What is known as the prime geodesic theorem is a result that descrives the asymptotic distribution of prime geodesics on a hyperbolic manifold.
The prime geodesic theorem is analogous to the prime number theorem. Just as the zeros of the Riemann zeta function control the error term in the prime number theorem, so the zeros of the Selberg zeta function control the error term in the prime geodesic theorem.
Let be a hyperbolic manifold of dimension . Let
be its fundamental group.
For each element there exists one closed geodesic in representing it. By standard convenient abuse of notation we write also for that geodesic. Write
for the length of the geodesic ;
for what is called its norm;
for the number of primitive elements such that .
The prime geodesic theorem is a statement of the form that this number satisfies
where
is the logarithmic integral function;
are the zeros of the Selberg zeta function in the interval
is some term, estimates of which are the main content of this class of theorems.
The original estimates are due to Atle Selberg, Hejhal, Huber, and Peter Sarnak.
Further developments include
Koyama Prime geodesic theorem for arithmetic compact surfaces, Int Math Res Notices (1998) 1998 (web)
Maki Nakasuji, Prime Geodesic Theorem for Higher-Dimensional Hyperbolic Manifold, Transactions of the AMS, Vol. 358, No. 8, 2006 (JSTOR, pdf)
K. Soundararajan, Matthew P. Young, The Prime Geodesic Theorem,J. Reine Angew. Math. 676 (2013), 105-120 (arXiv:1011.5486)
Discussion for the case of simple geodesics in hyperbolic surfaces contains
Igor Rivin, Simple Curves on Surfaces, Geometriae Dedicata August 2001, Volume 87, Issue 1-3, pp 345-360 (web)
Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Pages 97-125 from Volume 168 (2008), Issue 1 (web)
Last revised on December 10, 2014 at 22:08:23. See the history of this page for a list of all contributions to it.