prime number theorem

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prime number theorem

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Noun

prime number theorem (plural prime number theorems)

1. (number theory) The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.
   • 1932, A. E. Ingham, Albert Edward Ingham, The Distribution of Prime Numbers, Cambridge University Press, page 39:

     But we cannot infer from them the equivalence in any sense of these two propositions, since we have used in our proof of the prime number theorem a subsidiary theorem on the order of magnitude of ${\displaystyle \zeta '(s)/\zeta (s)}$.

   • 1974 , Harold M. Edwards, Riemann's Zeta Function, 2001, Dover, page 182,

     The problem of locating the roots ${\displaystyle p}$ of ${\displaystyle \zeta }$, and consequently the problem of estimating the error in the prime number theorem, is closely related to the problem of estimating the growth of ${\displaystyle \zeta }$ in the critical strip ${\displaystyle \{0\leq \mathrm {Re} \ s\leq 1\}}$ as ${\displaystyle \mathrm {Im} \ s\rightarrow \infty }$.

   • 2016, Benjamin Fine, Gerhard Rosenberger, Number Theory: An Introduction via the Density of Primes, 2nd edition, Springer (Birkhäuser), page 145:

     In 1859, Riemann attempted to give a complete proof of the prime number theorem using the zeta function for complex variables s. Although he was not successful in proving the prime number theorem he established many properties of the zeta function and showed that the prime number theorem depended on the zeros of the zeta function.

2. (number theory) Any theorem that concerns the distribution of prime numbers.
   • 1996, Illinois Journal of Mathematics, volume 40, University of Illinois Press, page 245:

     In [6], [11], abstract prime number theorems are proved under a variety of conditions.

Usage notes

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

Translations

See also

• prime-counting function

Further reading

• Prime-counting function on Wikipedia.Wikipedia
• Chebyshev function on Wikipedia.Wikipedia
• Riemann hypothesis on Wikipedia.Wikipedia
• On the Number of Primes Less Than a Given Magnitude on Wikipedia.Wikipedia
• Dirichlet prime number theorem on Wikipedia.Wikipedia
• Prime Number Theorem on Wolfram MathWorld
• Euclidean prime number theorem on Encyclopedia of Mathematics
• Bombieri prime number theorem on Encyclopedia of Mathematics