# Erdős–Wintner theorem

Jump to: navigation, search

A result in probabilistic number theory characterising those additive functions that possess a limiting distribution.

## Limiting distribution

A distribution function $F$ is a non-decreasing function from the real numbers to the unit interval [0,1] which is right-continuous and has limits 0 at $-\infty$ and 1 at $+\infty$.

Let $f$ be a complex-valued function on natural numbers. We say that $F$ is a limiting distribution for $f$ if $F$ is a distribution function and the sequence $F_N$ defined by

$$F_n(t) = \frac{1}{N} | \{n \le N : |f(n)| \le t \} |$$

converges weakly to $F$.

## Statement of the theorem

Let $f$ be an additive function. There is a limiting distribution for $f$ if and only if the following three series converge: $$\sum_{|f(p)|>1} \frac{1}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)^2}{p} \ .$$

When these conditions are satisfied, the distribution is given by $$F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ .$$