# Talk:Erdős–Kac theorem

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## Random x

This idea of thinking of x as random is not a good one. In analytic number theory the convention is that x is real and n is an integer. Seeing w(x) is confusing, as w is defined on the integers. Also people might interpret the limit as taken over the integers n. (Which is a different statement - altough easily seen to be equivalent). Also the sign P indicating probability is confusing. I think the conventional statement as in the French version is much better (and much more standard btw). —Preceding unsigned comment added by 132.206.124.96 (talkcontribs)

## n or N

Before someone gets confused, both

${\displaystyle \lim _{N\rightarrow \infty }{\frac {1}{N}}\left|\left\{n\leq N:a\leq {\frac {\omega (n)-\ln \ln N}{\sqrt {\ln \ln N}}}\leq b\right\}\right|=\int _{a}^{b}\varphi (u)\,du}$

as in Mathworld and

${\displaystyle \lim _{N\rightarrow \infty }{\frac {1}{N}}\left|\left\{n\leq N:a\leq {\frac {\omega (n)-\ln \ln n}{\sqrt {\ln \ln n}}}\leq b\right\}\right|=\int _{a}^{b}\varphi (u)\,du}$

as here are correct, due to the slow growing of ${\displaystyle ln(ln(n))}$. (Mark Kac: Statistical Independence in Probability, Analysis and Number Theory, Wiley, New York 1959.) --Erzbischof (talk) 09:20, 24 July 2008 (UTC)

## N(0,1) or N(loglog n, (loglog n)^½)

${\displaystyle Q={\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}}}$ is N(0,1) distributed, Q is normal distributed with mean loglog(n) and standard deviation \sqrt{\log \log n}.--Erzbischof (talk) 19:36, 23 June 2009 (UTC)

The normal distribution is sometimes denoted N(mean, variance) in this case we would have N(loglog n, loglog n) as the variance is the square of the standard deviation.Mikewarbz (talk) 12:14, 29 June 2009 (UTC)