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Title: On the summation of the singular series
Author: Arenas, A. (Àngela), 1955-
Keywords: Geometria algebraica
Universitat de Barcelona. Institut de Matemàtica
Issue Date: 1986
Publisher: Universitat de Barcelona
Series/Report no: Mathematics Preprint Series; 45
Abstract: The singular series has great importance in the study of the number of representations of a rational integer as a sum oí integral squares (cf. (3)). As is known (cf. (3), [81) the sum of the singular series is jusi the average number r(n, gen Jk) of representations of a positive integer n by the genus of the identity quadralic form in k variables. Bateman [2) calculated the sum of the singular series in the cases k = 3.4, following Hardy and Hecke methocl~. His results can though more easily be obtained by using Siegel's formula for the evaluation of r(n, gen lk). In this paper we derive in sorne special cases a formula for r(n, gen lk)• from Siegel's formula. which covers those considered by Bateman. We use Gauss-Weber sums to evaluate the 2-adic densities, which, in Siegel's rnethod, causes the main difficulties. Our considerations in the case k • 24 also yield the celebrated Ramanujan's formula about the number of representations of an integer as sum of 24 squares. I wish to thank Professor P. Bayer for her encouragement in doing this paper.
Note: Preprint enviat per a la seva publicació en una revista científica: Manuscripta Math 57, 469–475 (1987). []
Note: Reproducció digital del document original en paper [CRAI Biblioteca de Matemàtiques i Informàtica - Dipòsit Departament CAIXA 31.33]
Appears in Collections:Preprints de Matemàtiques - Mathematics Preprint Series

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