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Prime Counting Function

Tool for counting prime numbers via the prime-counting function denoted pi(n) which counts the prime numbers less than or equal to a real number n.

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Prime Counting Function -

Tag(s) : Arithmetics, Mathematics

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# Prime Counting Function

## Prime Count π(n)

Tool for counting prime numbers via the prime-counting function denoted pi(n) which counts the prime numbers less than or equal to a real number n.

### How to calculate pi(n)?

For small numbers, the easiest method to count all the first primes less than $$n$$ is to use the Eratosthenes sieve to quickly list prime numbers.

Example: $$\pi(100) = 25$$ so there are 25 prime numbers less than 100.

### How to calculate an approximation of pi(n)?

The value of pi(n) approaches $$n / \ln(n)$$ when $$n$$ is very big:

$$\lim_{ n \to + \infty } \pi(n) = \frac{ n }{ \ln(n) }$$

This formula is also called the prime number theorem.

### What is pi(n) for?

The calculation of pi(n) allow to locate a prime number with respect to another, knowing its rank in the list of prime numbers.

If pi(a) < pi(b) then a < b.

### How to get an estimation of the nth prime number?

A consequence of the prime number theorem is that the nth prime number $$p_n$$ is close to $$n \ln(n)$$ (and closer when $$n$$ is very large) $$p_n \sim n \ln (n)$$