Sums of primes, after Hardy and Littlewood 1923, Partitio Numerorum III, Theorems B to F proved without Hypothesis R
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…
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