Small gaps between primes (1311.4600v3)

Abstract: We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.

• The paper refines the GPY sieve method to prove that a positive proportion of admissible k-tuples yield infinitely many primes under the prime k-tuples conjecture, advancing analytic number theory.
• The research provides an unconditional gap bound of 600 between consecutive primes and, assuming the Elliott-Halberstam conjecture, a tighter bound of 12, demonstrating improved numerical results.
• The methodology extends to various subsequences of primes, offering new insights into prime configurations with potential applications in cryptography and further theoretical research.

Insights into "Small gaps between primes" by James Maynard

James Maynard's paper "Small gaps between primes" presents a significant advancement in analytic number theory, particularly pertaining to the distribution of prime numbers. Utilizing a refined version of the GPY sieve method, originally developed by Goldston, Pintz, and Yıldırım, Maynard introduces novel techniques that not only overcome previous constraints but also yield impactful numerical results regarding gaps between consecutive prime numbers.

Core Contributions

1. Refinement of the GPY Sieve Method: Maynard refines the GPY sieve, which focuses on prime tuples and small gaps between primes. This refinement allows for the demonstration that a positive proportion of admissible $k$-tuples satisfy the prime $k$-tuples conjecture, which posits that such tuples should produce infinitely many primes if admissible. This represents a significant theoretical verification of this conjecture for certain cases, supported by empirical methods.
2. Numerical Results for Prime Gaps:
   • General Bound: Maynard establishes that $\liminf_{n \to \infty} (p_{n+1} - p_n) \le 600$ without assuming conjectures beyond standard analytic results. This result is notable because it provides an unconditional upper bound on the differences between consecutive primes.
   • Conditional Bound: Assuming the Elliott-Halberstam conjecture, an even tighter bound $\liminf_{n \to \infty} (p_{n+1} - p_n) \le 12$ is achieved. This highlights the potential power of related distribution conjectures in analytic number theory.
3. Implications for Prime-Tuple Configurations: The methodology outlined in the paper ensures that for any integer $m$, $\liminf_{n \to \infty} (p_{n+m} - p_n) < \infty$, offering new insights into the behavior and distribution of prime numbers within bounded gaps.
4. Generality and Extensions: Maynard’s method also demonstrates applications to different subsequences of prime numbers, extending its utility and suggesting versatility in addressing additional conjectures within number theory, such as those involving primes in arithmetic progressions.

Theoretical and Practical Implications

The implications of Maynard’s paper are multifaceted. Theoretically, this work advances our understanding of the distribution of prime numbers, confirming certain behaviors predicted by conjectural generalizations of the Hardy-Littlewood $k$-tuple conjecture. On a practical level, improvements in our understanding of prime gaps can influence computational approaches that rely on the properties of prime numbers, such as in cryptography.

Prospective Future Developments

Maynard's approach opens up several avenues for future research:

• Refinements of the Sieving Process: Further optimization of the sieve method could yield even tighter bounds on successive prime gaps or address even larger classes of $k$-tuples.
• Examining Conjectural Bounds: With support for the Elliott-Halberstam conjecture, additional research might move towards verifying this and similar conjectures with greater numerical precision or partial results.
• Application to Subsequence Primes: Applying this methodology to specific arithmetical progressions or conditional random walks might also prove fruitful.

By overcoming traditional limitations associated with the GPY method and leveraging profound advancements secured over the last decade, this work enriches both the theoretical framework and practical methodologies in analytic number theory.