It may or may not be results of the type you are looking for, but one have deduced asymptotic formulae for the average value of the digit sum $S_b(n)$ considered a function of $n$ and considered a function of $b$. (See this post.) Specifically, one has$$\sum \limits_{n=1}^{N}S_b(n)\sim\frac{(b-1)\log(N)}{2N \log b} \quad \text{as} \; n\to\infty.$$For the other agument, one has the following result:$$\sum_{b=1}^{n}S_b(n)=(1-\frac{\pi^2}{12})n^2+C\frac{n^2}{\log(n)}+o\left(\frac{n^2}{\log(n)}\right) \quad \text{as} \; n\to \infty,$$where $C$ is the constant $1-\frac{\pi^2}{24}-\frac{1}{2}\sum_{t=2}^{\infty}\frac{\log(t)}{t^2}=0.119\ldots$.
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