The second Hardy-Littlewood conjecture is true

TL;DR

This paper proves the long-standing second Hardy-Littlewood conjecture unconditionally by providing explicit bounds, confirming the inequality involving prime counting functions and prime locations.

Contribution

It revisits Udrescu's partial result, modifies it to obtain explicit bounds, and proves the conjecture unconditionally, also reformulating it in terms of prime positions.

Findings

The conjecture holds for all sufficiently large integers.

Explicit bounds on the validity region are established.

Reformulation relates prime counting to prime positions.

Abstract

The second Hardy-Littlewood conjecture, that $\pi(x)+\pi(y) \geq \pi(x+y)$ for integers $x$ and $y$ with $\min\{x,y\}\geq 2$, was formulated in 1923. It continues to attract attention to this day, almost 100 years later. In 1975 Udrescu proved that this conjecture holds for $(x,y)$ sufficiently large, but without an explicit effective bound on the region of validity. We shall revisit Udrescu's result, modifying it to obtain explicit effective bounds, ultimately proving that the second Hardy-Littlewood conjecture is in fact unconditionally true. Furthermore we note that constraints on the prime counting function imply, (and are implied by), constraints on the location of the primes, and re-cast Segal's 1962 equivalent reformulation of the second Hardy-Littlewood conjecture in the more symmetric (and perhaps clearer) form that for integers $i$ and $j$ with $\min\{i,j\} \geq 2$ one has $p_{i+j-1} \geq p_i + p_j -1$.