[Paper Review] Primes in arithmetic progressions to large moduli II: Well-factorable estimates

James Maynard|arXiv (Cornell University)|Jun 12, 2020

Analytic Number Theory Research11 references4 citations

TL;DR

This paper establishes a new mean value theorem for primes in arithmetic progressions with triply well-factorable weights, extending the range of permissible moduli from $x^{4/7 - \epsilon}$ to $x^{3/5 - \epsilon}$, significantly improving prior results by Bombieri, Friedlander, and Iwaniec. The method leverages advanced factorization techniques and bilinear form estimates to handle larger moduli in sieve-theoretic applications.

ABSTRACT

We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.

Motivation & Objective

• To extend the range of moduli for which mean value estimates for primes in arithmetic progressions hold, beyond the $x^{4/7 - \epsilon}$ bound of Bombieri, Friedlander, and Iwaniec.
• To establish a new mean value theorem for primes in arithmetic progressions with triply well-factorable weights, enabling stronger sieve-theoretic estimates.
• To demonstrate that the current method reaches its limit at $x^{3/5 - \epsilon}$, suggesting no further gain from stronger factorability conditions like quadruply well-factorable weights.
• To provide a quantitative improvement in the level of distribution for sieve weights, particularly for the linear sieve and its variants.
• To bridge the gap between the Bombieri-Vinogradov theorem and the Elliott-Halberstam conjecture by handling larger moduli under weaker conditions on the weights.

Proposed method

• Introduces the concept of triply well-factorable sequences, which generalize well-factorable sequences by allowing factorization into three parts with bounded coefficients.
• Employs a refined bilinear form method to handle sums over arithmetic progressions with triply well-factorable weights.
• Applies a greedy factorization strategy across three levels $D_1, D_2, D_3$ to decompose multiplicative structures in the sieve weights.
• Uses case analysis based on the size of prime products to ensure all configurations are covered under the required bounds.
• Applies dyadic decomposition and dyadic intervals to control error terms and maintain uniformity across different scales.
• Verifies that the resulting bounds on weighted sums of error terms in prime counting functions remain $O(x / (\log x)^A)$, matching the strength of Bombieri-Friedlander-Iwaniec but with a larger modulus range.

Experimental results

Research questions

• RQ1 Can the range of moduli for which mean value estimates for primes in arithmetic progressions hold be extended beyond $x^{4/7 - \epsilon}$ when using well-factorable weights?
• RQ2 What is the maximal modulus size for which such estimates remain valid under stronger factorability conditions, such as triply well-factorable weights?
• RQ3 Does the method of triply well-factorable weights allow for a quantitative improvement in the level of distribution for sieve weights compared to the standard well-factorable case?
• RQ4 Is $x^{3/5 - \epsilon}$ the theoretical limit of this method, or could further refinements extend it further?
• RQ5 Can the linear sieve, which is not triply well-factorable, still benefit from the improved bounds via its factorization properties?

Key findings

• The paper establishes a new mean value theorem for primes in arithmetic progressions with triply well-factorable weights, valid for moduli up to $x^{3/5 - \epsilon}$, improving the prior bound of $x^{4/7 - \epsilon}$.
• The main result shows that for any fixed $a \in \mathbb{Z}$ and $A, \epsilon > 0$, the weighted sum $\sum_{q \leq Q, (a,q)=1} \lambda_q \left( \pi(x;q,a) - \frac{\pi(x)}{\phi(q)} \right)$ is bounded by $O_{a,A,\epsilon}(x / (\log x)^A)$ when $Q \leq x^{3/5 - \epsilon}$ and $\lambda_q$ is triply well-factorable of level $Q$.
• The method is shown to be optimal in the sense that extending the modulus range to $x^{7/12 + \delta}$ would require additional constraints on the parameters, indicating $x^{3/5 - \epsilon}$ is the natural limit of the current approach.
• The result applies directly to the factorable variant of $\beta$-sieve weights with $\beta \geq 2$, which are linear combinations of triply well-factorable sequences, thus improving their level of distribution.
• Despite the linear sieve ($\beta = 1$) not being triply well-factorable, the paper shows that its factorization properties still allow for the application of the new bounds, suggesting broader applicability.
• The paper demonstrates that no further improvement is gained by requiring quadruply well-factorable weights, indicating that the triply well-factorable condition is sufficient and optimal for this method.