# 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 '11 at 18:40
• @Emre: No, it can't. Take, e.g., the number $0.101001000100001000001...$. – Hendrik Vogt Jul 16 '11 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. – André Nicolas Jul 17 '11 at 0:22
• @Hendrik: The theta function, huh? I'm not sure that has been proven transcendental... Liouville's might be a better example... – J. M. isn't a mathematician Jul 17 '11 at 1:20
• @André: Yes, that's what I actually had in mind - thanks a lot! – Hendrik Vogt Jul 17 '11 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. – Kevin Apr 11 '19 at 0:56