Suites limite-périodiques et théorie des nombres. IX (Q759797)

Let \(f\) be a multiplicative arithmetical function, and define \[ {\mathcal F}(s)=\sum f(n) \cdot n^{-s},\quad {\mathcal G}(s)=\sum | f(n)| \cdot n^{-s}. \] The author sketches a new proof (based on Jessen's theorem in \(L^ 1(E,d\mu)\), where \(E\) is a certain compactification of the semigroup \(N\) of positive integers) of a theorem closely related to Elliott-Daboussi's theorem on mean-values of multiplicative functions. His main result is: The conditions (A) For any \(c>0\) the series \[ \sum (f(p)-1)\cdot p^{-1}, \quad \sum_{| f(p)-1| <c}p^{-1}\cdot | f(p)-1|^ 2, \quad \sum_{| f(p)-1| \geq c}p^{-1}\cdot | f(p)|^{\alpha} \] (where \(\alpha\geq 1)\) and \(\sum_{p}\) \(\sum_{k\geq 2}p^{-k}\cdot | f(p^ k)|^{\alpha}\) are convergent and \(\sum^{\infty}_{k=0}p^{-k}\cdot f(p^ k)\neq 0\) for any prime \(p\). (T) \(\lim_{\sigma \to 1+}\zeta (\sigma)^{-1}\cdot {\mathcal F}(\sigma)=A\neq 0\), \(\limsup_{\sigma \to 1+}\zeta (\sigma)^{-1}\cdot {\mathcal G}(\sigma)=B<\infty\), \(\forall \varepsilon >0\) \(\exists \eta >0\) : if \(\lim_{\sigma \to 1+}\zeta (\sigma)^{-1}\) \(\sum^{\infty}_{1}| \gamma (n)| n^{-\sigma}<\eta\) then \[ \limsup_{\sigma \to 1+}\zeta (\sigma)^{-1}\cdot \sum^{\infty}_{1}| f(n)|\, | \gamma (n)|\, n^{-\sigma}<\varepsilon, \] (L) The sequence of functions \(F_ y(t)=\prod_{p\leq y}f(t_ p)\) (defined a.e. on \(E\)) converges \(d\mu\)-a.e. and in \(L^ 1(E,d\mu)\) to some integrable function \(F(t)\) such that \(\int_{E}F\cdot d\mu =A\neq 0,\) are equivalent. Notation. \(f(t_ p)=f(p^{\alpha})\) if \(t_ p=p^{\alpha}\), \(\alpha\geq 0\), \(t=(t_ p)_{p\, \text{prime}}\). \[ \Gamma =\{\gamma: {\mathbb N}\to {\mathbb R},\quad \gamma =\sum_{\text{finite}}\quad \lambda_ dI_ d,\quad \lambda_ d\in {\mathbb R}\}, \] where \(I_ d(n)=1\) if \(d\mid n\), and 0 otherwise. For parts VII and VIII, see ibid. 59, 26--28, 164--166 (1983; Zbl 0528.10038).