A smooth shift approach for a Ramanujan expansion

Giovanni Coppola

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arXiv · 1901.01584

A smooth shift approach for a Ramanujan expansion

Giovanni Coppola

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Abstract

All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast \mu$ supported in $Q-$smooth numbers, have a kind of unique Ramanujan expansion; also, these Ramanujan coefficients decay very well to $0$ and have two explicit expressions (in the style of Carmichael and Wintner). This general result, then, is applied to the shift-Ramanujan expansions, i.e., the expansions for correlations with respect to the shift, whence the title.

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Giovanni Coppola. 2019-01-06. A smooth shift approach for a Ramanujan expansion. https://arxiv.org/abs/1901.01584

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