Abstract Analytic Number Theory

Well, what is “abstract” analytic number theory? The book’s back cover provides an answer:

The three part treatment [given by Knopfmacher] applies classical analytic number theory [methods] to a wide variety of mathematical subjects not usually treated in an arithmetical way. The first part deals with arithmetical semigroups and algebraic enumeration problems. Part Two addresses arithmetical semigroups with analytical properties of classical type; and the final part explores analytical properties of other arithmetical systems.

Aha! We’re getting closer. What about some samplings from each of the three parts? Fine: here is a pair of titles: §2 of Chapter 1 is titled “Categories satisfying theorems of the Krull-Schmidt type,” and §1 of Chapter 2 is titled “The Dirichlet algebra of an arithmetical semigroup.” Not your usual analytic number theory fare. But then in §7 of the second chapter Knopfmacher discusses “\(\zeta\)-formulae” and we find stuff like this: “A function \(f \in \mathrm{Dir}(G)\) is a \(\mathrm{PIM}\)-function if and only if \(f\) possesses a generalized \(\zeta\)-function.” Here \(G\) is an arithmetical semigroup, \(\mathrm{Dir}(G)\) is the algebra of arithmetical functions on \(G\) (these being functions from \(G\) to the complex numbers: c’est tout) and \(\mathrm{PIM}\) (or \(\mathrm{PIM}(G)\)) stands for the set of prime-independent functions among all the multiplicative functions on \(G\).” So, yes, the forest is getting denser, but there is certainly some connection discernible with classical notions: we find on p. 59, indeed, that if \(G\) is the semigroup of positive integers and \(f\) is the identity, then the ensuing \(\zeta\)-function is Riemann’s.

All right, what about Part Two? Well, Chapter 6 is enticingly titled “The Abstract Prime Number Theorem,” and we find the lead role in the play appearing already on p. 154, to the effect that if \(G\) is an arithmetical semigroup, such that, with \(N_G(x)\) denoting the number of elements in \(G\) having norm at most \(x\), there exist positive \(A\), \(\delta\), and \(0\leq \eta <\delta\), with \(N_G(x) = Ax^\delta _ O(x^\delta)\) as \(x\to\infty\), then \(\pi_G(x) \sim x^\delta/[\delta(\log x)]\): a very familiar business, given that, of course, \(\pi_G(x)\) counts primes in the expected way. The more things change the more they stay the same, at least sort of.

Part Two goes on to a (for me, at least) particularly tantalizing discussion of Fourier analysis: on p. 206, Knopfmacher throws Ramanujan expansions into the mix. And then Part Three deals with some relative esoterica like counterparts of earlier results for arithmetical additive semigroups and arithmetical formations. I will just mention these without further elaboration, as the point is already made, I think: Knopfmacher’s book is not going to be everyone’s cup of tea, given the severity of the level of abstraction he presents, but it generalizes a very classical and fecund subject, viz. the vaunted analytic theory of numbers of Riemann and Dirichlet, and before them, Euler and Gauss. That said, however, the generalizations Knopfmacher presents are exciting in their own right, even as they evince welcome connections with and parallels to their classical precursors. To have these results available in such contexts as the theory of arithmetical semigroups is a boon indeed.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.