Normalization of Beurling generalized primes with Riemann Hypothesis

Abstract. Two Beurling generalized number systems are given such that the primes of both systems up to any assigned large number are exactly all the rational primes not exceeding the number and both consist of only rational numbers of the form \(n+\ell/2^n\). The counting functions of g-integers of both satisfy \(N(x)=x+x^{1/2}O(\exp\{c(\log\,x)^{2/3}\})\). The first realizes the RH and the second realizes exactly the classical zero-free region of the Riemann zeta function and the de la Vallée Poussin error term.