Quite deterministic nature of prime numbers, due to the complexity of the recurrent generating algorithms, is mimicking ‘randomness’ and stimulates to apply probabilistic instruments to analyze number-theoretic problems. The key issue in the probabilistic analysis in a number-theoretic framework remains an enigmatic connection between the deterministic nature of integer sequences related to prime numbers and their apparent complicated (‘unpredictable’ or ‘chaotic’) behavior interpreted as ‘randomness’. We derive multiplicative and additive models with recurrent equations for generating sequences of prime numbers based on the reduced Sieve of Eratosthenes Algorithm and analyze their asymptotic behavior with the help of Riemann Zeta probability distribution. This allows interpreting such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by prime numbers, representing paths of stochastic dynamical systems. We analyze in this work an additive continuous-time probabilistic model of counting function of primes pi(n) in terms of diffusion approximation of non-Markov random walks. We assume that ‘updating’ terms eta in the recurrent equation pi(n(k+1)) - pi(n(k)) = eta(n(k+1) follow Zeta probability distribution and calculate infinitesimal characteristics of the random walk, which approximate coefficients of the corresponding stochastic differential equation. Computer modeling illustrates graphically an impressive fitting of trajectories for the original counting function, the calculated trend function, and the Brownian approximation.