Difference between revisions of "Prime number theorem"

From Conservapedia
Jump to: navigation, search
(Fixed error)
Line 1: Line 1:
The '''prime number theorem''' is an estimate of the number of primes below any given number:
+
'''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 a function providing the number of primes less than or equal to ''n'', for any positive number ''n''. The prime number theorem states that
+
Let π(''n'') be a 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) = \lim_{n\to\infty}n/\ln(n)</math>
+
:<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>
  
 
[[category:mathematics]]
 
[[category:mathematics]]

Revision as of 06:19, 31 August 2007

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 a 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

.

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