A smooth shift approach for a Ramanujan expansion

2019/01/06 by Coppola, Giovanni
#11A25 #11K65 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1901.01584

Abstract

All arithmetical functions F satisfying Ramanujan Conjecture, i.e., F(n)≪_εn^ε, and with Q-smooth divisors, i.e., with Eratosthenes transform F':=F∗ μ 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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