[Paper Review] On the singular series in the prime k-tuple conjecture

J. Pintz|arXiv (Cornell University)|Apr 7, 2010

Analytic Number Theory Research3 references3 citations

TL;DR

This paper establishes sharp quantitative bounds on the average behavior of the singular series in the prime k-tuple conjecture, showing that for any admissible k-tuple H ⊂ [1, H], the average ratio Sₕ(H) of singular series over shifts h ∈ [1, H] converges to 1 as H → ∞, with explicit error rates depending on k and ε. The key result is that Sₕ(H) = 1 + O(ε) when H ≥ exp(k¹ᐟᵝ), significantly strengthening prior mean-value estimates and providing a simpler, more effective tool for applications in small prime gaps and almost-prime problems.

ABSTRACT

In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the length of the interval $H$ is sufficiently large as a function of $k$). This result of Gallagher played a key role in our original joint work with D. A. Goldston and C. Y. Yildirim, where we showed the existence of infinitely many small gaps between consecutive primes. In the present work some weaker variants of Gallagher`s result are also proved (in an even easier way) which are still sufficient for the above mentioned applications.

Motivation & Objective

• To refine and strengthen the mean-value estimates of the singular series used in the proof of bounded gaps between primes.
• To provide a simpler, more effective lower bound for the singular series ratio that suffices for applications in small prime gaps and almost-prime problems.
• To analyze the convergence rate of the average singular series ratio Sₕ(H) = (1/H)∑ₕ₌₁ᴴ ℱ(ℋ∪{h})/ℱ(ℋ) as H → ∞ for fixed k.
• To establish explicit quantitative bounds on the convergence of Sₕ(H) to 1, depending on k and ε, using elementary estimates on product over primes.

Proposed method

• Define the singular series ℱ(ℋ) = ∏ₚ (1 − νₚ/p) / (1 − 1/p)^k, where νₚ is the number of distinct residues of ℋ modulo p.
• Analyze the ratio ℱ(ℋ ∪ {h}) / ℱ(ℋ) by splitting the product over primes into three parts: p ≤ y, p > y and p | Δ, and p > y and p ∤ Δ, with y = (5/6) log H and Δ = ∏(h − hᵢ).
• Use the Prime Number Theorem and bounds on ∑_{p|Δ} 1/p to show that ∏₂ (over p > y dividing Δ) is bounded by exp(O(ε)) when ∑_{p|Δ} log p ≤ 2ky.
• Establish that ∏₃ (over p > y not dividing Δ) is 1 + O(k/(y log y)) = 1 + O(ε) for y = (5/6) log H.
• Use periodicity modulo P = ∏_{p≤y} p to average ∏₁(h) over h ∈ [1, P], showing the average is exactly 1 via combinatorial counting of residue classes.
• Combine the estimates for ∏₁, ∏₂, and ∏₃ to prove Sₕ(H) = 1 + O(ε) under the condition H ≥ exp(k¹ᐟᵝ), and derive weaker but absolute bounds via trivial estimates on ∏₂.

Experimental results

Research questions

• RQ1 How fast does the average singular series ratio Sₕ(H) converge to 1 as H → ∞ for a fixed k-tuple ℋ?
• RQ2 Can a simpler, more effective lower bound for the singular series ratio be derived that suffices for applications in bounded prime gaps?
• RQ3 What is the optimal dependence of the convergence rate on k and H, particularly in terms of exponential thresholds?
• RQ4 How do the contributions from small and large primes in the singular series product interact in the average?
• RQ5 Can the global mean-value result of Gallagher be recovered and strengthened via pointwise estimates on individual shifts?

Key findings

• For any admissible k-tuple ℋ ⊂ [1, H], the average singular series ratio Sₕ(H) = (1/H)∑ₕ₌₁ᴴ ℱ(ℋ ∪ {h}) / ℱ(ℋ) satisfies Sₕ(H) = 1 + O(ε) when H ≥ exp(k¹ᐟᵝ), with ε > 0 arbitrary.
• The convergence rate is quantified: if H ≥ exp(k¹ᐟᵝ), then Sₕ(H) = 1 + O(ε), showing that the average ratio approaches 1 uniformly for large H.
• A weaker but absolute lower bound is established: Sₕ(H) ≥ c₁ for H ≥ exp(c₂ k / log k), with absolute constants c₁, c₂ > 0.
• The average of the ratio ℱ(ℋ ∪ {h}) / ℱ(ℋ) over h ∈ [1, P] (with P = ∏_{p≤y} p) is exactly 1 due to symmetric residue distribution, proving the central identity in the proof.
• The error terms in the product decomposition are controlled using bounds on ∑_{p|Δ} 1/p and the Prime Number Theorem, leading to the O(ε) error under the stated H threshold.
• The result strengthens Gallagher’s mean-value estimate by providing pointwise control on the average ratio, which is sufficient for applications in bounded gaps between primes and almost-prime problems.