On the Second Hardy-Littlewood Conjecture

2025/03/04 by Chahal, Bittu, Elma, Ertan, Fellini, Nic +2
#11N05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.02766

Abstract

The second Hardy-Littlewood conjecture asserts that the prime counting function π(x) satisfies the subadditive inequality π(x+y)\leqslant π(x)+π(y) for all integers x,y\geqslant 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 x_ε \geqslant 2 such that for all x\geqslant x_ε and y in the range ((2+ε)√(x)log²x)/(8π)\leqslant y\leqslant x, the inequality π(x+y)\leqslant π(x) + π(y) holds.

Related

• The second Hardy-Littlewood conjecture is true
• Inequalities For The Primes Counting Function
• Some Results on a Conjecture of Hardy and Littlewood
• Hardy-Littlewood inequality for primes
• Averaged Form of the Hardy-Littlewood Conjecture