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The counting prime numbers function, called $ \pi(n) $, aims to count the prime numbers less than or equal to a 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) $$ p_n \sim n \ln (n) $$

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