{"id":"palomar/PALOMAR-2026-10-22-000022","url":"https://github.com/sthubbar/goldbach-lean/tree/3ba1189ed38f0a913828af00d72196eebd68777f/entries/hardylittlewood1923","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-22-000022-v1.json","repo":"https://github.com/sthubbar/goldbach-lean","tags":["math.NT","msc:11P32","msc:11P55","msc:11N13","additive-number-theory","primes","goldbach-type-problems","lean"],"score":0,"title":"Sums of primes, after Hardy and Littlewood 1923, Partitio Numerorum III, Theorems B to F proved without Hypothesis R","trust":"high","domain":"math.NT","source":"palomar","authors":["Steve Hubbard"],"reviews":0,"subject":"math.NT","summary":"A sorry-free Lean 4 proof, with no hypothesis, of Theorems B, C, D, E and F of G. H. Hardy and J. E. Littlewood, Some problems of Partitio Numerorum III, Acta Math. 44, 1923. Hardy and Littlewood proved these theorems on their Hypothesis R, a form of the generalized Riemann hypothesis for Dirichlet L-functions; here they are proved unconditionally, from Vinogradov's estimate for exponential sums over primes and the Siegel-Walfisz theorem, so the argument is not Hardy and Littlewood's own for Theorems B, D and E. Formalized: Theorem B, the asymptotic for the weighted count nu_r of n as an…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-22","reactions":{},"formalized":true,"formalizes":[{"title":"Some problems of 'Partitio Numerorum'; III: On the expression of a number as a sum of primes","identifier":"doi:10.1007/BF02403921","relationship":"adapts"}]}