It's possible to calculate the frequency of distribution of digits of $\pi$?

Question

It's possible using mathematical formula to calculate frequency of distribution of digits of $\pi$ or other constant?

I know that there are already plenty of data available with statistics and you can extract that information, but it's actually possible to calculate it using mathematics? If yes, how?

E.g. how many zeros are in the first 1 million digits of $\pi$ or similar.

Answer 1

For $\pi$, no way is known other than by computing the digits and counting.

Rational numbers have periodic decimal expansions and so the number of times any particular digit shows up can easily be counted. Irrational numbers can be explicitly constructed to have a decimal expansion with any given distribution of digit frequencies. It's still an open question whether "natural" irrational numbers like $\pi$ or $e$ have equally-distributed digits, though substantial numerical evidence supports that hypothesis and you'd be hard-pressed to find a mathematician willing to bet against it.

Answer 2

From Wolfram:

It is not known if $\pi$ is normal (Wagon 1985, Bailey and Crandall 2001), although the first 30 million digits are very uniformly distributed (Bailey 1988).

In other terms, it appears that the distribution of the digits of $\pi$ (in its decimal expansion) is still unknown.