## 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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