Prime number theorem

English

Noun

1. Template:Lb The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.
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   • 1974 [Academic Press], Template:W, 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 $\{0\le \mathrm{R}\mathrm{e}s\le 1\}$ as $\mathrm{I}\mathrm{m}s\to \mathrm{\infty }$.

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2. Template:Lb Any theorem that concerns the distribution of prime numbers.
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Usage notes

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

See also

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Further reading

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• Prime Number Theorem on Template:W
• Euclidean prime number theorem on Template:W
• Bombieri prime number theorem on Template:W