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On the number of perfect binary quadratic forms. (English) Zbl 1133.11310

Summary: A perfect form is a form \(f=mx^2+ny^2+kxy\) with integral coefficients \((m,n,k)\) such that \(f(\mathbb{Z}^2)\) is a multiplicative semigroup. The growth rate of the number of perfect forms in cubes of increasing side \(L\) in the space of the coefficients is known for small cubes, where all perfect forms are known. A form is perfect if its coefficients belong to the image of a map, \(Q\), from \(\mathbb{Z}^4\) to \(\mathbb{Z}^3\). This property of perfect forms allows us to estimate from below the growth rate of their number for larger values of \(L\). The conjecture that all perfect forms are generated by \(Q\) allows us to reformulate results and conjectures on the numbers of the images \(Q(\mathbb{Z}^4)\) in cubes of side \(L\) in terms of the numbers of perfect forms. In particular, the proportion of perfect elliptic forms in a ball of radius \(R\) should decrease faster than \(R^{-3/4}\) and the proportion of all perfect forms in a ball of radius \(R\) should decrease faster than \(2/\sqrt{R}\).

MSC:

11E16 General binary quadratic forms
11E12 Quadratic forms over global rings and fields
11N99 Multiplicative number theory
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References:

[1] Aicardi F., ”On Trigroups and Semigroups of Binary Quadratic Forms Values and of Their Associated Linear Operators.” (2003) · Zbl 1186.11020
[2] DOI: 10.1007/s00574-003-0001-8 · Zbl 1044.11016
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