A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes

In this paper we work out a Riemann–von Mangoldt type formula for the summatory function := , where is an arithmetical semigroup (a Beurling generalized system of integers) and is the corresponding von Mangoldt function attaining with a prime element and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function , belonging to , to the number of zeroes of in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.

ON MURATA DENSITIES

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci.56(7) (1980) 351–353; On some fundamental relations among certain asymptotic densities, Math. Rep. Toyama Univ.4(2) (1981) 47–61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).

The function field abstract prime number theorem

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties(i) ∂(l) = 0,∂(p) > 0 for p∈P;(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

Arithmetical Semigroup Rings

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].