scholarly journals Selberg zeta-function associated to compact Riemann surface is prime

2021 ◽  
pp. 213-218
Author(s):  
Ramūnas Garunkštis
Keyword(s):  
Riemann Surface ◽  
Zeta Function ◽  
Compact Riemann Surface ◽  
Selberg Zeta Function

scholarly journals An inequality for the Selberg zeta-function, associated to the compact Riemann surface

2019 ◽  
Vol 27 (3) ◽  
pp. 37-44
Author(s):  
Igoris Belovas
Keyword(s):  
Riemann Surface ◽  
Zeta Function ◽  
Compact Riemann Surface ◽  
Selberg Zeta Function ◽  
The Absolute

AbstractWe consider the absolute values of the Selberg zeta-function, associated to the compact Riemann surface, at places symmetric with respect to the line ℛ(s) = 1/2. We prove an inequality for the Selberg zeta-function, extending the result of R. Garunkštis and A. Grigutis.

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.

scholarly journals Normal modes in thermal AdS via the Selberg zeta function

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Victoria Martin ◽  
Andrew Svesko
Keyword(s):  
Zeta Function ◽  
Heat Kernel ◽  
Normal Mode ◽  
Kernel Method ◽  
Normal Modes ◽  
Quasinormal Mode ◽  
Partition Functions ◽  
Selberg Zeta Function ◽  
Exact Agreement ◽  
Spin Fields

The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin ss fields on hyperbolic quotient spacetimes \mathbb{H}^{3}/\mathbb{Z}ℍ3/ℤ are related via the Selberg zeta function. We extend that analysis to thermal \text{AdS}_{2n+1}AdS2n+1 backgrounds, with quotient structure \mathbb{H}^{2n+1}/\mathbb{Z}ℍ2n+1/ℤ. Specifically, we demonstrate the zeros of the Selberg function encode the normal mode frequencies of spin fields upon removal of non-square-integrable modes. With this information we construct the 1-loop partition functions for symmetric transverse traceless tensors in terms of the Selberg zeta function and find exact agreement with the heat kernel method.

Sign in / Sign up

Export Citation Format

Share Document