Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new generalized prime random approximation procedure and some of its applications

Frederik Broucke, Jasson Vindas

Abstract

We present a new random approximation method that yields the existence of a discrete Beurling prime system $\mathcal{P}=\{p_{1}, p_{2}, \dotso\}$ which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function $F$. This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704]. We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed.

A new generalized prime random approximation procedure and some of its applications

Abstract

We present a new random approximation method that yields the existence of a discrete Beurling prime system which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function . This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704]. We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed.

Paper Structure

This paper contains 4 sections, 7 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Balazard's question for general Dirichlet series and the existence of well-behaved Beurling numbers of type $[0,1/2,1/2]$
  4. A Beurling number system with highly regular primes but integers with large oscillation

Key Result

Theorem 1.1

Let $f$ be a non-negative $L^{1}_{loc}$-function supported on $[1,\infty)$ satisfying Then there exists an unbounded sequence of real numbers $1<p_1<p_2<\dots<p_j<\dots$ such that for any real $t$ and any $x\ge1$

Theorems & Definitions (16)

  • Theorem 1.1: Diamond, Montgomery, Vorhauer DiamondMontgomeryVorhauer, Zhang Zhang2007
  • Theorem 1.2
  • Proposition 1.4
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['th: discretization']}
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • ...and 6 more