On the asymptotics of Kempner-Irwin sums

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.