In number theory, the second Hardy-Littlewood conjecture concerns the number of primes in intervals. If π(x) is the number of primes up to and including x then the conjecture states that

π(x + y) ≤ π(x) + π(y)

where x, y ≥ 2.

This means that the number of primes from x + 1 to x + y is always less than or equal to the number of primes from 1 to y. This is probably false in general as it is inconsistent with the first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x and y.

fr:Seconde conjecture de Hardy-Littlewood