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An example in Beurling's theory of generalised primes

Faez Al-Maamori, Titus Hilberdink (2015)

Acta Arithmetica

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 - ( ζ ' ₀ ( s ) ) / ( ζ ₀ ( s ) ) = ζ ( s ) - 1 , 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...

Chebyshev bounds for Beurling numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

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 ∫ 1 ∞ | N ( x ) - A x | d x / x 2 < ∞ 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.

Diversity in inside factorial monoids

Ulrich Krause, Jack Maney, Vadim Ponomarenko (2012)

Czechoslovak Mathematical Journal

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

Jack Maney, Vadim Ponomarenko (2012)

Czechoslovak Mathematical Journal

Let M be a (commutative cancellative) monoid. A nonunit element q ∈ M is called almost primary if for all a , b ∈ M , q ∣ a b implies that there exists k ∈ ℕ such that q ∣ a k or q ∣ b k . 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...

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