Erdős–Wintner theorem

In mathematics, specifically probabilistic number theory, a real-valued function on the integers is additive if for coprime , . The Erdős–Wintner theorem gives a condition under which a real-valued additive function has a limiting distribution

In particular, it asserts that the limiting function exists if the three following series pertaining to prime numbers converge:

When this happens, the characteristic function of the limiting distribution is equal to

where is the imaginary unit. Furthermore, the limiting distribution is continuous if and only if the series

diverges, otherwise the distribution is purely discrete. It is an analogue of Kolmogorov's three-series theorem.

References