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.
Pace Nielsen
- 18.7k
- 4
- 75
- 137