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# Contents

## Idea

A prime geodesic is a primitive closed geodesic in a hyperbolic manifold.

## Properties

### Analogy with prime numbers

Prime geodesics in hyperbolic geometry are in several ways analogous to prime numbers and prime ideals in arithmetic geometry.

Notably they appear in the Selberg zeta function (see there) in the same way as prime ideals appear in the Artin L-function, and moreover their asymptotics is controled in an analogous way by this function via the prime geodesic theorem as that of prime numbers is by the prime number theorem and the Riemann zeta function.

A table of analogies is in (Brown 09, p. 9). Some are also mentioned in Wikipedia, prime geodesic – Number theory.

### Prime geodesic theorem

The prime geodesic theorem (e.g. Soundararajan-Young 13) describes the asymptotic distribution of prime geodesics in analogy to the prime number theorem.

• Wikipedia, Prime geodesic

• Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

On the prime geodesic theorem (see there for more):

• K. Soundararajan, Matthew P. Young, The Prime Geodesic Theorem, J. Reine Angew. Math. 676 (2013), 105-120 (arXiv:1011.5486)

• Maki Nakasuji, Prime Geodesic Theorem for Higher-Dimensional Hyperbolic Manifold, Transactions of the AMS, Vol. 358, No. 8