It is currently believed that the second conjecture is likely false, but it hasn't been proven quite yet. There is an interval of size 3159 which is not prevented from having more primes than the initial segment of 3159 integers, which is how the first Hardy-Littlewood conjecture would refute the second one. See this wiki article for more information.

It is currently believed that the second conjecture is likely false, but it hasn't been proven quite yet. There is an interval of size 3159 which is not prevented from having more primes than the initial segment of 3159 integers, which is how the first Hardy-Littlewood conjecture would refute the second one. See this wiki article for more information.

The second conjecture has been shown to likely be false, but it hasn't been proven quite yet. However, as I understand it, it has been shown that there are ranges of integers where more primes can fit in those intervals than can fit into the initial segment of the integers. See this wiki article for more information.

It is currently believed that the second conjecture is likely false, but it hasn't been proven quite yet. There is an interval of size 3159 which is not prevented from having more primes than the initial segment of 3159 integers, which is how the first Hardy-Littlewood conjecture would refute the second one. See this wiki article for more information.