Erdős–Wintner theorem

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 $$ F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ . $$

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