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Digit functions of integer sequences. (English) Zbl 0549.10041
For a given sequence \(\{a_n\}\) of natural numbers, let \(d_1(a_n),d_2(a_n),\ldots\). denote their first, second, …digits to some base \(b\). The authors study the joint distributions of the random variables \(d_j(x)\) and show that, for a large class of sequences \(\{a_n\}\), they are correlated but tend to equiprobability. They also prove that in certain cases, the non-uniform behavior of \(d_1\) persists for all \(j\). Thus the paper generalizes known results on “Benford behavior” \((\text{prob}\{d_1=a\}=\log_b(1+1/a),\ a=1,\ldots,b-1)\) in several ways.
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A63 Radix representation; digital problems