nLab prime number theorem

Idea

What is known as the prime number theorem is a description of the asymptotic distribution of prime numbers. It says that the number of primes less than a positive real number $x$ is asymptotic to $x/\log(x)$, where $\log x$ is the natural logarithm.

Proof

See proof of the prime number theorem, which is based on a well-known account by Don Zagier (who in turn was explicating a short proof due to Donald J. Newman) – see the references there.

Related concepts

• An analog of the theorem exists for prime geodesics in hyperbolic manifolds, accordingly called the prime geodesic 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.

References

• Wikipedia, Prime number theorem