Restricted partions: the polynomial case

We prove a restricted inverse prime number theorem for an arithmetical semigroup with polynomial growth of the abstract prime counting function. The adjective “restricted” refers to the fact that we consider the counting function of abstract integers of degree $\le t$ whose prime factorization may only contain the first $k$ abstract primes (arranged in nondescending order of their degree). The theorem provides the asymptotics of this counting function as $t,k\to\infty$. The study of the discussed asymptotics is motivated by two possible applications in mathematical physics: the calculation of the entropy of generalizations of the Bose gas and the study of the statistics of propagation of narrow wave packets on metric graphs.