We introduce the notion of stochastic zeta function, which for some classes of processes share many properties with their counterparts in analytic number theory. These zeta functions, defined in terms of the barcode, are related to a dual variable counting the number of bars in the bars in the barcode of length $\geq \varepsilon$. A "prime number theorem" can then be proved for this dual variable in terms of the zeta function of the considered process.