A new class of Correlations insisting on Ramanujan expansions

Abstract

Studying Correlations with Ramanujan Expansions, we arrive to present the new class of, say, Two-Seasons Correlations, abbr. T-S, as a natural set expressing some of the features of, say, H-L-like Correlations; these are the ones that mimic the H-L ($=$Hardy-Littlewood) Correlation with shift $2k$, needed to study $2k-$twin primes following Hardy \& Littlewood Conjecture. After introducing the $3-$Hypotheses Correlations in a previous paper, we add two other, very natural, hypotheses: the fifth is a technical one, simplifying calculations; but the fourth is called 'Parity', since it deals with the parity of natural numbers we play with. In particular, we may build (devoting to this 'our mainstream', here) a single Correlation that satisfies these '5 Axioms', thus a T-S one, that 'entangles two different Correlations' (whence Two-Seasons: T-S) depending on $a$ ($=$ the shift) parity. For $a$ even, our 'Artifact' mimics the H-L Correlation, in fact $a=2k$; but, while H-L Correlation is 'negligible', say, on $a$ odd, our Artifact seems to compare at least in the order of magnitude to H-L Correlation on $a$ even, being linked to another additive problem. Namely, on $a$ even, the Artifact 'counts', say, classic solutions to: $p_1+a=p_2$, in odd primes $p_1,p_2$; while, on $a$ odd, it 'counts' solutions to: $p_1+a=2^j p_2$, again with odd primes $p_1,p_2$ and with $j\in \N$ (satisfying the natural arithmetic constraints). More in general, our T-S Correlations 'entangle' two different Diophantine equations.