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• W2962730002 abstract "For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a cofinite Kleinian group acting on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {H}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we study the prime geodesic theorem on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M equals normal upper Gamma minus double-struck upper H cubed> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mi class=MJX-variant mathvariant=normal>∖<!-- ∖ --></mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>M=Gamma backslash mathbb {H}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal upper Gamma Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E_{Gamma }(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the error in the counting of prime geodesics with length at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=log upper X> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>log X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For the Picard manifold, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma equals normal upper P normal upper S normal upper L left-parenthesis 2 comma double-struck upper Z left-bracket i right-bracket right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>P</mml:mi> <mml:mi mathvariant=normal>S</mml:mi> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Gamma =mathrm {PSL}(2,mathbb {Z}[i])</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we improve the classical bound of Sarnak, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal upper Gamma Baseline left-parenthesis upper X right-parenthesis equals upper O left-parenthesis upper X Superscript 5 slash 3 plus epsilon Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>5</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E_{Gamma }(X)=O(X^{5/3+epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal upper Gamma Baseline left-parenthesis upper X right-parenthesis equals upper O left-parenthesis upper X Superscript 13 slash 8 plus epsilon Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>13</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>8</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E_{Gamma }(X)=O(X^{13/8+epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the process we obtain a mean subconvexity estimate for the Rankin–Selberg <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=application/x-tex>L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function attached to Maass–Hecke cusp forms. We also investigate the second moment of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal upper Gamma Baseline left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E_{Gamma }(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a general cofinite group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma> <mml:semantics> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we show that it is bounded by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis upper X Superscript 16 slash 5 plus epsilon Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>16</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>5</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(X^{16/5+epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2962730002 date "2018-12-28" @default.
• W2962730002 modified "2023-09-24" @default.
• W2962730002 title "Prime geodesic theorem in the 3-dimensional hyperbolic space" @default.
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