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Integer powers and Benford’s law. (English) Zbl 1068.11047
Benford’s law tells us that the first digit of many numerical data sets given in decimal form is not equally distributed of \(\{1,2,\dots, 9\}\). F. Benford gave in 1938 a mathematical reasoning for this fact and proved that the probability of the the first digit being equal to \(d\) is \(\log(1+ d^{-1})\) for \(d\in\{1,\dots, 9\}\).
In the paper the exact probability distribution of the first digit of integer powers up to an arbitrary but fixed number of digits is derived. Based on its asymptotic distribution, it is proved that it approaches Benford’s law very closely for sufficiently high powers.

11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11B83 Special sequences and polynomials
11K31 Special sequences
62E20 Asymptotic distribution theory in statistics