Examples of Beurling prime systems

Year, pages: 2017, 321 - 344

A generalized prime system $\mathcal{P}$ is a sequence of positive reals $p1, p2,p3,...$ satisfying $1< {p1}≤ {p2}≤ … ≤ {pn} ≤ …$ and for which ${p}n\rightarrow∞$ as ${n \rightarrow∞}$. The $\{pn\}$ are called generalized primes (or Beurling primes) with the products ${pa11}· {pa22}… {pakk}$ (where $k \in \mathbb N$ and $a1, a2, …, ak \in \mathbb N \cup \{0\}$) forming the generalized integers (or Beurling integers). In this article we generalise Balanzario's result [BALANZARIO, E.: \textit{An example in {B}eurling's theory of primes}, Acta Arith. \textbf{87} (1998), 121–139] by adapting his method to show that for any $0<α<1$ there is a continuous g-prime system for which \begin{equation} Π_{\mathcal{P}}(x)={ li}(x)+ O(x\e-( log x)α),\label{nurr1} \end{equation} and \begin{equation} \mathcal {N}_{\mathcal {P}}(x)= ρ x + Ω\pm(x\e-c( log x)β),\label{nue1} \end{equation} for some $ρ, c>0$ hold with $β=α$. We use the method developed by Diamond, Montgomery and Vorhauer [DIAMOND, H.— —MONTGOMERY, H.—VORHAUER, U.: \textit{{B}eurling primes with large oscillation}, Math. Ann. \textbf{334} (2006), 1–36] and Zhang [ZHANG, W.: \textit{{B}eurling primes with {RH} and {B}eurling primes with large oscillation}, Math. Ann. \textbf{337} (2007), 671–704] to prove (by using some measure theoretical results) that there is a \emph{discrete} system of Beurling primes satisfying \eqref{nurr1} and \eqref{nue1} which is similar to the continuous system. Finding discrete example is typically more challenging since one cannot control the various growth rates (of $π_{\mathcal {P}}(x),$ $\mathcal {N}_{\mathcal {P}}(x)$ and $ζ_{\mathcal {P}} (s)$) so easily.

Al-Maamori, F. 2017. Examples of Beurling prime systems. In Mathematica Slovaca, vol. 67, no.2, pp. 321-344. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0269

Al-Maamori, F. (2017). Examples of Beurling prime systems. Mathematica Slovaca, 67(2), 321-344. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0269