Digamma function and general Fischer series in the theory of Kempner sums

Abstract

The harmonic sum of the integers which are missing $p$ given digits in a base $b$ is expressed as $b\log(b)/p$ plus corrections indexed by the excluded digits and expressed as integrals involving the digamma function and a suitable measure. A number of consequences are derived, such as explicit bounds, monotony, series representations and asymptotic expansions involving the zeta values at integers, and suitable moments of the measure. In the classic Kempner case of $b=10$ and $9$ as the only excluded digit, the series representation turns out to be exactly identical with a result obtained by Fischer already in 1993. Extending this work is indeed the goal of the present contribution.