Some examples in the theory of Beurling's generalized prime numbers (Q2833610)

Beurling's primes are systems of real numbers \(P=\{p_k\}_{k\geq 1}\) with \(1<p_1<\cdots<p_k<\cdots\) such that \(p_k\to\infty\). Their associate integers are \(n_0=1<n_1<\cdots<n_k<\cdots\) all elements in the multiplicative semigroup generated by \(P\). Beurling's problem is to determine the behavior of NEWLINE\[ N(x)=\sum_{n_k\leq x} 1 \] NEWLINE NEWLINEsuch that \(\pi_P(x)=\sum_{p_k\leq x} 1\) to satisfy PNT, namely \(\pi_P(x)\sim x/\log x\) as \(x\to\infty\). \textit{A. Beurling} [Acta Math. 68, 255--291 (1937; JFM 63.0138.01)] showed that NEWLINE\[ N(x)=ax+O\left(\frac{x}{(\log x)^{\gamma}}\right)\tag{1} \] NEWLINE NEWLINEwhere \(a>0\) and \(\gamma>3/2\) suffices for PNT to hold, while \textit{J.-P. Kahane} [J. Théor. Nombres Bordx. 9, No. 2, 251--266 (1997; Zbl 0905.11042)], answering a conjecture of Bateman and Davenport (see [\textit{P. T. Bateman} and \textit{H. G. Diamond}, MAA Stud. Math. 6, Studies Number Theory, 152--210 (1969; Zbl 0216.31403)]), showed that the \(L^2\)-hypothesis NEWLINE\[ \int_1^{\infty} \left(\frac{(N(t)-a(t))^2}{t}\right)^2 \frac{dt}{t}<\infty\tag{2} \]NEWLINE implies PNT. In this paper, the authors construct, for every \(\alpha\in (1,3/2)\), a prime number system \(P_{\alpha}\) such that NEWLINE\[ N_{P_{\alpha}}(x)=a_{\alpha} x+O\left(\frac{x}{(\log x)^n}\right),\qquad n=1,2,\ldots \] NEWLINE but NEWLINE\[ \int_1^{\infty} \left(\frac{(N_{P_{\alpha}}(t)-a_{\alpha} t)^2}{t}\right)^2 \frac{dt}{t}=\infty, \] NEWLINENEWLINEyet PNT still holds for \(P_{\alpha}\) with remainder \(x/(\log x)^{\alpha}\), namely NEWLINE\[ \pi_{P_{\alpha}}(x)=\frac{x}{\log x}+O\left(\frac{x}{(\log x)^{\alpha}}\right). \] NEWLINE NEWLINEIn other words, \(P_{\alpha}\) satisfies (1) but not (2) yet PNT holds. The proof use heavy complex analysis.