A theorem on the \(k\)-adic representation of positive integers (Q2781329)

Let \(k>1\) be a fixed integer. Any positive integer \(x\) can be uniquely represented in base \(k\) as NEWLINE\[NEWLINEx = a_1 k^{n_1} + a_2 k^{n_2} + \dots + a_t k^{n_t} ,NEWLINE\]NEWLINE where \(n_1 > n_2 > \dots > n_t \geq 0\) are integers and \(a_1,a_2,\dots,a_t \in \{0,1,\dots,k-1\}\). Define NEWLINE\[NEWLINE\alpha (x) = \sum_{i=1}^t a_i \quad \text{and} \quad A(x) = \sum_{y \leq x} \alpha (y).NEWLINE\]NEWLINE The author proves that NEWLINE\[NEWLINEA(x)={k-1\over 2\log k}x\log x+\theta (x)x,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE- {5k-4 \over 8} \leq \theta (x) \leq {k+1 \over 2} .NEWLINE\]NEWLINE The author seems to be unaware of the theorem of \textit{H. Delange} [Enseign. Math. (2) 21, 31--47 (1975; Zbl 0306.10005)] that gives an exact expression for the error in terms of an explicit fractal function.