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.