scholarly journals On the distribution of the a-values of the Selberg zeta-function associated to finite volume Riemann surfaces

2017 ◽  
Vol 173 ◽  
pp. 64-86 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Raivydas Šimėnas
Keyword(s):  
Zeta Function ◽  
Finite Volume ◽  
Riemann Surfaces ◽  
Selberg Zeta Function

scholarly journals ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

2016 ◽  
Vol 228 ◽  
pp. 21-71 ◽  
Author(s):  
JAY JORGENSON ◽  
LEJLA SMAJLOVIĆ
Keyword(s):  
Riemann Surface ◽  
Vertical Distribution ◽  
Zeta Function ◽  
Finite Volume ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Horizontal Distance ◽  
Distribution Of Zeros ◽  
Selberg Zeta Function ◽  
Hyperbolic Riemann Surface

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

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.

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