2021/01/09 by Matt Visser, Visser, Matt
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2101.03283
openalex publication_date 2021/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The second Hardy-Littlewood conjecture, that π(x)+π(y) ≥ π(x+y) for integers x and y with min\x,y\≥ 2, was formulated in 1923. It continues to attract attention to this day, almost 100 years later. In 1975 Udrescu proved that this conjecture holds for (x,y) sufficiently large, but without an explicit effective bound on the region of validity. We shall revisit Udrescu's result, modifying it to obtain explicit effective bounds, ultimately proving that the second Hardy-Littlewood conjecture is in fact unconditionally true. Furthermore we note that constraints on the prime counting function imply, (and are implied by), constraints on the location of the primes, and re-cast Segal's 1962 equivalent reformulation of the second Hardy-Littlewood conjecture in the more symmetric (and perhaps clearer) form that for integers i and j with min\i,j\ ≥ 2 one has pi+j-1 ≥ pi + pj -1.