The prime number theorem for Beurling's generalized numbers. New cases
\textit{A. Beurling} [Acta Math. 68, 255--291 (1937; Zbl 0017.29604)] has defined generalized prime numbers \(p_k\) and integers \(n_k\) \((k\geq 1)\). He proved the prime number theorem (PNT) NEWLINE\[NEWLINE\pi(x):= \sum_{p_k<x} 1\sim{x\over\log x}NEWLINE\]NEWLINE under the condition NEWLINE\[NEWLINEN(x):= \sum_{n_k<x} 1= ax+ O\Biggl({x\over\log^\gamma x}\Biggr),NEWLINE\]NEWLINE where \(a>0\), \(\gamma>{3\over 2}\). The authors extend Beurling's theorem by replacing this condition by an asymptotic in Cesàro sense NEWLINE\[NEWLINE\int^x_1 {N(t)-at\over t} \Biggl(1-{t\over x}\Biggr)^m \,dt= O\Biggl({x\over\log^\gamma x}\Biggr).NEWLINE\]NEWLINE The proof works by Tauberian theorems and is close to Landau-Ikehara's approach to the PNT. The authors construct a number system such that \(\gamma={3\over 2}\) and the PNT does not hold. Therefore, the condition \(\gamma>{3\over 2}\) is sharp.
scientific article; zbMATH DE number 6023952