Kempner-like Harmonic Series

Abstract

Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums ∑ ′ 1 / n where the integers n in the summation have “restricted” digits. First we give a short proof that lim k → ∞ ( ∑ s 2 ( n ) = k 1 / n ) = 2 log 2 , where s 2 ( n ) is the sum of the binary digits of the integer n. Then we give a generalization that 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 lim k → ∞ ∑ s b ( n ) = k 1 / n = ( 2 log b ) / ( b − 1 ) . Finally we indicate that other generalizations could be studied: the sum of digits in base 2 could be replaced with, e.g., the function a 11 ( n ) of—possibly overlapping—11 in the base-2 expansion of n, for which one can obtain lim k → ∞ ∑ a 11 ( n ) = k 1 / n = 4 log 2 .