On the asymptotics of Kempner-Irwin sums

TL;DR

This work establishes that the Kempner–Irwin sums $I(b,d,k)$, which prune the harmonic series to integers with exactly $k$ occurrences of digit $d$ in base $b$, admit an asymptotic expansion $I(b,d,k)-b\log(b)$ to all orders in $1/b$ for fixed $d$ and $k$. The authors develop a digamma-based integral framework and analyze the moments of associated ellipsephic measures $\mu_k$, employing Euler–Maclaurin and recurrence relations to derive explicit coefficients involving $\zeta(n)$. They treat all cases ($d=0$ and $d>0$, with $k=0$ and $k>0$) and provide four or five-term expansions, together with rigorous numerical validation. The results unify and extend prior tabulations (e.g., $K(b,0)$) and offer precise asymptotics for a broad family of digit-restricted harmonic sums, with potential applications in analytic number theory and high-precision computations for digit-constrained sequences.

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.

Theorems & Definitions (19)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• Proposition 5
• proof
• Proposition 6
• proof