mersenneforum.org - View Single Post - Sum of all integer digits of all primes between 1 an n

AntonVrba · 2006-09-20, 12:26

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

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

Conjecture
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?

Regards
Anton Vrba

2006-09-20, 12:26	#1
AntonVrba Jun 2005 2·7² Posts	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 $\frac{(b-1)(b^m-1)}{2\, ln(b)}\,$ Conjecture 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? Regards Anton Vrba