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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). [https://doi.org/10.1007/BF01168672] |
Note: | Reproducció digital del document original en paper [CRAI Biblioteca de Matemàtiques i Informàtica - Dipòsit Departament CAIXA 31.33] |
URI: | http://hdl.handle.net/2445/151628 |
Appears in Collections: | Preprints de Matemàtiques - Mathematics Preprint Series |
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