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Prime Number Theorem (view source)
Revision as of 09:55, November 16, 2008
, 09:55, November 16, 2008raw paste from prime number theorem
The '''prime number theorem''' is the name given to several theorems that provide estimates of the number of primes less than or equal to any given number:
Let π(''n'') be the [[prime counting function]] providing the number of primes less than or equal to ''n'', for any positive number ''n''. The simplest form of the prime number theorem states that
:<math>\pi(n)\sim\frac{n}{\ln n}</math>.
That is, as n tends to infinity, the [[relative error]] between π(''n'') and ''n''/(ln ''n'') tends to zero. This can be expressed using limit notation as
:<math>\lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1</math>
The '''Prime Number Theorem''' is one of the most famous theorems in mathematics. It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.
The '''Prime Number Theorem''' is one of the most famous theorems in mathematics. It states that the number of primes not exceeding n is asymptotic to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e.