Distribution of leading digits of numbers (Q2835472)

Let \(d_1>0,\dots ,d_s\) be the digits in the base \(b\geq 2\), \(r>0\) and for every \(w\in [0,1]\) we define NEWLINE\[CARRIAGE_RETURNNEWLINE g_w(x)=\frac {1}{b^{w/r}}\cdot \frac {b^{x/r}-1}{b^{1/r}-1}+\frac {\min (b^{x/r},b^{w/r})-1}{b^{w/r}} CARRIAGE_RETURNNEWLINE\]NEWLINE for \(x\in [0,1]\). In this paper is proved that NEWLINE\[CARRIAGE_RETURNNEWLINE \begin{aligned} &\lim_{i\to \infty}\frac {\#\{n\leq N_i;\, \text{first } s\;\text{digits of } n^r \text{ are } d_1d_2\dots d_s\}}{N_i}\\ &=g_w\big (\log_b d_1\cdot d_2\dots (d_s+1)\big)-g_w\big (\log_b d_1\cdot d_2\dots d_s\big) \end{aligned} CARRIAGE_RETURNNEWLINE\]NEWLINE assuming \(\lim_{i\to \infty}\bigl \{\log_b(N_i^r)\bigr \}=w\). Since \(g_w(x)\neq x\), the sequence \(n^r\) does not satisfy the Benford's law, but \(\lim_{r\to \infty}g_w(x)=x\) for every \(w\in [0,1]\), thus \(n^r\) tends to the Benford's law. This is a qualitative proof of results given by \textit{S. Eliahou} et al. [Acta Math. Hung. 139, No. 1--2, 49--63, (2013; Zbl 1299.60004)]. The authors also prove that the same function \(g_w(x)\) can be used for \(p_n^r\), where \(p_n\) is the sequence of all primes and \(\lim_{i\to \infty}\bigl \{\log_b(p^r_{N_i})\bigr \}\to w\). In the proof the theory of distribution functions of sequences is used.