  Let $I(b,d,k)$ be the subseries of the harmonic series keeping only those
denominators having exactly $k$ occurrences of the digit $d$ in base $b$. We
prove the existence of an asymptotic development to all orders in descending
powers of $b$, for $d$ and $k$ fixed, of $I(b,d,k)-b \log(b)$ or rather, if
$k=0$ and $d>0$, of $I(b,d,0)- (b\log(b) - b\log(1+1/d))$. We give explicitly,
according to cases, the first four or first five terms. The coefficients of
these developments involve the values of the Riemann zeta function at positive
integers.
