On the second Hardy-Littlewood conjecture.

The article focuses on the second Hardy–Littlewood conjecture, which posits a subadditive inequality for the prime counting function \( p(x) \), stating that \( p(x+y) \leq p(x) + p(y) \) for all integers \( x, y \geq 2 \). By connecting this subadditivity to the error term in the Prime Number Theorem, the authors provide unconditional improvements on the range of \( y \) for which the inequality holds and, assuming the Riemann Hypothesis, establish a wider range where the conjecture is valid. They also quantify the size of the exceptional set where the inequality fails and extend analogous results to primes in arithmetic progressions under the Generalized Riemann Hypothesis. The work includes explicit bounds and conditions under which the subadditivity inequality can be verified, contributing to the understanding of prime distribution in intervals.