In this paper 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.

An example in Beurling's theory of generalised primes

Given $$\beta \in (0,1)$$
 , we show the existence of a Beurling generalized number system whose integer counting satisfies $$N(x) = ax + O\bigl (x\exp (-c\log ^{\beta } x)\bigr )$$
 for some $$a>0$$
 and $$c>0$$
 , and whose prime counting function satisfies $$\pi (x) = {{\,\mathrm{Li}\,}}(x) + \varOmega \bigl (x\exp (-c'(\log x)^{\frac{\beta }{\beta +1}})\bigr )$$
 for some $$c'>0$$
 . This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the prime number theorem with Malliavin-type remainder.

Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the system of generalized integers NP generated by P satisfies ∑m⩽x,m∈N1=ax+O(xδ), a&gt;0, 0⩽δ&lt;1, and the function ζP(s) in some strip lying in σ^&lt;σ&lt;1, σ^&gt;δ, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. 

On Value Distribution of Certain Beurling Zeta-Functions

Fur ein SystemG vonBeurlingschen Zahlen wird die Gleichverteilung der ‘ganzen’ Elemente in der FormNG(x)≈cx fur einc>0 hergeleitet unter der Voraussetzung, das die zugehorigen Primelemente so verteilt liegen, das$\pi _G (x) = \frac{x}{{log x}} + 0\left( {\frac{x}{{log x log _2^\gamma x}}} \right)$ gilt mit γ>1. Dieses Ergebnis ist insofern scharf, als es SystemeG gibt mit γ=1 undNG(x)≈cx fur keinc>0.

Zur Gleichverteilung von Beurlingschen Zahlen

Solved, Semi-Solved, and Unsolved Problems in Generalized Integers: A Survey

The prime number theorem for generalized primes

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The prime number theorem for generalized primes