Selberg Zeta Function

Let  run over all distinct primitive ordered periodic geodesics, and let  denote the positive length of , then the Selberg zeta function is defined as

for .

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.