In a recent article [J. d' Analyse Math131, 207, 2017], Olsen intoduced a generalized notion of multifractal pressure, and also a multifractal dynamical zeta function, which essentilly consists in considering not all configurations, but those which are ”multifractally relevant”. In this way more precise information about the multifractal spectrum analyzed is encoded by the multifractal pressure and the multifratcal zeta function. He applied the theory for dynamical systems modelled by finite alphabet shifts, in particular for self conformal iterated systems. Here we continue with this line considering dynamical systems given by countable Markov shifts.