  We study three functions which are power series in the variable $z$,
Dirichlet series in the variable $s$ and with coefficients given by
arithmetical functions. A strong point is to relate these functions to some
Hilbert spaces. Three main ingredients are used: an estimate of Davenport on
sums of M\"obius functions, a result of Lucht on convolutions of arithmetical
Dirichlet series and the introduction of an operation $\otimes$ on power
series, naturally associated with the mentioned Hilbert spaces.
