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On Benford’s law: the first digit problem. (English) Zbl 0642.10007
Probability theory and mathematical statistics, Proc. 5th Jap.-USSR Symp., Kyoto/Jap. 1986, Lect. Notes Math. 1299, 158-169 (1988).
[For the entire collection see Zbl 0626.00026.]
The authors develop several results concerning Benford’s law, which in one form can be stated as the probability of a decimal number beginning with the digit k \((k=1,2,...,9)\) equals \(\log (k+1)/k.\) Their principal theorem establishes that if the characteristic equation of a linear recurrence sequence \(\{U_ n\}_{n=1,2,...}\) has exactly two roots of maximum modulus which satisfy certain conditions then \(\{U_ n\}\) obeys Benford’s law. The result refines and extends earlier work by some of the four authors and others.
An application of this earlier work is given in the current paper: Let \(\sigma\) be a quadratic irrational whose simple continued fraction expansion is ultimately periodic, then the denominators of the convergents satisfy Benford’s law. A final section demonstrates the sequence of primes also satisfies the same law if the weight function (log p)/p is used in the density definition.
No reference is given to the work of Bombieri, P. Diaconis [Ann. Probab. 5, 72-81 (1977; Zbl 0364.10025)], D. I. A. Cohen [J. Combinat. Theory, Ser. A 20, 367-370 (1976; Zbl 0336.10052)] and particular D. I. A. Cohen and T. M. Katz [J. Number Theory 18, 261-268 (1984; Zbl 0549.10040)] who have a more general density result.
Reviewer: G.Lord

11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11B37 Recurrences
11A55 Continued fractions