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2016•Forum MathematicumOpen access

The hyperbolic lattice point problem in conjugacy classes

Dimitrios Chatzakos, Yiannis N. Petridis

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Abstract

Abstract For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces Γ \ ℍ ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound O ⁢ ( X 2 / 3 ) ${O(X^{2/3})}$ , due to Good. For SL 2 ⁢ ( ℤ ) ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

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Abstract For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces Γ \ ℍ ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound O ⁢ ( X 2 / 3 ) ${O(X^{2/3})}$ , due to Good. For SL 2 ⁢ ( ℤ ) ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

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Available abstract

Abstract For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces Γ \ ℍ ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound O ⁢ ( X 2 / 3 ) ${O(X^{2/3})}$ , due to Good. For SL 2 ⁢ ( ℤ ) ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

Key concepts: Mathematics, Conjugacy class, Lattice (music), Quadratic equation, Riemann hypothesis, Pure mathematics, Hyperbolic group, Backslash

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