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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

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# Prime Counting Function

## Nth Prime Calculator

### What is the counting prime function? (Definition)

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: $\pi(100) = 25$ as 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 $n$ tends to infinity (ie. $n / \ln(n)$ is a good approximation of $pi(n)$ when $n$ is very large)

$$\pi(n)\ \underset{ n \to \infty }{ \sim } \frac{n}{\ln(n)}$$

This formula is also called the prime number theorem.

### What is pi(n) for?

The calculation of pi(n) allows 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) $$p_n \underset{ n \to \infty }{ \sim } n \ln (n)$$

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Prime Counting Function on dCode.fr [online website], retrieved on 2023-09-27, https://www.dcode.fr/prime-number-pi-count

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