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Taking r > 0, let π 2r (x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π 2r (x) ∼ 2C 2r li 2 (x) with an explicit constant C 2r > 0. There seems to be no good conjecture for the remainders ω 2r (x) = π 2r (x)−2C 2r li 2 (x) that corresponds to Riemann's formula for… (More)

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