An Approach to the Selberg Trace Formula via the Selberg Zeta-Function

1987 ◽  
Author(s):  
Jürgen Fischer
Keyword(s):  
Zeta Function ◽  
Trace Formula ◽  
Selberg Trace Formula ◽  
Selberg Zeta Function

scholarly journals The Selberg trace formula for modular correspondences

1990 ◽  
Vol 117 ◽  
pp. 93-123
Author(s):  
Shigeki Akiyama ◽  
Yoshio Tanigawa
Keyword(s):  
Zeta Function ◽  
Kernel Function ◽  
Trace Formula ◽  
Conjugacy Classes ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Selberg Trace Formula ◽  
Weight One ◽  
Space Of Cusp Forms

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

scholarly journals Selberg's trace formula on thek-regular tree and applications

2003 ◽  
Vol 2003 (8) ◽  
pp. 501-526 ◽  
Author(s):  
Audrey Terras ◽  
Dorothy Wallace
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Trace Formula ◽  
Geodesic Flow ◽  
Prime Number Theorem ◽  
Selberg Trace Formula ◽  
Regular Tree ◽  
Graph Theoretic ◽  
Upper Half Plane ◽  
The Laplace Operator

We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications. This paper includes a review of the basic geometry of ak-regular treeΞ(symmetry group, geodesics, horocycles, and the analogue of the Laplace operator). A detailed discussion of the spherical functions is given. The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms). Two versions of the pretrace formula for a finite connectedk-regular graphX≅Γ\Ξare given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of lengthrinX(without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow onX(which is analogous to a result of Wallace for a compact quotient of the Poincaré upper half plane). Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function ofX, leading to a graph theoretic analogue of the prime number theorem.

From classical periodic orbits to the quantization of chaos

1992 ◽  
Vol 437 (1901) ◽  
pp. 693-714 ◽  
Keyword(s):  
Periodic Orbit ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Periodic Orbits ◽  
Trace Formula ◽  
Riemann Surfaces ◽  
Chaotic Systems ◽  
Critical Line ◽  
Selberg Zeta Function ◽  
Periodic Orbit Theory

We present a quantitative analysis of Selberg’s trace formula viewed as an exact version of Gutzwiller’s semiclassical periodic-orbit theory for the quantization of classically chaotic systems. Two main applications of the trace formula are discussed in detail, (i) The periodic-orbit sum rules giving a smoothing of the quantal energy-level density. (ii) The Selberg zeta function as a prototype of a dynamical zeta function defined as an Euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a Dirichlet series. It is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an ‘orbit-selection function’. Numerical results are presented for the free motion of a point particle on compact Riemann surfaces (Hadamard-Gutzwiller model), which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos. Our results demonstrate clearly the crucial role played by the long periodic orbits. An exact rule for quantizing chaos is derived for such systems where the Dirichlet series representing the Selberg zeta function converges on the critical line. Explicit formulae are given for the computation of the abscissae of absolute and conditional convergence, respectively, of these dynamical Dirichlet series. For the two Riemann surfaces considered, it turns out that one can cross the entropy barrier, but that the critical line cannot be reached by a convergent Dirichlet series. It would seem that this is the main reason why the Riemann-Siegel lookalike formula, recently conjectured by M. V. Berry and J. P. Keating, fails in generating the lower-lying quantal energies for these strongly chaotic systems.

scholarly journals A Selberg trace formula for hypercomplex analytic cusp forms

2015 ◽  
Vol 148 ◽  
pp. 398-428 ◽  
Author(s):  
D. Grob ◽  
R.S. Kraußhar
Keyword(s):  
Trace Formula ◽  
Cusp Forms ◽  
Selberg Trace Formula
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