This is just a random thought that came to my mind.

What is the fraction of even numbers in decimal expansion of $\pi$ and $e$?

Is there any research on this and it’s possible generalisations?

Edit: I checked the first 100 digits and found number of even digits in $\pi$ and $e$ to be $50$ and $58$ respectively.

Is there any reason we should believe that the fraction should converge to exactly $50$%?

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Aditya Sharma
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    See https://en.wikipedia.org/wiki/Normal_number – PM 2Ring Dec 04 '20 at 18:27
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    No one knows. For all anyone knows, it's possible that only two digits appear in the decimal expansion (beyond some point). it is widely conjectured that each digit occurs with equal frequency, but this is not known. – lulu Dec 04 '20 at 18:27
  • Ok, thanks for your inputs! – Aditya Sharma Dec 04 '20 at 18:30
  • @lulu, you mean after a finite string of decimals we can have only two digits in some order indefinitely? – Aditya Sharma Dec 04 '20 at 18:33
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    This is an area where there is a large gap between what is conjectured and what can be proven. For instance, it is conjectured that all algebraic irrationals are normal in every base; but *no* algebraic irrational has been proven to be normal in *any* base. – mjqxxxx Dec 04 '20 at 18:33
  • @JMoravitz All I have realised is that there is-no answer to the question..;) – Aditya Sharma Dec 04 '20 at 18:36
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    @AdityaSharma Yes, in principle. Again, nobody expects this to be the case, but we can't prove much. We know that more than one digit must occur infinitely often, since otherwise the decimal would be rational (and we know that neither $\pi$ nor $e$ is rational). – lulu Dec 04 '20 at 18:37
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    @AdityaSharma "*...we can have only two digits in some order indefinitely*" There are certainly numbers like that, infinitely many of them even, including some rational examples, some irrational examples, and even some transcendental examples. To emphasize, we don't actually *believe* that $\pi$ eventually hits such a point, however it hasn't been actually proven that it doesn't. To show off computing power, people try to break the record for digits of $\pi$ calculated and we're well past 30 trillion digits at this point... it remains effectively normal up to that point so far. – JMoravitz Dec 04 '20 at 18:37
  • @lulu, but according to my understanding of your statement, that would imply the relative fractions of the other 8 digits to be zero (as they are present only finitely) – Aditya Sharma Dec 04 '20 at 18:40
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    As I have said (a few times) nobody believes that only two digits appear after some point. I am just using that possibility (however unlikely it is) to illustrate how little we know. – lulu Dec 04 '20 at 18:42
  • @JMoravitz, thanks a lot for that comment, my doubts are cleared now. – Aditya Sharma Dec 04 '20 at 18:43
  • @lulu, yes, even I had trouble believing it initially which is why I reasked the same question. Thanks a lot! – Aditya Sharma Dec 04 '20 at 18:45

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