## Distribution of the sum-of-digits function of random integers: a survey.(English)Zbl 1327.60029

In the present survey, the authors investigate probabilistic properties of the sum-of-digits function. They present four methods attacking the problem as well as an extension to general numeration systems.
The authors start with a historical background on the sum-of-digits function and its investigations. They provide several “formulas” which were used over time to consider the arithmetic mean of the sum-of-digits function. Then, they consider the variance, higher moments and limit distribution of the sum-of-digits function. Throughout their considerations, they provide several methods attacking these problems. At the end of the first part, they consider the asymptotic distribution of the sum-of-digits function, again describing the different approaches used over time.
In the second part, the authors provide new results. Let $$X_n$$ denote the number of ones in the binary representation of a random integer, where each of the integers $$\{0,1,\dots,n-1\}$$ is chosen with equal probability $$\frac1n$$. Then, among other results, they show that $\sum_{0\leq k \leq\lambda}\left| \mathbb{P}\left(X_n=k\right)-\sum_{0\leq r<m} (-1)^ra_r(n)2^{-\lambda}\Delta^r\binom{\lambda}{k}\right|$
$=\frac{h_m\left|a_m(n)\right|}{(\log_2n)^{m/2}}+\mathcal{O}\left((\log n)^{-(m+1)/2}\right),$ for $$m=1,2,\dots$$, where $$a_r(n)$$ are constants, which can be explicitly calculated, and $$\Delta$$ is the difference operator.
The final part deals with general numeration systems and applications. In particular, the authors apply the methods from the first and second part to Gray codes and general codings satisfying a relation on the numbers of ones in the representation of $$n$$ and $$2n$$.

### MSC:

 60C05 Combinatorial probability 60F05 Central limit and other weak theorems 62E17 Approximations to statistical distributions (nonasymptotic) 11N37 Asymptotic results on arithmetic functions 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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### References:

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