is obtained as x→ 0 and N→∞ such that Nx is finite. The form of this expansion holds for all values of Nx+θ and, in particular, in the neighbourhood of integer values of Nx+θ. A simple bound for the remainder in the expansion is derived. Numerical results are presented to demonstrate the accuracy of the expansion and the sharpness of the bound.

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• On quadratic Gauss sums and variations thereof

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