Terence Tao in his brilliant book Solving Mathematical Problems: a Personal Perspective states (page 17):
It is highly probable (though not proven!) that the digit-sum of $2^{n}$ is approximately $(4.5 \log_{10} 2)n≈1.355n$ for large $n$.
This problem sounds very interesting to me! Do you know something more about this problem? What is its exact formulation (replacing word approximately with a limit)? Is it still not proven?
I found only this which does not satisfy me enough.
Thank you very much for your answers and have a nice week!
As discussed in the comments:
Informally, this is a statement about the distribution of the digits in $2^n$. If we imagined that they were distributed uniformly, then the claim would follow at once: The average value of a randomly selected digit is $\frac 92=4.5$ and the number of digits in $2^n$ is $\lceil n\log_{10} 2\rceil$.
Of course, it isn't at all clear that this assumption is justified, nor is it clear how one might go about proving it. Indeed, the units place is obviously not uniform (it must be even) nor is the lead digit uniform (the low digits are favored disproportionately, see, e.g., this). Of course, a couple of digits at the ends do not have any material impact on the overall distribution of the units.