Examples of beurling prime systems

Abstract

Abstract A generalized prime system 𝓟 is a sequence of positive reals p 1, p 2, p 3, … satisfying 1 < p 1 ≤ p 2 ≤ ⋯ ≤ p n ≤ ⋯ and for which p n → ∞ as n → ∞. The {p n } are called generalized primes (or Beurling primes) with the products p 1 a 1 ⋅ p 2 a 2 … p k a k ${p^{a_{1}}_{1}}\cdot {p^{a_{2}}_{2}}\dots {p^{a_{k}}_{k}}$ (where k ∈ ℕ and a 1, a 2, ⋯, a k ∈ ℕ ∪ {0}) forming the generalized integers (or Beurling integers). In this article we generalise Balanzario’s result [BALANZARIO, E.: An example in Beurling’s theory of primes, Acta Arith. 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 Π P ( x ) = l i ( x ) + O ( x e − ( log ⁡ x ) α ) , $$ \Pi_{\mathcal {P}}(x)={\rm li}(x)+ O(x\text{e}^{-(\log x)^\alpha}), $$ (0.1) and N P ( x ) = ρ x + Ω ± ( x e − c ( log ⁡ x ) β ) , $$ \mathcal {N}_{\mathcal {P}}(x)= \rho x + \Omega_{\pm}(x\text{e}^{-c(\log x)^\beta}), $$ (0.2) We use the method developed by Diamond, Montgomery and Vorhauer [DIAMOND, H.—MONTGOMERY, H.—VORHAUER, U.: Beurling primes with large oscillation, Math. Ann. 334 (2006), 1–36] and Zhang [ZHANG, W.: Beurling primes with RH and Beurling primes with large oscillation, Math. Ann. 337 (2007), 671–704] to prove (by using some measure theoretical results) that there is a discrete system of Beurling primes satisfying (0.1) and (0.2) which is similar to the continuous system. Finding discrete example is typically more challenging since one cannot control the various growth rates (of π 𝓟(x), 𝓝𝓟(x) and ζ 𝓟(s)) so easily.