[Paper Review] A new generalized prime random approximation procedure and some of its applications

Frederik Broucke, Jasson Vindas|arXiv (Cornell University)|Feb 16, 2021

Meromorphic and Entire Functions12 references4 citations

TL;DR

This paper introduces a novel generalized prime random approximation procedure that constructs Beurling prime systems with exceptional control over the discrepancy between the prime counting function and a given target function F, achieving |πₚ(x) − F(x)| ≤ 2. The method improves upon the DMVZ probabilistic scheme by enabling tighter error bounds, which are then applied to resolve open problems on Dirichlet series with unique zeros, well-behaved generalized number systems, and Beurling zeta functions with large oscillations.

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.

Motivation & Objective

• To develop a refined random approximation method that produces generalized prime systems with significantly improved error control over the classical DMVZ method.
• To address open questions in Beurling number theory, particularly concerning the existence of Dirichlet series with a unique zero in their half-plane of convergence.
• To construct well-behaved generalized number systems that satisfy recent open conditions posed by Hilberdink and Neamah.
• To improve upon prior constructions of Beurling systems with regular primes but highly irregular integers by achieving bounded discrepancy in the prime counting function.

Proposed method

• The method constructs a discrete generalized prime system 𝒫 = {p_j} by assigning random variables with a new distribution rule, differing fundamentally from the DMVZ approach to allow tighter control on πₚ(x) − F(x).
• It applies a modified probabilistic scheme that ensures |πₚ(x) − F(x)| ≤ 2 for all x ≥ 1, under the condition that F is non-decreasing, right-continuous, unbounded, and satisfies the Chebyshev bound F(x) ≪ x/log x.
• The construction is extended to approximate measures dF that are not necessarily absolutely continuous, allowing application to singular or non-smooth target functions.
• For continuous F, the method produces a strictly increasing sequence with |πₚ(x) − F(x)| ≤ 1, enhancing precision in smooth settings.
• The method relies on a new bound on exponential sums: |∑_{p_j ≤ x} p_j^{−it} − ∫₁ˣ u^{−it} dF(u)| ≪ √x + √(x log(|t|+1)/log(x+1)), which preserves the strength of the DMVZ estimate while enabling tighter counting control.
• The procedure is adapted to construct specific prime systems by applying it to target functions Π_c(x) and π_c(x), which are built from the logarithmic integral and oscillatory error terms.

Experimental results

Research questions

• RQ1 Can a Dirichlet series with exactly one zero in its half-plane of convergence be constructed?
• RQ2 Do there exist well-behaved generalized number systems that satisfy the recent conditions posed by Hilberdink and Neamah?
• RQ3 Can a Beurling prime system be constructed with regular primes but extremely irregular integers, while maintaining bounded discrepancy in the prime counting function?
• RQ4 Is it possible to achieve |πₚ(x) − F(x)| ≤ 2 for any unbounded, right-continuous, non-decreasing F satisfying the Chebyshev bound?

Key findings

• The method constructs a generalized prime system 𝒫 such that |πₚ(x) − F(x)| ≤ 2 for all x ≥ 1, significantly improving upon the √x error bound of the DMVZ method.
• For continuous F, the method produces a strictly increasing sequence with |πₚ(x) − F(x)| ≤ 1, ensuring better regularity and precision.
• The method successfully resolves Balazard's question by constructing a Dirichlet series with exactly one zero in its half-plane of convergence.
• It constructs a generalized number system with regular primes and highly irregular integers, improving upon a prior construction with uncontrolled discrepancy.
• The method enables the construction of a Beurling zeta function with no zeros in the right half-plane {Re s > 0}, matching the behavior of the classical Riemann zeta function in this respect.
• The method's core bound on exponential sums is preserved: |∑_{p_j ≤ x} p_j^{−it} − ∫₁ˣ u^{−it} dF(u)| ≪ √x + √(x log(|t|+1)/log(x+1)), ensuring analytic control for zeta function properties.