how many $1$s are there in the first n digits of $\pi$? Any good approximation of its distribution? How about the place of the $n$th $1$? Are these two questions related?
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2I think about $[n/10],$ no? – Leox Dec 22 '13 at 19:10
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The answer is not known, but it is conjectured that $\pi$ is simply normal in base $10$, and from that you would expect one tenth of the digits to be $1$s, and the $n$-th $1$ to be found near digit number $10n$, for huge values of $n$, in some sense.
Jeppe Stig Nielsen
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There is no known answer to this question.
Yiorgos S. Smyrlis
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The exact number, as a function of $n$ is not known. There are partial answers on the question of the distribution of digits. – Yiorgos S. Smyrlis Dec 22 '13 at 19:26
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1@DietrichBurde It is not even known if there are infinitely many $1$s or not. – Jeppe Stig Nielsen Dec 22 '13 at 19:33
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See http://math.stackexchange.com/questions/51829/distribution-of-the-digits-of-pi – Yiorgos S. Smyrlis Dec 22 '13 at 19:44
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Actually, yes, Kavim, since $\pi$ is suspected (though not yet proven) to be a normal number. As n grows large, the ratio is approximately $\dfrac1{10}$ for each digit. All computations done so far, even up to over a trillion digits, confirm this conjecture.
Lucian
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