Can anything be stated about the distribution of the digits of Pi, i.e., if I were to sample n digits of Pi, can anything be said about the probability to observe certain digits, or is there any reason to assume that they would not be evenly distributed?
This is purely a curiosity question.
Distribution of the digits of Pi
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probability-distributions
pi
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0The same question applies to any transcendental number. – 2011-07-16
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4@Emre: No, it can't. Take, e.g., the number $0.101001000100001000001...$. – 2011-07-16
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1I did not know this has been proved transcendental. Has it? If we let the number of $0$'s between $1$'s grow like $n!$, then the resulting number is certainly transcendental. – 2011-07-17
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0@Hendrik: The theta function, huh? I'm not sure that has been proven transcendental... Liouville's might be a better example... – 2011-07-17
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0@André: Yes, that's what I actually had in mind - thanks a lot! – 2011-07-17
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0@J.M.: see above. – 2011-07-17
1 Answers
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It is suspected that $\pi$ is a normal number, i.e. that its digits in any base $b$ are uniformly distributed in a certain precise sense (the link explains in more detail). However, this has not been proven yet. In fact, there is relatively little we know about the distribution of the digits of $\pi$; take a look at these posts (here and here) on MathOverflow.