An elementary property of correlations

TL;DR

This paper investigates the shift-Ramanujan expansion of functions satisfying the Ramanujan Conjecture to derive explicit formulas for shifted convolution sums, providing a new approach to problems like twin primes under certain hypotheses.

Contribution

It introduces a method to obtain explicit formulas for shifted convolution sums using Ramanujan expansions and the Delange Hypothesis, with applications to twin primes.

Findings

Derives explicit formulas for shifted convolution sums under Delange Hypothesis.

Provides a new proof approach for Hardy-Littlewood Conjecture on twin primes.

Connects Ramanujan expansions with prime number conjectures.

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.