Kempner-like harmonic series

Abstract: Inspired by a question asked on the list { t mathfun}, we revisit {em Kempner-like series}, i.e., harmonic sums

s u m^{'} 1 / n

where the integers

n

in the summation have ``restricted digits. First we give a short proof that $l i m_{k o i n f t y} (s u m_{s_{2} (n) = k} 1 / n) = 2 l o g 2$ , where $s_{2} (n)$ is the sum of the binary digits of the integer $n$ . Then we propose two generalizations. One generalization addresses the case where $s_{2} (n)$ is replaced with $s_{b} (n)$ , the sum of $b$ -ary digits in base $b$ : we prove that $l i m_{k o i n f t y} s u m_{s_{b} (n) = k} 1 / n = (2 l o g b) / (b - 1)$ . The second generalization replaces the sum of digits in base $2$ with any block-counting function in base $2$ , e.g., the function $a (n)$ of -- possibly overlapping -- $11$ 's in the base- $2$ expansion of $n$ , for which we obtain $l i m_{k o i n f t y} s u m_{a (n) = k} 1 / n = 4 l o g 2$ .

Mathematics Subject Classification ID

Combinatorics on words (68R15) Permutations, words, matrices (05A05) Convergence and divergence of series and sequences (40A05) Radix representation; digital problems (11A63) Automata sequences (11B85)

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