On the second Hardy-Littlewood conjecture

The second Hardy–Littlewood conjecture asserts that the prime counting function π(x) satisfies the subadditive inequality π(x+y)⩽π(x)+π(y) for all integers x,y⩾2. By linking the subadditivity of π(x) to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of y for which π(x) is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all ϵ>0, there exists xϵ⩾2 such that for all x⩾xϵ and y in the range (2+ϵ)x√log2x8π⩽y⩽x, the inequality π(x+y)⩽π(x)+π(y) holds.