Sur certaines fonctions définies par les chiffres des entiers. (On certain functions defined by digits of integers) (Q1092947)

The author is interested in results concerning q-multiplicative functions. For example, applying his main result (Theorem 1) he obtains: If f satisfies the assumptions of theorem 1, then \[ \lim_{x\to \infty}(x^{-1}\sum_{n\leq x}f(n))\cdot (\prod_{r\leq (\log x)/(\log q)}[q^{-1}\sum^{q-1}_{a=0}\quad f(aq^ r)]^{-1})=1, \] and the mean-value M(f) exists and is nonzero, iff the two series \(\sum_{r\geq 0}\sum^{q-1}_{a=0}(f(aq^ r)-1)\) and \(\sum_{r\geq 0}\sum^{q- 1}_{a=0}(f(aq^ r)-1)^ 2\) converge. For the formulation of theorem 1 some terminology is necessary. Denote by \(I_{a,q^ k}: {\mathbb{N}}\to \{0,1\}\) the characteristic function of the residue class \(\{\) n; \(n\equiv a mod q^ k\}\), \(\Gamma =Lin_{{\mathbb{R}}}\{I_{a,q^ k}\}\), and by \(\Gamma^+\) those linear combinations in \(\Gamma\) with negative coefficients. Define \(\ell_ f: \Gamma^+\to {\mathbb{C}}\) by \[ \ell_ f(I_{a,q^ r}) = <f,I_{a,q^ r}> = f(a)(\prod_{k<r}\sum^{q- 1}_{a'=0}f(a' q^ k))^{-1}. \] Then the main reslt is: If (*) \(\forall \epsilon >0\), \(\exists \eta >0:\forall \gamma \in \Gamma^+(<1,\gamma >\leq \eta \to <f,\gamma >\leq \epsilon)\), then the series \(\sum_{r\geq 0}\sum^{q-1}_{a=0}(f(aq^ r)-1)^ 2\) is convergent. For the proof the author introduces the compact space \(\lim proj {\mathbb{Z}}/q^ k {\mathbb{Z}}\approx E=\{0,1,...,q-1\}^{{\mathbb{N}}}.\) Using a suitable product measure \(d\mu\), the theorems of Radon-Nikodym, of Jessen, the `three series theorem' and the Kakutani theorem are used to obtain the conclusion.