The prime number theorem for Beurling's generalized numbers

AbstractLet P = {pj}j=1∞, where 1 < p1 ≤ p2 ≤ …, pj → ∞. Following Beurling [Acta Math, 1937], we call such a sequence a system of generalized (g−) primes. Let N denote the multiplicative semigroup generated by P. Set N(x) = NP(x) = {n ∈ N : n ≤ x}. Beurling proved that if NP satisfies the asymptotic relation (1) NP(x) = Ax + O(x log−γ x) with some numbers A > 0 and γ > 32, then the conclusion of the prime number theorem (P.N.T.) is valid for the system P. He gave an example of a g-prime system which satisfies (1) with γ = 32, but for which the P.N.T. does not hold. The following theorem lies in the narrow range between the abovementioned results of Beurling. Let NP(x) = Ax + O{x (log x)−3/2 exp (−[log log x]a)} for some numbers A > 0 and α > 13. Then the P.N.T. holds for P