Prime counting function

The 60 first values of π(n)

In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x. It is written as [math]\displaystyle{ \pi(x) }[/math], but it is not related to the number π. Some key values of the function include [math]\displaystyle{ \pi(1)=0 }[/math], [math]\displaystyle{ \pi(2)=1 }[/math] and [math]\displaystyle{ \pi(3)=2 }[/math].

In general, if [math]\displaystyle{ p_n }[/math]stands for the n-th prime number, then [math]\displaystyle{ \pi(p_n)=n }[/math].

Prime Counting Function Media

• 

  The values of π(n) for the first 60 positive integers

• 

  Graph showing ratio of the prime-counting function π(x) to two of its approximations, x/log x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x/log x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.

• 

  Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function

Related pages

• Ramanujan prime