  A celebrated conjecture of Hardy and Littlewood provides with an asymptotic
formula for the counting function of the twin primes. We give an unconditional
proof of such a formula by means of a finite Ramanujan expansion of the
counting function expressed in terms of the von Mangoldt function and its
incomplete form. In a completely analogous way, we solve the conjugate
conjecture on the representations of any even integer as the sum of two prime
numbers.
