On the Second Hardy-Littlewood Conjecture

The second Hardy-Littlewood conjecture asserts that the prime counting function π(x) satisfies the subadditive inequality {align*} π(x+y) π(x)+π (y) {align*} 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 x_ε and y in the range {align*} {(2+ε)√xlog^2x}{8π} y x, {align*} the inequality π(x+y) π(x) + π(y) holds.