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Journal Articles Hardy-Ramanujan Journal Year : 2019

An elementary property of correlations

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Giovanni Coppola
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Abstract

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.
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Dates and versions

hal-01986597 , version 1 (18-01-2019)

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Giovanni Coppola. An elementary property of correlations. Hardy-Ramanujan Journal, 2019, Atelier Digit_Hum, pp.68 - 76. ⟨10.46298/hrj.2019.5108⟩. ⟨hal-01986597⟩

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