The asymptotic oscillations of moments related to Dirichlet series with missing digits

We prove that the (suitably rescaled) moments of certain discrete measures on the unit interval, which are related to the numerical evaluation of zeta series with missing digits in radix $b$, are asymptotically $1$-periodic in the base $b$ logarithm of the index, i.e. asymptotically invariant under multiplication by $b$ of the index.