An asymptotic formula for elements of a semigroup of integers
An example in Beurling's theory of generalised primes
We prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes. The example has its generalised Chebyshev function given by [x]-1, and associated zeta function ζ₀(s) given via , where ζ is Riemann’s zeta function. We study the behaviour of the corresponding Beurling integer counting function N(x), producing O- and Ω- results for the ’error’ term. These are strongly influenced by the size of ζ(s) near...
An example in Beurling's theory of primes
Analytic and arithmetic theory of semigroups with divisor theory
Analytic number theory in formations based on additive arithmetical semigroups.
Arithmetical properties of finite rings and algebras, and analytical number theory. V. Categories and relative analytic number theory.
Arithmetical semigroups I: direct factors.
Arithmetical semigroups II: sieving by large and small prime elements. Sets of multiples.
Arithmetical Semigroups III: Elements With Prime Factors in Residue Classes.
Arithmetical Semigroups V: Multiplicative Functions.
Asymptotic density in combined number systems.
Chebyshev bounds for Beurling numbers
The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.
Chebyshev type estimates in prime number theory.
Comportement asymptotique du produit des k premiers nombres premiers généralisés
Der Satz von der Zerlegung Finslerscher Zahlen in Primfaktoren.
Distinct Degree Factorizations in Additive arithmetical semigroups.
Diversity in inside factorial monoids
In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary...
Diversity in monoids
Let be a (commutative cancellative) monoid. A nonunit element is called almost primary if for all , implies that there exists such that or . We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties (homogeneity and strong homogeneity) that measure how far an almost primary element is from being primary. Finally, as an application...
Ein Beitrag zur abstrakten Primzahltheorie.
"Factorisatio numerorum" in arithmetical semigroups