On the asymptotics of Kempner-Irwin sums

TL;DR

This paper investigates the asymptotic behavior of Kempner-Irwin sums, which are subseries of the harmonic series filtered by digit occurrence, providing detailed expansions involving the zeta function.

Contribution

It establishes the existence of complete asymptotic expansions for these sums in powers of the base, explicitly including multiple terms with coefficients linked to the zeta function.

Findings

Asymptotic expansions exist for all orders in powers of the base.

Explicit formulas involve the Riemann zeta function at integers.

Results depend on the digit and occurrence parameters.

Abstract

Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers.