Editing Prime geodesic (section)

==Applications of prime geodesics==
The importance of prime geodesics comes from their relationship to other branches of mathematics, especially [[dynamical systems]], [[ergodic theory]], and [[number theory]], as well as [[Riemann surface]]s themselves. These applications often overlap among several different research fields.

===Dynamical systems and ergodic theory===
In dynamical systems, the [[closed geodesic]]s represent the [[Periodic function|periodic]] [[Group action (mathematics)|orbits]] of the [[Geodesic#Geodesic flow|geodesic flow]].

===Number theory===
In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the [[prime number theorem]]. To be specific, we let π(''x'') denote the number of closed geodesics whose norm (a function related to length) is less than or equal to ''x''; then π(''x'') ~ ''x''/ln(''x''). This result is usually credited to [[Atle Selberg]]. In his 1970 Ph.D. thesis, [[Grigory Margulis]] proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, [[Peter Sarnak]] proved an analogue of [[Chebotarev's density theorem]].

There are other similarities to number theory &mdash; error estimates are improved upon, in much the same way that error estimates of the prime number theorem are improved upon. Also, there is a [[Selberg zeta function]] which is formally similar to the usual [[Riemann zeta function]] and shares many of its properties.

Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that [[prime ideal]]s in the [[ring of integers]] of a [[number field]] can be split (factored) in a [[Galois extension]]. See [[Covering map]] and [[Splitting of prime ideals in Galois extensions]] for more details.

===Riemann surface theory===
Closed geodesics have been used to study Riemann surfaces; indeed, one of [[Riemann]]'s original definitions of the [[genus (mathematics)|genus]] of a surface was in terms of simple closed curves. Closed geodesics have been instrumental in studying the [[eigenvalue]]s of [[Laplacian]] [[operator (mathematics)|operator]]s, [[arithmetic group|arithmetic Fuchsian group]]s, and [[Teichmüller space]]s.