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Old 2006-09-20, 12:26   #1
AntonVrba's Avatar
Jun 2005

2·72 Posts
Default Sum of all integer digits of all primes between 1 an n

With the discovery of M44 (congratulations GIMPS) I pondered over the thought if it is possible to count/calculate all the ones of all the primes between 2 and M44. It is a bit off-topic but the result is interesting.

To calculate or estimate the number of ones, I set about as follows.

Define c_b(n) as the sum of all base-b integer digits between 1 and n and can be expressed as.
c_b(n)=\sum_{k = 0}^{m}\,\,{d_k}\,\left( b^k\, \frac{{d_k} + k\, \left( b - 1\right)  - 1}{2} + {mod (n,\,{b^k})} + 1\right)

Above has the spot values c_b\left( b^m-1 \right) =\frac{m\left( b-1\right)b^m}{2}

Now assume m to be large then m=log_b(b^m-1)=ln(b^m-1)/ln(b) and proportioning c_b(b^m-1) to the number of primes between 1 and b^m-1 which is approximated in the Prime Number Theorem as (b^m-1)/ln(b^m-1) we obtain the unexpected result that the cumulative sum of all base-b integer digits of all the primes between 1 and prime b^m-1 approximates to

 \frac{(b-1)(b^m-1)}{2\, ln(b)}\,

The ratio\kappa_b defined as "the sum of all base-b digits of all the primes between 1 and n" to "n", converges to the constant  \frac{b-1}{2\, ln(b) for increasing n.

\kappa_2=0.7213476 and \kappa_{10}=1.9543251

A computation check confirms above tendency already at relatively small values of n.

Is above already known or have I introduced a new constant?

In parctice how will the constant depart from above definition?

Anton Vrba
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