Beurling generalized numbers (Q2829451)

Generalized (briefly g-) numbers have their roots in the \textit{A. Beurling}'s paper [Acta Math. 68, 255--291 (1937; Zbl 0017.29604)] where he initiated a study of the dependence of the validity of a prime number theorem type theorem in multiplicative systems of real numbers when their additive structure is not taking into account. More precisely, he started with a set of real numbers \({\mathcal P}=\{1<p_1\leq p_2\leq p_3\leq\dots\}\) such that \(\lim_{i\to\infty}p_i=\infty\) and took the multiplicative semigroup \({\mathcal N}\) generated by \({\mathcal P}\). The question he asked was what we can deduce from the analytical behavior of the counting function \(N_{\mathcal P}(x)=\sum_{n\in {\mathcal N}}1\) in direction of the analytical behavior of the `prime' counting function \(\pi_{\mathcal P}=\sum_{p\in{\mathcal P}}1\) and vice verse. For instance, if \({\mathcal P}\) is the set of all odd primes, then \(N_{\mathcal P}(x)\sim x/2\) but still \(\pi_{\mathcal P}(x)\sim x/\log x\). Beurling found a surprising `barrier' for the validity of the PNT: If \(N_{\mathcal P}=Ax+O(x/\log^\gamma x)\), \(A>0\), then \(\pi_{\mathcal P}\sim x/\log x\) provided \(\gamma>3/2\). Buerling proved that if \(g=3/2\) then the conclusion need not hold. The simplest non-trivial application of the Beurling approach is the Landau's prime ideal theorem.NEWLINENEWLINENEWLINEAnalytically, define two functions \(N(x)\) and \(\Pi(x)\) connected through a function \(\zeta(s)\) by the relations \(\zeta(s)=\int_{1-}^\infty u^{-s}dN(u)\) and \(\log\zeta(s)=\int_1^\infty u^{-s}d\Pi(u)\). The question is how far the asymptotic behavior of \(\Pi(x)\) as \(x\to\infty\) determines that of \(N(x)\), and vice versa.NEWLINENEWLINENEWLINEIn the book under review, the authors study continuous analogues of g-numbers. The book presents a very interesting contribution to the areas where the interested reader can find answers to the questions of the following type (as formulated by the authors in the introduction): (1) How sensitive is the prime number theorem to the additive structure of the underlying integers, (2) How important is this additive structure for validity of Riemann hypothesis, (3) Is the counting measure of integers representable as a certain exponential, (4) Is there a continuous analogue of prime number theory, (5) What is the crux of the problem to improve de la Valleé Poussin PNT error estimate?NEWLINENEWLINENEWLINEThe book has 18 chapters, each of them is sufficiently `thin' to describe the intended material in sufficient depth and still reflecting the substance of the propounded topic. The approach which the authors use in the book shows the following reformulation of the core of proofs of Chebyshev's inequalities taken from Chapter 3: \(LdN=dN\ast d\psi\), where \(dN\) is the integer counting measure, \(d\psi=Ld\Pi\) is the weighted prime and prime power counting measure and \(L\) is the log operator. Applications of the above Mellin's formulas shows that this product formula holds in all g-number systems of `polynomial growth' (Theorem 3.1). The Chebyshev bounds are subject of study in Chapters 9 (elementary theory), 10 (analytic methods), and 11 (optimality of bounds). To the prime number theorem there are devoted Chapters 13 (Beurling PNT), 14 (equivalences to the PNT), 15 (Kahane's PNT), 16 (PNT with remainder), and 17 (optimality of the remainder term). As there follows from a survey by the first author and \textit{P. Bateman} [MAA Stud. Math. 6, Studies Number Theory, 152--210 (1969; Zbl 0216.31403)] even if there are generalized prime structures which are quite close to PNT, RH fails in them. Thus an attack to the RH will require more than an assumption that the positive integers form a multiplicative semigroup which counting function is close to \(x\).NEWLINENEWLINENEWLINEDespite this, the present book is an overdue survey of a fascinating branch of analytic number theory. The book masterly covers a wide and important range of topics related to PNT. The text includes a number of sensitively selected accompanying examples. All this only shows that the book is written by leading masters in the subject, and it can only be recommended to all who are interested in a more wider background for the validity of PNT.