General elementary methods meeting elementary properties of correlations

Abstract

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.