A PNT equivalence for Beurling numbers
The objective of the paper is to investigate equivalences between the prime number theorem and the sharp Merten's formula for Beurling generalized numbers introduced in [\textit{A. Beurling}, Acta Math. 68, 255--291 (1937; Zbl 0017.29604)]. Beurling generalized numbers form a multiplicative semigroup \(\mathcal{N}\subset\mathbb{R}\) generated by Beurling generalized prime system \(\mathcal{P}\) formed by an unbounded sequence of real number \(1<p_1\leq p_2\leq\dots\). Write \(N_{\mathcal{P}}(x)\) for the number of such generalized integers in \([1,x]\), and define \(\pi_{\mathcal{P}}(x)\) and \(\psi_{\mathcal{P}}(x)\) in analogy to the definitions of these function for the usual set of primes. The main results of the paper says: Suppose that the PNT holds and that for some \(c>0\) we have \(|N_{\mathcal{P}}(x)-cx|\leq xD(x)\) for \(x\geq1\), where \(D\) is right continuous, monotone decreasing, and satisfies \(\int_1^\infty x^{-1}D(x)\,dx<\infty\). Then there follows the sharp Mertens relation \(\int_1^x (1/t)\,d\psi_{\mathcal{P}}(t)=\log x+c_1+o(1)\) for some constant \(c_1\). The authors show that this sharp Merten's relation implies the PNT, the PNT alone is not sufficient for the validity of the sharp Merten's relation, however. For its validity even PNT with the hypotheses \(N_{\mathcal{P}}=cx+O(x/\log x)\) is not sufficient, as shown by the authors in terms of a continuous example of Beurling generalized numbers.
scientific article; zbMATH DE number 6074841
scientific article