Conditional on RH: the number of Goldbach representations on average, after Languasco and Zaccagnini
Conditional on RH: a sorry-free Lean 4 proof of Theorem 1 of A. Languasco and A. Zaccagnini, The number of Goldbach representations of an integer, Proc. Amer. Math. Soc. 140 2012, 795 to 804. With G(n) the sum over m1 + m2 = n of Lambda(m1) Lambda(m2), Mathlib's RiemannHypothesis, taken as an explicit hypothesis of the theorem and never as an axiom, implies that the sum over n = 2. The weaker earlier error terms of Fujii, O((N log N)^{4/3}), and of Bhowmik and Schlage-Puchta, O(N log^5 N), follow and are stated too. The zero sum counts zeros with multiplicity, and its absolute convergence is…
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