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Re: sum of all integer digits of all primes between 1 and p = k*p

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  • Joshua Zucker
    ... It seems to me that this conjecture is true, and your proof is valid, if you assume that primes are equally likely to consist of any digit? But while we
    Message 1 of 2 , Oct 10, 2006
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      --- In primenumbers@yahoogroups.com, "Anton" wrote:
      > Conjecture:
      > The ratio k_b defined as "the sum of all base-b digits
      > of all the primes between 1 and n" to "n",
      > converges to the constant (b-1)/(2 ln(b)) for increasing n.
      >
      > Proof:
      > Sum of all base-b integer digis of all primes between 1 and b^m-1
      > = (m*b^m*(b-1)/2)*pi(b^m-1)/(b^m-1)

      It seems to me that this conjecture is true, and your proof is valid,
      if you assume that primes are equally likely to consist of any digit?
      But while we might believe that's true in the long run, it seems like
      in the short run it certainly won't be true. (E.g. primes in base 10
      won't have so many even digits in them, or 5s, in the short run; but
      by the time they have a lot of digits, maybe that last-digit fact
      won't matter much any more).

      Is there a known proof somewhere that large primes asymptotically have
      equal proportion of each digit (in any base)?

      It seems very likely to be true, but way beyond what I'm capable of
      proving or even intelligently investigating.

      Thanks,
      --Joshua Zucker
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