Let 
run over all distinct primitive ordered periodic geodesics, and let 
denote the positive length of 
, then the Selberg zeta function is defined as
![Z(s)=product_({p})product_(k=0)^infty[1-e^(-tau(p)(s+k))],](/images/equations/SelbergZetaFunction/NumberedEquation1.svg) |
for 
.
Selberg Zeta Function
See also
Selberg Trace FormulaExplore with Wolfram|Alpha
References
d'Hoker, E. and Phong, D. H. "Multiloop Amplitudes for the Bosonic Polyakov String." Nucl. Phys. B 269, 205-234, 1986.d'Hoker, E. and Phong, D. H. "On Determinants of Laplacians on Riemann Surfaces." Commun. Math. Phys. 104, 537-545, 1986.Fried, D. "Analytic Torsion and Closed Geodesics on Hyperbolic Manifolds." Invent. Math. 84, 523-540, 1986.Selberg, A. "Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series." J. Indian Math. Soc. 20, 47-87, 1956.Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439-465, 1987.Referenced on Wolfram|Alpha
Selberg Zeta FunctionCite this as:
Weisstein, Eric W. "Selberg Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SelbergZetaFunction.html