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In this paper we give some generalizations and improvements of the Pavlović result on the Holland–Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in 
${{\text{R}}^{m}}$.
Two arithmetic functions 
$f$ and 
$g$ form a Möbius pair if 
$f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers 
$n$. In that case, 
$g$ can be expressed in terms of 
$f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members 
$f$ and 
$g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both 
${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and 
${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.
