Binary Goldbach under EH(θ) and a Thin-Set BDH Mean-Square via a Hybrid Minor-Arc Method

2020/06/11 by Maynard, James · 6 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2006.06572

Abstract

Assuming the Elliott–Halberstam hypothesis EH(θ) for primes at some level θ>1/2, together with a Barban–Davenport–Halberstam–type mean-square bound restricted to a thin family of moduli, we prove that every sufficiently large even integer N is the sum of two primes. This removes the μ-twisted hypothesis used by Huang–Li. The method is hybrid on the minor arcs, where a positive-density good family is handled in L² by unconditional mean-square inputs (Maynard; BFI), while the thin bad family is handled by a short-arc L² reduction with an explicit 1/q weight paired with the above BDH-type bound. Major arcs use Siegel–Walfisz for q≤(log N)^B, while minor arcs use Dirichlet neighborhoods |α−a/q|≤1/(qQ1) with Q1=N^1/2+η, and an unsmoothing step yields R2(N)≥1 for all large even N.

Citations

• Primes in arithmetic progressions to large moduli
• Prime Numbers in Short Intervals and a Generalized Vaughan Identity

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