Remarks on number theory over additive arithmetical semigroups

UDC 511 We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding ζ -functions. We use these results to prove prime number theorems and a Selberg formula for such semigroups.

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).