A smooth shift approach for a Ramanujan expansion

Abstract

All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n){\ll }_{\epsilon }{n}^{\epsilon }$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F^{\prime }:=F\ast \mu$ 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.

Giving counterexamples for the Reef to hold, we disprove Conjectures 1 & 2 (see version 2)

References in corpus (1)

• Finite Ramanujan expansions and shifted convolution sums of arithmetical functions

Cited by in corpus (2)

• Finite and infinite Euler products of Ramanujan expansions
• Multiplicative Ramanujan coefficients of null-function