Prime geodesic

Let ${\displaystyle X}$ be a hyperbolic surface. A closed geodesic ${\displaystyle \gamma }$ on ${\displaystyle X}$ is called prime or primitive if it is not an iterate ${\displaystyle \gamma _{0}^{n}}$ of another closed geodesic ${\displaystyle \gamma _{0}}$ with ${\displaystyle n\geq 2}$.^[1] Equivalently, a closed geodesic is prime if it traverses its image exactly once.

Every closed geodesic on a hyperbolic surface is an iterate of a unique prime geodesic.^[1][2]

Note: A geodesic is not considered closed merely because its image crosses itself. To be closed, it must return to its starting point with the same tangent direction; prime geodesics are those closed geodesics for which this first return is not itself a repetition of a shorter closed geodesic.

If ${\displaystyle X=\Gamma \backslash \mathbb {H} }$ is a hyperbolic surface presented as a quotient of the hyperbolic plane by a Fuchsian group ${\displaystyle \Gamma \subset \mathrm {PSL} (2,\mathbb {R} )}$, then each hyperbolic element of ${\displaystyle \Gamma }$ has an invariant geodesic in ${\displaystyle \mathbb {H} }$, called its axis. The projection of this axis to ${\displaystyle X}$ is a closed geodesic.^[2]

With the usual conventions, this gives a correspondence between closed geodesics on ${\displaystyle X}$ and conjugacy classes of hyperbolic elements of ${\displaystyle \Gamma }$.^[2][4] Under this correspondence, prime geodesics are exactly the conjugacy classes represented by primitive hyperbolic elements, that is, elements that are not nontrivial powers of other elements of ${\displaystyle \Gamma }$.^[2]

If ${\displaystyle P}$ is a prime geodesic of length ${\displaystyle \ell (P)}$, its norm is usually defined by

${\displaystyle N(P)=e^{\ell (P)}.}$

This normalization is standard in statements of the prime geodesic theorem and in the definition of the Selberg zeta function.^[3][5]

Let

${\displaystyle \pi _{X}(x)=\#\{P:P{\text{ is a prime geodesic on }}X,\ N(P)\leq x\}.}$

For a finite-area hyperbolic surface, the prime geodesic theorem states that

${\displaystyle \pi _{X}(x)\sim {\frac {x}{\log x}}\qquad (x\to \infty ).}$

Equivalently, the number of prime geodesics of length at most ${\displaystyle L}$ is asymptotic to ${\displaystyle e^{L}/L}$ as ${\displaystyle L\to \infty }$.^[6][3]

This theorem is an analogue of the prime number theorem. More refined versions include error terms, weighted counting functions analogous to the Chebyshev functions, and arithmetic refinements for special surfaces such as the modular surface.^[3][7]

Prime geodesics enter the theory of hyperbolic surfaces through the Selberg zeta function, an Euler product taken over prime geodesics:

${\displaystyle Z_{X}(s)=\prod _{P}\prod _{k=0}^{\infty }\left(1-N(P)^{-s-k}\right),}$

where ${\displaystyle P}$ ranges over the prime geodesics on ${\displaystyle X}$.^[5]

The analytic properties of ${\displaystyle Z_{X}(s)}$ are closely related to spectral data of the Laplace–Beltrami operator on the surface, and the prime geodesic theorem can be proved using this relationship together with the Selberg trace formula.^[6][3]