  A celebrated theorem of Delange gives a sufficient condition for an
arithmetic function to be the sum of the associated Ramanujan expansion with
the coefficients provided by a previous result of Wintner. By applying the
Delange theorem to the correlation of the von Mangoldt function with its
incomplete form, we deduce an inequality involving the counting function of the
prime numbers in arithmetic progressions. A remarkable aspect is that such an
inequality is equivalent to the famous conjectural formula by Hardy and
Littlewood for the twin primes.
