[Paper Review] Long gaps between primes

Kevin Ford, Ben Green+3 more|arXiv (Cornell University)|Dec 16, 2014

Analytic Number Theory Research23 references13 citations

TL;DR

This paper establishes a new lower bound for the maximal gap between consecutive primes up to X, proving that $ G(X) \gg \frac{\log X \log_2 X \log_4 X}{\log_3 X} $, significantly improving upon previous results. The authors achieve this by generalizing a hypergraph covering theorem of Pippenger and Spencer using the Rödl nibble method and applying it within a multidimensional prime-detecting sieve framework to construct long intervals free of primes.

ABSTRACT

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

Motivation & Objective

• To improve the quantitative lower bound on the largest gap $ G(X) $ between consecutive primes up to $ X $, surpassing prior results that had not fully leveraged the power of multidimensional sieves.
• To generalize the hypergraph covering theorem of Pippenger and Spencer to enable more flexible and effective sieve constructions in the context of prime gaps.
• To provide an effective, quantitative improvement over previous bounds, including those of the first three and fifth authors and the fourth author, by combining sieve methods with advanced covering techniques.
• To lay the groundwork for future work on chains of consecutive large prime gaps by formulating key propositions in greater generality than strictly necessary for Theorem 1.

Proposed method

• The authors generalize the Pippenger-Spencer hypergraph covering theorem using the Rödl nibble method to allow for more flexible and robust covering of sets with controlled error terms.
• They apply a multidimensional prime-detecting sieve with carefully chosen weights to identify long intervals devoid of primes, leveraging the structure of linear forms in primes.
• The method involves constructing a sieve weight function $ w_{k,\mathcal{L},B,R}(n) $ that detects integers in arithmetic progressions with controlled density, using the dual form of the sieve to estimate the number of primes in shifted intervals.
• Key estimates rely on the singular series $ \mathfrak{S} $, the singular integral $ J_k $, and the dual singular integral $ I_k $, which control the expected number of solutions to systems of linear equations in primes.
• The proof uses a hybrid approach combining the sieve-theoretic framework of [32] with the linear forms method of [13], particularly through the use of admissible tuples and residue classes to cover intervals.
• A crucial step involves showing that the sieve weights remain stable under perturbations of the linear forms, with error terms controlled via $ O(1/\log_{2}^{10}x) $ bounds.

Experimental results

Research questions

• RQ1 Can the lower bound for the maximal prime gap $ G(X) $ be improved beyond the $ \frac{\log X \log_2 X}{\log_3 X} $ threshold established in prior work?
• RQ2 Can a generalized hypergraph covering theorem be constructed that allows for effective control over sieve error terms in multidimensional prime detection?
• RQ3 To what extent can the Rödl nibble method be adapted to provide quantitative improvements in sieve-theoretic constructions for prime gaps?
• RQ4 How can the methods of multidimensional sieves and linear forms in primes be unified to yield stronger lower bounds on large prime gaps?

Key findings

• The paper establishes the new lower bound $ G(X) \gg \frac{\log X \log_2 X \log_4 X}{\log_3 X} $ for all sufficiently large $ X $, representing a significant quantitative improvement over previous results.
• The implied constant in the bound is effective, meaning the result is not only asymptotic but also computationally meaningful.
• The authors prove a generalized hypergraph covering theorem that extends the Pippenger-Spencer result and is of independent interest in extremal combinatorics.
• The method successfully combines the sieve-theoretic framework of [32] with the linear forms approach of [13], enabling tighter control over error terms and improved estimates.
• The proof relies on a refined analysis of the singular series $ \mathfrak{S} $, the dual singular integral $ I_k $, and the main term $ J_k $, which are essential for estimating the number of prime solutions in shifted intervals.
• The authors demonstrate that the sieve weights remain stable under small perturbations of the linear forms, with error terms bounded by $ O(1/\log_{2}^{10}x) $, which is sufficient to absorb lower-order contributions.