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episciences.org
Hardy-Ramanujan Journal
2804-7370
2019-01-23
10.46298/hrj.2019.5108
5108
journal article
An elementary property of correlations
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.
https://hrj.episciences.org/5108/pdf
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]