Large gaps between primes (1408.5110v2)

Abstract: We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.

• The paper by James Maynard demonstrates that the gaps between consecutive prime numbers can be arbitrarily large, confirming a long-standing conjecture by Erdős.
• The proof significantly enhances classical methods using advanced sieve techniques and innovative applications of probability, building on recent work on small prime gaps.
• This work refines the understanding of prime distribution and provides a basis for potential algorithmic advancements in computational number theory and cryptography.

Analysis of "Large gaps between primes" by James Maynard

James Maynard's paper titled "Large gaps between primes" addresses a classical question in analytic number theory that has intrigued mathematicians for many years: the size of gaps between consecutive prime numbers. The paper provides a comprehensive mathematical framework, significantly advancing upon the methods originally developed by Erdős and Rankin.

Key Contributions

The central result of the paper is the demonstration that pairs of consecutive primes can have arbitrarily large differences, a result that refines our understanding of the distribution of prime numbers. Specifically, Maynard shows that for any fixed real number $t$, there are pairs of consecutive primes less than $x$ such that their differences exceed

$$t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}.$$

This resolves a well-known question posed by Erdős, who conjectured about the potential magnitude of these gaps.

Methodological Advances

Maynard's proof incorporates advanced sieve methods, building on recent advancements regarding small gaps between primes. The paper adapts techniques similar to those used by methods targeting small primes but applies them innovatively to large gaps.

Highlights of the Methodology:

1. Erdős-Rankin Framework Enhancement: The paper enhances the classical Erdős-Rankin construction for large gaps, incorporating sieve ideas to elevate the bound on the maximal prime gap.
2. Utilization of Probability and Sieve Methods: An innovative use of the probabilistic method alongside a tailored sieve (akin to those used in the Green-Tao theorem) is key to the approach.
3. Collaborative Results: The paper’s results align with concurrent findings by Ford, Green, Konyagin, and Tao, who independently achieved similar outcomes using different methods. Their approach utilizes results on linear equations in primes, which contrasts with Maynard's focus on small gaps between primes.

Numerical and Theoretical Implications

The implications of this research are profound for both theoretical and practical aspects of number theory. Theoretically, the results deepen the understanding of prime distribution. Practically, it provides a basis for algorithmic advancements in prime number theory, which could impact computational mathematics, cryptography, and numerical analysis.

Speculation on Future Research

Given the substantial progress made in proving large gaps between primes, future research may delve further into the characteristics of prime gaps and their application. Subsequent investigations might explore the generalization of these results to broader classes of prime-related problems or extend the use of these methods to other unresolved questions in analytic number theory.

Conclusion

Maynard's work on large gaps between primes stands as a pivotal achievement, enhancing classical concepts with modern insights. His combination of probabilistic methods and intricate sieve techniques breaks new ground in addressing a longstanding problem. As such, this paper represents a substantial contribution to the field of number theory, with its methodology likely to influence future research directions.