  This is a kind of survey on properties of correlations of two very general
arithmetic functions, mainly from the point of view of Ramanujan expansions. In
fact, our previous papers on these links had, as a focus, the "Ramanujan
coefficients" of these correlations and the resulting "R.e.e.f.", i.e.,
Ramanujan exact explicit formula. This holds, actually, under a variety of
sufficient conditions, mainly under two conditions of convergence involving
correlations' "Eratosthenes Transform", namely what we call "Delange
Hypothesis" and "Wintner Assumption" (the former implying the latter). We
proved Hardy-Littlewood Conjecture on $2k-$twin primes, in particular, from the
first of these two (that implies convergence of classic Ramanujan expansion,
whence the R.e.e.f.); more recently, we gave a more general proof, from second
condition, entailing the R.e.e.f. again, but this time from another method of
summation for Ramanujan expansions, we detailed in "A smooth summation of
Ramanujan expansions"; in which paper (see 8th ver.) we also started to give
few elementary methods for correlations. Which we deepen here, adding recent,
elementary and entirely new ones.
