Erdős–Wintner theorem

Short description: Theorem in number theory

In number theory, a real-valued function $f(n)$ on the integers is additive if $f(mn)=f(m)+f(n)$ for coprime m, n. The Erdős–Wintner theorem gives a condition under which a real-valued additive function has a limiting distribution $$F(y)=\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\mathrm{\#}\{n<N:f(n)⩽y\}.$$ In particular, it asserts that the limiting function exists if the three following series pertaining to prime numbers converge: $$\sum _{|f(p)|>1}\frac{1}{p},\sum _{f(p)⩽1}\frac{f(p)}{p},\sum _{|f(p)|⩽1}\frac{f(p{)}^2}{p}.$$ When this happens, the characteristic function $v(t)$ of the limiting distribution $F$ is equal to $$v(t)=\prod _p(1-\frac{1}{p})(1+\sum _{m=1}^{\mathrm{\infty }}{p}^{-m}\exp (itf(p^m))),$$ where $i$ is the imaginary unit. Furthermore, the limiting distribution is continuous if and only if the series $$\sum _{f(p)\ne 0}\frac{1}{p}$$ diverges, otherwise the distribution is purely discrete. It is an analogue in probabilistic number theory of Kolmogorov's three-series theorem.

The condition that $f(n)$ is additive can be rephrased as $$f(p_1^{e^1}\cdots p_r^{e^r})=f(p_1^{e^1})+\cdots +f(p_r^{e^r})$$ where $e_1,\dots ,e_r$ represent the positive integers, and $p_1,\dots ,p_r$ represent distinct primes.