[Paper Review] New equidistribution estimates of Zhang type, and bounded gaps between primes

D. H. J. Polymath|arXiv (Cornell University)|Feb 4, 2014

Analytic Number Theory Research33 citations

TL;DR

This paper establishes new equidistribution estimates for primes in arithmetic progressions modulo large smooth squarefree numbers, leveraging Chinese Remainder Theorem-compatible congruence classes. It achieves an exponent of distribution of $\frac{1}{2} + \frac{7}{300}$, advancing the state of the art in sieve methods and contributing to the bounded gaps between primes problem.

ABSTRACT

We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} + \frac{7}{300}$.

Motivation & Objective

• To improve the current best-known exponent of distribution for primes in arithmetic progressions with large smooth squarefree moduli.
• To extend equidistribution results to congruence classes that satisfy the Chinese Remainder Theorem conditions.
• To contribute to the long-standing problem of bounded gaps between consecutive primes by strengthening sieve-theoretic tools.
• To refine the use of multiplicative functions and exponential sums in the context of the Bombieri–Vinogradov theorem.
• To provide quantitative improvements in the level of distribution that can be used in applications such as Zhang's theorem on bounded gaps.

Proposed method

• Utilizes advanced estimates for exponential sums over arithmetic progressions with large smooth squarefree moduli.
• Applies the Chinese Remainder Theorem to ensure compatibility of congruence classes across coprime moduli.
• Employs a refined version of the dispersion method to control error terms in the distribution of primes.
• Integrates techniques from multiplicative number theory to handle the structure of smooth squarefree moduli.
• Optimizes the use of the large sieve and Weil-type bounds to improve the exponent of distribution.
• Applies a dyadic decomposition of the summation range to manage the dependence on the modulus size.

Experimental results

Research questions

• RQ1 What is the best possible exponent of distribution for primes in arithmetic progressions modulo large smooth squarefree numbers?
• RQ2 How can the Chinese Remainder Theorem be effectively used to extend equidistribution estimates to composite moduli?
• RQ3 To what extent can the error terms in the Bombieri–Vinogradov theorem be reduced using advanced exponential sum estimates?
• RQ4 Can the exponent of distribution be improved beyond $\frac{1}{2} + \frac{1}{6}$ in the context of Zhang-type theorems?
• RQ5 What is the quantitative impact of these improved distribution estimates on the bounded gaps between primes?

Key findings

• The paper achieves an exponent of distribution of $\frac{1}{2} + \frac{7}{300}$ for primes in arithmetic progressions modulo large smooth squarefree numbers.
• This exponent improves upon previous results in the context of Zhang-type estimates, particularly for moduli with favorable multiplicative structure.
• The improvement is achieved by refining the analysis of exponential sums under CRT-compatible congruence conditions.
• The method applies to moduli that are smooth and squarefree, which are particularly useful in sieve-theoretic applications.
• The result strengthens the foundation for subsequent improvements in the bounded gaps between primes problem.
• The distribution estimates are effective for moduli up to a power of the square root of the level, consistent with the requirements of the GPY and Zhang methods.