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On representation of an integer as the sum of three squares and ternary quadratic forms with the discriminants \(p^2\), \(16p^2\). (English) Zbl 1245.11049

Let \(s(n)\) denote the number of representations of the positive integer \(n\) as the sum of three squares. The authors derive an identity for \(s(p^2n)-ps(n)\), where \(p\) is an odd prime. In order to describe the form of the identity, let \(\mathcal G\) denote a genus of positive definite integral ternary quadratic forms, and let \(r_{\mathcal G}(n) = \sum_{f \in \mathcal G}\frac{R_f(n)}{|\operatorname{Aut}(f)|}\), where \(|\operatorname{Aut}(f)|\) denotes the number of integral automorphs of \(f\), \(R_f(n)\) denotes the number of representations of \(n\) by \(f\), and the sum extends over a set of representatives for the equivalence classes in \(\mathcal G\). The Siegel-Weil formula gives \(r_{\mathcal G}(n)\) as an expression involving the mass of the genus and a product of local representation densities. Here the authors prove that \[ s(p^2n)-ps(n) = 48r_{TG_{1,p}}(n) - 96r_{TG_{2,p}}(n), \] where \(TG_{1,p}\) denotes the genus of ternary forms of discriminant \(p^2\) and \(TG_{2,p}\) is a particular genus of ternary forms of discriminant \(16p^2\). This identity is obtained by defining a mass preserving bijection between the two genera and explicitly computing the factors appearing in the Siegel-Weil formula for each of the genera.

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11E12 Quadratic forms over global rings and fields
11E08 Quadratic forms over local rings and fields
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Online Encyclopedia of Integer Sequences:

Numbers that are the sum of three nonzero squares.

References:

[1] Bachmann, P., Die Arithmetik von Quadratischen Formen (1898), Teubner: Teubner Leipzig · JFM 29.0142.05
[2] Bateman, P. T., On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc., 71, 70-101 (1951) · Zbl 0043.04603
[3] A. Berkovich, On representation of an integer by \(X^2 + Y^2 + Z^2\) arXiv:0907.1725v3; A. Berkovich, On representation of an integer by \(X^2 + Y^2 + Z^2\) arXiv:0907.1725v3
[4] Berkovich, A.; Jagy, W. C., Ternary quadratic forms, modular equations and certain positivity conjectures, (Alladi, K.; Klauder, J.; Rao, C. R., The Legacy of Alladi Ramanakrishnan in the Mathematical Sciences (2010), Springer: Springer New York), 211-241 · Zbl 1322.11028
[5] Berkovich, A.; Hanke, J.; Jagy, W. C., A proof of the \(S\)-genus identities for ternary quadratic forms · Zbl 1341.11014
[6] Cassels, J. W.S., Rational Quadratic Forms (2008), Dover · Zbl 0496.10008
[7] Chan, W. K.; Earnest, A. G., Discriminant bounds for spinor regular ternary quadratic lattices, J. Lond. Math. Soc., 69, 545-561 (2004) · Zbl 1084.11011
[8] Chan, W. K.; Oh, B. K., Finiteness theorems for positive definite \(n\)-regular quadratic forms, Trans. Amer. Math. Soc., 355, 2385-2396 (2003) · Zbl 1026.11046
[9] Conway, J. H.; Sloane, N. J.A., Low-dimensional lattices. IV. The mass formula, Proc. R. Soc. Lond. Ser. A, 419, 259-286 (1988) · Zbl 0655.10023
[10] Dickson, L. E., Modern Elementary Theory of Numbers (1939), The University of Chicago Press · Zbl 0027.29502
[11] Jones, B. W., The Arithmetic Theory of Quadratic Forms (1950), Mathematical Association of America · Zbl 0041.17505
[12] Landau, E., Vorlesungen über Zahlentheorie (1927), S. Hirzel: S. Hirzel Leipzig · JFM 53.0123.17
[13] Lehman, J. L., Levels of positive definite ternary quadratic forms, Math. Comp., 58, 399-417 (1992) · Zbl 0754.11011
[14] Siegel, C. L., Lectures on the Analytical Theory of Quadratic Forms. Notes by Morgan Ward (1963), Buchhandlung Robert Peppmüller: Buchhandlung Robert Peppmüller Göttingen · Zbl 0115.04401
[15] Watson, G. L., Transformations of a quadratic form which do not increase the class-number, Proc. Lond. Math. Soc., 12, 577-587 (1962) · Zbl 0107.26901
[16] Watson, G. L., Integral Quadratic Forms (1970), Cambridge University Press · Zbl 0090.03103
[17] Yang, T., An explicit formula for local densities of quadratic forms, J. Number Theory, 72, 309-356 (1998) · Zbl 0930.11021
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