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][1] for more information.


  [1]: http://en.wikipedia.org/wiki/Second_Hardy%25E2%2580%2593Littlewood_conjecture