Prime Number Theorem

The average relative frequency of the prime numbers decreases as the integers become larger. Equivalently, the probability that any given integer is prime becomes smaller for larger integers. The Prime Number Theorem makes this precise:

Thm: let \(p(n)\) denote the number of primes less than \(n\). Then \[\lim_{n\rightarrow \infty} \displaystyle\frac{p(n)}{\frac{n}{\log n}} = 1\]

This tells us that \(\frac{n}{\log n}\) becomes an increasingly accurate estimation of the number of primes less than \(n\). Equivalently, the probability that a randomly chosen number between 1 and \(n\) is prime is approximately \(\frac{1}{\log n}\); or, also equivalently, that the \(m^{th}\) prime will be approximately \(m\log m\).

This theorem is closely related to (and its discovery was inspired by) the Riemann Zeta Function.