An elementary property of correlations

Abstract

We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan expansion is with respect to a>0). We prove that, assuming Delange Hypothesis (DH) for the expansion, we get say Ramnujan exact explicit formula (R.e.e.f.). A noteworthy case, of course, is $f=g=\Lambda$, the von Mangoldt function, so $C_{\Lambda,\Lambda}(N,2k)$, for natural $k$, regards $2k-$twin primes; assuming $(DH)$ for them, we get (from corresponding R.e.e.f.) the proof, easily, of Hardy-Littlewood Conjecture for them.