Prime counting function
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],[1] 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].[2]
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 Explicit Formula.gif
Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function
Related pages
References
- ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-07.
- ↑ Weisstein, Eric W. "Prime Counting Function". mathworld.wolfram.com. Retrieved 2020-10-07.