Hardy-Ramanujan Journal |

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.

Source : oai:HAL:hal-01986597v1

Published on: January 23, 2019

Accepted 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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