[Paper Review] Primes in intervals of bounded length

Andrew Granville|arXiv (Cornell University)|Oct 30, 2014

Analytic Number Theory Research40 references4 citations

TL;DR

This paper synthesizes breakthrough advances in bounded gaps between primes, extending Zhang's and Maynard-Tao's work to prove that for any integer $ m \geq 1 $, there are infinitely many intervals of length $ B_m = e^{8m+5} $ containing at least $ m $ distinct primes. It unifies and simplifies techniques from Zhang, Polymath8, and Maynard-Tao, demonstrating that bounded gaps of size 246 are achievable with combined methods, and highlights the need for new ideas to reach the twin prime conjecture’s goal of gap size 2.

ABSTRACT

The Twin Prime conjecture states that there are infinitely many pairs of distinct primes which differ by $2$. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound $B$ such that there are infinitely many pairs of distinct primes which differ by no more than $B$. This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable. Zhang even showed that one can take $B = 70000000$. Moreover, a co-operative team, \emph{polymath8}, collaborating only on-line, had been able to lower the value of $B$ to ${4680}$. They had not only been more careful in several difficult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof (and forms the basis for the proof presented herein). In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to $600$, by a substantially easier method. Both Maynard, and Terry Tao who had independently developed the same idea, were able to extend their proofs to show that for any given integer $m\geq 1$ there exists a bound $B_m$ such that there are infinitely many intervals of length $B_m$ containing at least $m$ distinct primes. We will also prove this much stronger result herein, even showing that one can take $B_m=e^{8m+5}$.

Motivation & Objective

• To unify and clarify the breakthrough methods of Zhang, Polymath8, and Maynard-Tao on bounded gaps between primes.
• To establish a quantitative bound $ B_m = e^{8m+5} $ such that there are infinitely many intervals of length $ B_m $ containing at least $ m $ distinct primes.
• To demonstrate that combining Zhang’s original method with Maynard-Tao’s framework reduces the best-known gap size to 246.
• To highlight the limitations of current techniques and the necessity of new ideas to achieve the twin prime conjecture’s goal of gap size 2.
• To provide a self-contained proof of Zhang’s key estimate on primes in short arithmetic progressions, foundational to the entire program.

Proposed method

• Adapting Zhang’s novel estimate on primes in short arithmetic progressions, which controls error terms in sieve methods.
• Applying the Maynard-Tao method of multidimensional sieves to detect multiple primes in short intervals using weighted sieve functions.
• Using the Polymath8 collaboration’s refinements to tighten error bounds and simplify Zhang’s original proof structure.
• Employing a generalized sieve framework that allows for variable shifts and multiple primes per interval, enhancing the detection of prime constellations.
• Introducing a new proof strategy that simplifies the original Zhang argument while maintaining strength, particularly in handling the fundamental open problem of bounded gaps.
• Establishing quantitative forms of the Maynard-Tao theorem with lower bounds on the number of $ n \in [x,2x] $ for which $ n + a_j $ contains many primes.

Experimental results

Research questions

• RQ1 Can the methods of Zhang, Maynard, and Tao be unified to prove the existence of infinitely many intervals of length $ B_m $ containing at least $ m $ distinct primes?
• RQ2 What is the best possible effective bound $ B_m $ for which there are infinitely many $ m $-tuples of primes in an interval of length $ B_m $?
• RQ3 To what extent can the Zhang-Maynard-Tao framework be optimized to reduce the best-known gap size below 246?
• RQ4 What are the quantitative implications of the Maynard-Tao theorem for the distribution of primes in short intervals?
• RQ5 What new analytical tools or number-theoretic insights are required to close the gap to 2 and resolve the twin prime conjecture?

Key findings

• For any $ m \geq 1 $, there exist infinitely many intervals of length $ B_m = e^{8m+5} $ containing at least $ m $ distinct primes.
• By combining Zhang’s method with the Maynard-Tao framework, the best-known bound on the minimal gap between infinitely many prime pairs is reduced to 246.
• The Maynard-Tao method can be generalized to show that for any $ m \geq 1 $, there are infinitely many $ m $-tuples of primes in intervals of length $ e^{8m+5} $.
• If all current techniques were pushed to their theoretical limits, the minimal gap size could potentially be reduced to 12 (or possibly 6), but new ideas are required to reach 2.
• The quantitative form of the Maynard-Tao theorem implies that for any $ x,y \geq 1 $, there are $ \gg x \exp(-\sqrt{\log x}) $ integers $ n \in (x,2x] $ such that $ (n,n+y] $ contains $ \gg \log y $ primes.
• The set of limit points $ \mathcal{L} $ of normalized prime gaps satisfies $ [0,c] \subset \mathcal{L} $ for some $ c > 0 $, and $ \mathcal{L} \cap [0,x] $ has measure at least $ x/49 $.