Theorem of Erdős- Kac

The theorem of Erdös - Kac is the first example of study of the weak convergence of frequencies of the additive functions, and is regarded as the revival of the probabilistic theory of the numbers. It is relating to the function $\backslash \; Omega\; (N)$ which indicates the number of factors first distinct from $n$. It was obtained by Erdös and Kac in 1939, then in 1958 by Rényi and Turan for an explicit version of the term of error.

The version presented here is the original one. For any reality $\backslash \; lambda$, one a:

$\backslash \; lim\_\; \{X\; \backslash \; rightarrow\; \backslash \; infty\}\; \backslash \; frac\; \{1\}\; \{X\}\; \backslash \; left\; |\; \backslash \; left\; \backslash \; \{N:\; N\; \backslash \; Leq\; X\; \backslash ,/\backslash ,\; \backslash \; Omega\; (N)\; \backslash \; Leq\; \backslash \; ln\; \backslash \; ln\; X\; +\; \backslash \; lambda\; \backslash \; sqrt\; \{\backslash \; ln\; \backslash \; ln\; X\}\; \backslash \; right\; \backslash \}\; \backslash \; right\; |\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{\backslash \; lambda\}\; e^\; \{-\; t^2/2\}\; \backslash ,\; dt.$

Category: Probabilities Category: Real analysis Erdos

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