Sur l'asymptotique des sommes de Kempner pour de grandes bases

Let $K$ be the sum of the reciprocals of the integers with no occurrence of the digit $b-1$ in base $b$. We show $K=b\log (b)-A\mathrm{/}b-B\mathrm{/}{b}^2-C\mathrm{/}{b}^3+O(1\mathrm{/}{b}^4)$ with $A=\zeta (2)\mathrm{/}2$, $B=(3\zeta (2)+\zeta (3))\mathrm{/}3$ and $C=(2\zeta (2)+4\zeta (3)+\zeta (4))\mathrm{/}4$.