On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

Details On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2 On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

2011-01-15

arxiv.org/abs/1101.2951v2

Mathematics

Number Theory

17 pages

Scientific paper

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.

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