Erd s–Kac theorem

Mathematical theorem in number theory

The Erd_s–Kac theorem is a fundamental result in number theory that describes the distribution of the number of distinct prime factors of a natural number. It is named after the mathematicians Paul Erd_s and Mark Kac, who first formulated and proved the theorem in the 1940s.

Statement of the theorem

The Erd_s–Kac theorem states that for a large natural number \( n \), the number of distinct prime factors \( \omega(n) \) of \( n \) is approximately normally distributed. More precisely, if \( \omega(n) \) denotes the number of distinct prime factors of \( n \), then for large \( n \), the random variable

\[ \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \]

converges in distribution to a standard normal distribution \( N(0,1) \).

Explanation

The theorem provides a probabilistic model for understanding the behavior of prime factors in large numbers. It suggests that the number of distinct prime factors of a number behaves like a random variable with a normal distribution, centered around \( \log \log n \) with a standard deviation of \( \sqrt{\log \log n} \).

Applications

The Erd_s–Kac theorem has applications in analytic number theory and probability theory. It is used to study the distribution of prime factors and has implications for the understanding of random integers and their properties.

Visual representation

Plot illustrating the normal distribution of the number of distinct prime factors as described by the Erd_s–Kac theorem.

The plot to the right illustrates the normal distribution of the number of distinct prime factors as described by the Erd_s–Kac theorem. The horizontal axis represents the standardized variable \( \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \), and the vertical axis represents the probability density.

Related pages

• Prime number
• Normal distribution
• Probabilistic number theory
• Paul Erd_s
• Mark Kac