Prime number theorem

Statement

Let $\pi(x)$ be the prime-counting function, and let $\ln$ be the natural logarithm. Then $$\lim_{x \to \infty} \frac{\pi(x)}{x / \ln x} = 1.$$

Remarks

We write this as $\pi(x) \sim x / \ln x$. An equivalent form: the $n$-th prime $p_n$ satisfies $p_n \sim n \ln n$. The logarithmic integral $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln t}$ gives a better approximation of $\pi(x)$, with the same limit. For example, $\pi(10^6) = 78498$ and $10^6 / \ln 10^6 \approx 72382$.

History

Legendre and Gauss conjectured the result from tables of primes, at the end of the 18th century. Hadamard and de la Vallée Poussin proved it independently in 1896, with methods of complex analysis. Selberg and Erdős found elementary proofs in 1949.

Explanations

L1 human

Near a large number $x$, about one number in $\ln x$ is prime. For numbers with $100$ digits, $\ln x \approx 230$, so about one number in $230$ is prime. The theorem makes this rough rule precise.