The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

Artin formalism for Selberg zeta functions of co-finite Kleinian groups

Let $Γ\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $Γ\subset {\rm SL}(2,\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $χ$ of $Γ$ one associates the Selberg zeta function $Z(s;Γ;χ)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tildeΓ$ is a finite index group extension of $Γ$ in ${\rm SL}(2,\mathbb C)$, and $π={\rm Ind}_Γ^{\tildeΓ}χ$ is the induced representation, then $Z(s;Γ;χ)=Z(s;\tildeΓ;π)$. In the second part of the paper we prove by a direct method the analogous identity for…

Kleinian群的首个Kronecker极限公式

A first Kronecker limit formula for Kleinian groups

本文为Kleinian群证明了首个Kronecker极限公式，推广Goldstein方法，利用Eisenstein级数Fourier展开构造Dedekind eta函数类似物，并研究其性质。

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Γ\backslash\mathbb{H}$ with cusps and all finite-dimensional unitary representations $χ$ of $Γ$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Γ,χ)$. Further, if $Γ$ is cofinite and $χ$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the s…

塞尔伯格ζ函数在σ=1处具有二阶矩

Second moments of general Dirichlet series and Selberg zeta functions at $σ= 1$

本文证明有限共形富克斯群的塞尔伯格ζ函数在σ=1处二阶矩存在，借助Broucke和Hilberdink的方法规避了Landau针对一般狄利克雷级数的分离条件。

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

On the prime geodesic theorem for hyperbolic 3-manifolds

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's $\frac{5}{3}$ to $\frac{3}{2}$. At the cost of excluding a set of finite logarithmic measure, the bound is further improved to $\frac{13}{9}$.

Mean square in the prime geodesic theorem

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $Γ$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=Γ\backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_Γ(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $Γ=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_Γ(X)=O(X^{5/3+ε})$, to $E_Γ(X)=O(X^{13/8+ε})$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $L$-function attached to Maass-Hecke cusp …

Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these se…

Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points

We consider the Ruelle zeta function $R(s)$ of a genus $g$ hyperbolic Riemann surface with $n$ punctures and $v$ ramification points. $R(s)$ is equal to $Z(s)/Z(s+1)$, where $Z(s)$ is the Selberg zeta function. The main result of this work is the leading behavior of $R(s)$ at $s=0$. If $n_0$ is the order of the determinant of the scattering matrix $φ(s)$ at $s=0$, we find that \begin{align*} \lim_{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2π)^{2g-2+n }\tildeφ(0)^{-1} \prod_{j=1}^v m_j, \end{align*}which says that $R(s)$ has order $2g-2+n-n_0$ at $s=0$, and its leading coe…

尖点处典范 Green 函数的界

Bounds for canonical Green's functions at cusps

本文证明了余有限 Fuchsian 子群在尖点处的典范 Green 函数的界，并以此推导出与 $Γ_0(N)$ 相关的典范 Green 函数的渐近表达式。

The transfer operator for the Hecke triangle groups

In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups G_q, q=3,4,..., which are non-arithmetic for q \not= 3,4,6. For this we make use of a Poincare map for the geodesic flow on the corresponding Hecke surfaces which has been constructed in arXiv:0801.3951 and which is closely related to the natural extension of the generating map for the so called Hurwitz-Nakada continued fractions. We derive simple functional equations for the eigenfunctions of the transfer operator which for eigenvalues rho =1 are expected…

Local average in hyperbolic lattice point counting

The hyperbolic lattice point problem asks to estimate the size of the orbit $Γz$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $Γ$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $\hbox{PSL}_2(Z)$. The result is that the error term can be improved to $O(X^{7/12+ε})$. The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimat…