Distribution of the digits of Pi

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.

• The same question applies to any transcendental number.
– Emre
Jul 16, 2011 at 18:40
• @Emre: No, it can't. Take, e.g., the number $0.101001000100001000001...$. Jul 16, 2011 at 21:42
• I 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. Jul 17, 2011 at 0:22
• @Hendrik: The theta function, huh? I'm not sure that has been proven transcendental... Liouville's might be a better example... Jul 17, 2011 at 1:20
• @André: Yes, that's what I actually had in mind - thanks a lot! Jul 17, 2011 at 7:54

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.
• There is also relatively little we know about normal numbers. It's fairly easy to construct "artificial" normal numbers like Champernowne's constant, but much harder to determine whether an arbitrary real number $\alpha$ is normal. Apr 11, 2019 at 0:56