nLab prime geodesic theorem

Idea

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.

Statement

Let $X$ be a hyperbolic manifold of dimension $d + 1$. Let

$$\Gamma \coloneqq \pi_1(X)$$

be its fundamental group.

For each element $\gamma \in \Gamma$ there exists one closed geodesic in $X$ representing it. By standard convenient abuse of notation we write $\gamma$ also for that geodesic. Write

• $l(\gamma)$ for the length of the geodesic $\gamma$;

• $N(\gamma) \coloneqq e^{l(\gamma)}$ for what is called its norm;

• $\pi_\Gamma(x)$ for the number of primitive elements $\gamma \in \Gamma$ such that $N(\gamma) \leq x$.

The prime geodesic theorem is a statement of the form that this number $\pi_\Gamma(x)$ satisfies

$$\pi_\Gamma(x) = li(x^d) + \underoverset{n = 0}{N}{\sum} li(x^{s_n}) + (error\;term)$$

where

• $li(x) = \underoverset{0}{x}{\int}\frac{d t}{ln t}$ is the logarithmic integral function;

• $\{s_1, \cdots, s_N\}$ are the zeros of the Selberg zeta function in the interval $(d/2,d)$

• $(error\;term)$ is some term, estimates of which are the main content of this class of theorems.

References

The original estimates are due to Atle Selberg, Hejhal, Huber, and Peter Sarnak.

• Peter Sarnak, Prime Geodesic Theorems, Stanford University, 1980

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)