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A result in [[probabilistic number theory]] characterising those [[additive function]]s 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

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

[[Weak convergence of probability measures|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
$$
F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ .
$$

==References==
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. ''Handbook of number theory I''. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9.  {{ZBL|1151.11300}} 
* Tenenbaum, Gérald ''Introduction to Analytic and Probabilistic Number Theory''. Cambridge studies in advanced mathematics '''46'''. Cambridge University Press (1995). ISBN 0-521-41261-7.  {{ZBL|0831.11001}}