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

as the sum of all base-b integer digits between 1 and n and can be expressed as.

Above has the spot values

Now assume

to be large then

and proportioning

to the number of primes between

and

which is approximated in the Prime Number Theorem as

we obtain the unexpected result that the cumulative sum of all base-b integer digits of all the primes between

and prime

approximates to

**Conjecture **
The ratio

defined as "the sum of all base-b digits of all the primes between 1 and n" to "n", converges to the constant

for increasing n.

and

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