In number theory, a real-valued function f ( n ) {\displaystyle f(n)} on the integers is additive if f ( m n ) = f ( m ) + f ( n ) {\displaystyle 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 ) = lim N → ∞ 1 N # { n < N : f ( n ) ⩽ y } . {\displaystyle F(y)=\lim _{N\to \infty }{\frac {1}{N}}\#\{n<N~:~f(n)\leqslant ~y\}.}
In particular, it asserts that the limiting function exists if the three following series pertaining to prime numbers converge:
∑ | f ( p ) | > 1 1 p , ∑ f ( p ) ⩽ 1 f ( p ) p , ∑ | f ( p ) | ⩽ 1 f ( p ) 2 p . {\displaystyle \sum _{|f(p)|>1}{\frac {1}{p}},~~\sum _{f(p)\leqslant 1}{\frac {f(p)}{p}},~~\sum _{|f(p)|\leqslant 1}{\frac {f(p)^{2}}{p}}.}
When this happens, the characteristic function v ( t ) {\displaystyle v(t)} of the limiting distribution F {\displaystyle F} is equal to
v ( t ) = ∏ p ( 1 − 1 p ) ( 1 + ∑ m = 1 ∞ p − m exp ( i t f ( p m ) ) ) , {\displaystyle v(t)=\prod _{p}\left(1-{\frac {1}{p}}\right)\left(1+\sum _{m=1}^{\infty }p^{-m}\exp(itf(p^{m}))\right),}
where i {\displaystyle i} is the imaginary unit. Furthermore, the limiting distribution is continuous if and only if the series
∑ f ( p ) ≠ 0 1 p {\displaystyle \sum _{f(p)\neq 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 ) {\displaystyle f(n)} is additive can be rephrased as
f ( p 1 e 1 ⋯ p r e r ) = f ( p 1 e 1 ) + ⋯ + f ( p r e r ) {\displaystyle f(p_{1}^{e^{1}}\cdots ~p_{r}^{e^{r}})=f(p_{1}^{e^{1}})+\cdots +f(p_{r}^{e^{r}})}
where e 1 , … , e r {\displaystyle e_{1},\ldots ,e_{r}} represent the positive integers, and p 1 , … , p r {\displaystyle p_{1},\ldots ,p_{r}} represent distinct primes.
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