[Paper Review] Prime pairs and Zeta's zeros

Jacob Korevaar|ArXiv.org|Jun 5, 2008

Analytic Number Theory Research27 references3 citations

TL;DR

This paper establishes a deep equivalence between the Hardy-Littlewood prime pair conjecture (PPC) and the boundary behavior of Dirichlet series involving the non-trivial zeros of the Riemann zeta function. Using a Tauberian approach and assuming the Riemann Hypothesis, it shows that the PPC holds if and only if certain zeta-zero-related functions exhibit pseudofunction boundary behavior at Re(s) = 1, linking prime pair distribution to the fine-scale statistics of zeta zeros.

ABSTRACT

There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of pi_{2r}(x), the number of prime pairs (p,p+2r) with p not exceeding x. However, it is still not known whether there are infinitely many prime pairs with given even difference! Using a strong hypothesis on (weighted) equidistribution of primes in arithmetic progressions, Goldston, Pintz and Yildirim have shown (2007) that there are infinitely many pairs of primes differing by at most sixteen. The present author uses a Tauberian approach to derive that the PPC is equivalent to specific boundary behavior of certain functions involving zeta's complex zeros. Under Riemann's Hypothesis and on the real axis, these functions resemble pair-correlation expressions. A speculative extension of Montgomery's classical work (1973) would imply that there must be an abundance of prime pairs.

Motivation & Objective

• To establish a precise equivalence between the prime pair conjecture (PPC) and the boundary behavior of Dirichlet series tied to zeta’s non-trivial zeros.
• To investigate whether the PPC can be derived from strong equidistribution hypotheses on primes in arithmetic progressions, using Tauberian theory.
• To explore the implications of the Riemann Hypothesis and Montgomery’s pair correlation conjecture for the abundance of prime pairs.
• To demonstrate that under the Riemann Hypothesis, the functions derived from zeta zeros resemble pair-correlation expressions on the critical line.
• To provide a conditional proof of the abundance of prime pairs based on the positivity of certain double sums involving zeta zeros and a sieving function.

Proposed method

• Applies a two-way Wiener–Ikehara Tauberian theorem to relate the asymptotic behavior of the prime pair counting function π₂ᵣ(x) to the boundary behavior of Dirichlet series D₂ᵣ(s) associated with zeta zeros.
• Uses the condition that the difference f(u+iv) − A/(u+iv) has a distributional limit as u ↘ 1, which must be a pseudofunction on finite intervals.
• Introduces a sieving function E^λ(ν) to analyze the sum ∑π₂ᵣ(x) over even differences 2r ≤ λ, leveraging its non-smoothness but effective behavior.
• Analyzes the double sum Σ²_λ(s) = ∑_{ρ,ρ'} Γ(ρ−s)Γ(ρ′−s) t^{2s−ρ−ρ′} cos(πρ/2)cos(πρ′/2) to show non-negativity when E^λ(t) ≥ 0.
• Employs the identity that (s−1/2)D₂ᵣ(s) → 0 as s ↘ 1/2 for angular approach, which is necessary for the pseudofunction condition.
• Combines estimates on D₂ᵣ(s) with the hypothesis that the average of π₂ₘⱼ(x) over a set of mⱼ is bounded below by c times the average over all 2r ≤ λ, to derive lower bounds on the limsup of π₂ₘ(x)/(x/log²x).

Experimental results

Research questions

• RQ1 Is the prime pair conjecture (PPC) equivalent to specific boundary behavior of Dirichlet series built from zeta’s non-trivial zeros?
• RQ2 Can the Riemann Hypothesis and the pseudofunction boundary condition jointly imply the existence of infinitely many prime pairs with a given even difference?
• RQ3 Does the positivity of the double sum Σ²_λ(s) for 1/2 < s < 1, under non-negative E^λ(t), lead to a conditional abundance of prime pairs?
• RQ4 To what extent does Montgomery’s pair correlation conjecture imply the abundance of prime pairs under the PPC?
• RQ5 Can a Tauberian approach with a non-smooth sieving function E^λ(ν) still yield meaningful asymptotic estimates for π₂ᵣ(x)?

Key findings

• The prime pair conjecture (PPC) is equivalent to the condition that the function g(w) = f(w) − A/w has a pseudofunction boundary limit at Re(w) = 1.
• Under the Riemann Hypothesis, the functions derived from zeta zeros resemble Montgomery’s pair-correlation expressions on the critical line Re(s) = 1/2.
• The limsup of (1/x)∑_{r≤m} ψ₂ᵣ(x) exceeds (2−ε)m, implying that the average number of prime pairs with difference 2r up to x is bounded below by a constant multiple of m.
• If the average of π₂ₘⱼ(x) over a set of mⱼ satisfies a lower bound condition, then the limsup of π₂ₘ(x)/(x/log²x) is bounded below by a positive constant c.
• The double sum Σ²_λ(s) is non-negative for 1/2 < s < 3/4 when E^λ(t) ≥ 0, due to the squared modulus structure of the sum over zeta zeros.
• The result δD₂ₘ((1/2)+δ) ≥ c(1/2 − 1/(4λδ)) − o(1) as δ↘0 and λ→∞ in a suitable sequence S implies that the residue at s=1/2 is bounded below, supporting abundance of prime pairs.