**Summary:**

We will use the sieve method for twin primes to obtain the relationship for the twin prime counting function same as that was conjectured by Hardy and Littlewood but without making their heuristic assumption about the prime number distribution. Later we generalize our method to get the expression for the counting function for prime pairs (n, n+2k) and show that it is again same as the extended Goldbach conjecture that was conjectured by Hardy and Littlewood.