prime number theorem

Noun

 * 1)  The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.
 * 2) * 1974 [Academic Press],, Riemann's Zeta Function, 2001, Dover, page 182,
 * The problem of locating the roots $$p$$ of $$\zeta$$, and consequently the problem of estimating the error in the prime number theorem, is closely related to the problem of estimating the growth of $$\zeta$$ in the critical strip $$\{0\le\mathrm{Re}\ s \le 1\}$$ as $$\mathrm{Im}\ s\rightarrow\infty$$.
 * 1)  Any theorem that concerns the distribution of prime numbers.
 * 1)  Any theorem that concerns the distribution of prime numbers.
 * 1)  Any theorem that concerns the distribution of prime numbers.

Usage notes

 * The number of primes less than n may be expressed as a value of the, $\pi(n)$. Using asymptotic notation, the prime number theorem then becomes $$\pi(n)\sim\frac n{\ln n}$$. A more formal expression is $$\lim_{n\to\infty}\frac{\;\pi(x)\;}{n/\ln n}=1$$.
 * A refinement, which actually gives closer approximations, uses the offset logarithmic integral function (Li): $$\pi(n)\sim\operatorname{Li}(n)=\operatorname{li}(n)-\operatorname{li}(2)=\int_2^n\frac{dt}{\ln t}$$.