General elementary methods meeting elementary properties of correlations

TL;DR

This paper surveys properties of correlations of arithmetic functions through Ramanujan expansions, focusing on convergence conditions, and presents new elementary methods for analyzing these correlations, including proofs of the Hardy-Littlewood conjecture on twin primes.

Contribution

It introduces new elementary methods for studying correlations of arithmetic functions and extends previous results on Ramanujan expansions and their convergence properties.

Findings

Proved Hardy-Littlewood conjecture on 2k-twin primes.

Established convergence of Ramanujan expansions under Delange Hypothesis.

Developed new elementary techniques for correlation analysis.

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.