Properties of the Beurling generalized primes
For a generalized prime number system \(P\) and \(x>0\) let \(N(x)\) be the number of elements of the multiplicative semigroup which are \(\leq x\). Suppose that \(N(x)=Ax+o(x)\) for some \(A>0\). The author generalizes a theorem of Mertens: \[ \lim_{x\to\infty}\frac{1}{\ln x}\prod_{p\leq x} \left(1-\frac{1}{p}\right)^{-1}=Ae^\gamma, \] where \(\gamma\) is Euler's constant. He conjectures that if \(P\) is different from the set of rational primes, then \[ \limsup_{x\to\infty}\frac{|N(x)-[x]|}{\ln x}>0. \] He proves that if the conjecture is true, then it is sharp in the following sense: for every \(c>0\) there exists a generalized prime number system other than the rational primes for which \[ |N(x)-[x]|<c\ln x \] for all \(x\geq 1\).
scientific article; zbMATH DE number 5817564