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So one night, I got bored, so, as you do, I made a depressingly inefficient python formula to calculate the density of the digits in a text file with a billion digits of pi in it. As I expected, it came out to roughly 10% per digit, and the precision increased proportionately to the number of digits I put in (so a billion digits was around a thousand times closer than a million was). Is there a way to prove this distribution approaches equality?

Erin Condron
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  • Did you find anything with Google? – Matti P. Oct 11 '19 at 07:51
  • https://math.stackexchange.com/questions/51829/distribution-of-the-digits-of-pi – Matti P. Oct 11 '19 at 07:51
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    This post on MathOverflow seems to answer your question. – Arthur Oct 11 '19 at 07:54
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    Welcome to Mathematics Stack Exchange. Your title says binary digits but your body says roughly 10% per digit !? – J. W. Tanner Oct 11 '19 at 08:46
  • If we assume that the distribution is like that of random digits, we will observe that the ratio gets closer and closer to $1/10$. The awfully many digits that have been calculated support this assumption, but noone knows whether this goes on forever. It is conjectured that $\pi$ is normal (meaning that every finite digit string occurs in the expected frequency if we look at more and more digits). But noone can rule out that eventually only two distinct digits occur, something like $01100110101110001...$ – Peter Oct 22 '19 at 17:41

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