Giovanni Coppola - An elementary property of correlations

hrj:5108 - Hardy-Ramanujan Journal, January 23, 2019 -
An elementary property of correlations

Authors: Giovanni Coppola

We study the "shift-Ramanujan expansion" to obtain a formulae for the shifted convolution sum $C_{f,g} (N,a)$ of general functions f, g satisfying Ramanujan Conjecture; here, the shift-Ramanujan expansion is with respect to a shift factor a > 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.

Published on: January 23, 2019
Submitted on: January 23, 2019
Keywords: correlation,shift Ramanujan expansion,2k-twin primes 2010 Mathematics Subject Classification 11N05,11P32,11N37, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]


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