Prime numbers are fascinating and fundamental to number theory. This page explores the proof of the Prime Number Theorem, which describes the distribution of prime numbers.

Proof Overview

The Prime Number Theorem states that the number of prime numbers less than or equal to a given number ( n ) is approximately ( \frac{n}{\ln(n)} ).

Key Steps

  1. Definition of Prime Numbers
    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

  2. Using the Logarithmic Integral
    The proof involves the use of the logarithmic integral function, denoted as ( Li(x) ), which is defined as the integral of the reciprocal of the natural logarithm from 0 to ( x ).

  3. Asymptotic Expansion
    The asymptotic expansion of ( Li(x) ) provides the approximation for the number of primes less than or equal to ( x ).

Proof Image

Here's an image that visually represents the distribution of prime numbers:

Prime Distribution Chart

Further Reading

For a more in-depth understanding of the Prime Number Theorem, you can explore the following resources: