On Beurling's prime number theorem (Q1338475)

The author studies a ``Beurling'' zeta function \[ \zeta (s)= \exp \Biggl( \int_1^\infty x^{-s} d\Pi (x) \Biggr)= 1+ \int_1^\infty x^{-s} dN(x) \] for \(\text{Re} (s)>1\), analytic in \(s\) for \(\text{Re} (s)>1\), where \(\Pi\) is a nondecreasing real function with \(\Pi (1) =0\), and \(N\) is a nondecreasing function with \(N(1) =0\), determined by \(\Pi\). Subject to varying asymptotic assumptions on \(N(x)\) as \(x\to \infty\), \textit{A. Beurling} [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] derived corresponding asymptotic conclusions about the function \(\Pi (x)\) as \(x\to \infty\). The starting point for such investigations was that sufficiently strong assumptions about \(N(x)\) imply analogues of the classical prime number theorem. The present author continues this line of investigation on the basis of more general assumptions about \(N(x)\). As with Beurling and other, later authors, the emphasis of the work is on the analytic techniques and results which varying analytic hypotheses lead to.