Value distribution of \(g\)-additive functions. (English) Zbl 1016.11029
The arithmetical function \(f\) is \(g\)-additive, if for all integers \(a\), \(\ell\), \(0\leq b < g^\ell\)
\[
f(ag^\ell +b) = f(ag^\ell) + f(b).
\]
The authors’ aim is a thorough study of [integer-valued resp. real-valued resp. complex-valued] \(g\)-additive arithmetical functions with respect to the existence of limit distributions, of limit distributions modulo \(m\), and of membership in spaces (\(\mathcal A^u\), \(\mathcal D^u\), \(\mathcal A^q\), and \(\mathcal D^q\) in the notation of W. Schwarz and J. Spilker [Arithmetical Functions, Cambridge University Press (1994; Zbl 0807.11001)]) of arithmetical functions, and they are interested in giving necessary and sufficient conditions.
The authors show:
(1) Every integer-valued \(g\)-additive function \(f\) has a limit distribution modulo \(m\); that means, the limit \[ c(f,k) = \lim_{N\to\infty} \frac 1N \cdot \#\bigl\{ n;\;0\leq n < N,\;f(n) \equiv k \bmod m \bigr\} \] exists.
(2) Write \[ a_r(f,k) = \tfrac 1g \sum_{0\leq e < g} \exp \left( 2\pi i \tfrac km f(eg^r)\right). \] Any \(g\)-additive integer-valued function \(f\) is uniformly distributed modulo \(m\) if and only if for every \(k\in [1,m-1]\) either \(a_r(f,k)=0\) for some \(r\) or \(a_r(f,k) \not= 1\) for infinitely many \(r\).
(3) The real-valued \(g\)-additive function is uniformly distributed modulo 1 if and only if for every positive integer \(h\) either \[ \sum_{0\leq e < g} \exp\bigl( 2\pi i h f(eg^r)\bigr) =0 \] for some integer \(r \geq 0\) or \[ S_2(f,e,h) : = \sum_{r\geq 0} \|h f(eg^r) \|^2 \] diverges for some \(e, 1\leq e<g\); \(\|\beta\|\) denotes the distance of the real \(\beta\) to the nearest integer.
(4) Necessary and sufficient conditions for the existence of a non-uniform limit distribution modulo 1 are given for \(g\)-additive real-valued functions.
(5) If the \(g\)-additive real-valued function \(f\) has a limit distribution modulo 1, then this limit distribution is continuous if and only if the set \[ \{r\geq 0; d\cdot f(eg^r) \text{ is not an integer for some } 0\leq e < g \} \] is infinite.
(6) Let \(1\leq q< \infty\). For a \(g\)-additive complex-valued function \(f\) the properties (a) \(f\in \mathcal A^q\), (b) \(f \in \mathcal D^q\), (c) The series \(\sum_{r\geq 0}\sum_{0\leq e<g} f(eg^r)\) and \(\sum_{r\geq 0}\sum_{0\leq e<g} |f(eg^r)|^2\) are convergent [according to Delange’s theorem the convergence of these two series is necessary and sufficient for real-valued \(g\)-additive functions to have a limit distribution], (d) \(\|f\|_q<\infty\), and the mean-values \(M(f)\) exists, (e) \(\operatorname{Re}(f)\) and \(\operatorname{Im}(f)\) do have limit distributions, are equivalent.
A similar criterion is given for membership in \(\mathcal A^u\) resp. \(\mathcal D^u\).
The proof makes use, for example, of a theorem of H. Delange [Acta Arith. 21, 285-298 (1972; Zbl 0219.10062)], of the mean-value theorem of Delange-Wirsing-Halász, of a Lemma of Elliott, and of a generalized Turán-Kubilius inequality for complex-valued \(g\)-additive functions.
The authors show:
(1) Every integer-valued \(g\)-additive function \(f\) has a limit distribution modulo \(m\); that means, the limit \[ c(f,k) = \lim_{N\to\infty} \frac 1N \cdot \#\bigl\{ n;\;0\leq n < N,\;f(n) \equiv k \bmod m \bigr\} \] exists.
(2) Write \[ a_r(f,k) = \tfrac 1g \sum_{0\leq e < g} \exp \left( 2\pi i \tfrac km f(eg^r)\right). \] Any \(g\)-additive integer-valued function \(f\) is uniformly distributed modulo \(m\) if and only if for every \(k\in [1,m-1]\) either \(a_r(f,k)=0\) for some \(r\) or \(a_r(f,k) \not= 1\) for infinitely many \(r\).
(3) The real-valued \(g\)-additive function is uniformly distributed modulo 1 if and only if for every positive integer \(h\) either \[ \sum_{0\leq e < g} \exp\bigl( 2\pi i h f(eg^r)\bigr) =0 \] for some integer \(r \geq 0\) or \[ S_2(f,e,h) : = \sum_{r\geq 0} \|h f(eg^r) \|^2 \] diverges for some \(e, 1\leq e<g\); \(\|\beta\|\) denotes the distance of the real \(\beta\) to the nearest integer.
(4) Necessary and sufficient conditions for the existence of a non-uniform limit distribution modulo 1 are given for \(g\)-additive real-valued functions.
(5) If the \(g\)-additive real-valued function \(f\) has a limit distribution modulo 1, then this limit distribution is continuous if and only if the set \[ \{r\geq 0; d\cdot f(eg^r) \text{ is not an integer for some } 0\leq e < g \} \] is infinite.
(6) Let \(1\leq q< \infty\). For a \(g\)-additive complex-valued function \(f\) the properties (a) \(f\in \mathcal A^q\), (b) \(f \in \mathcal D^q\), (c) The series \(\sum_{r\geq 0}\sum_{0\leq e<g} f(eg^r)\) and \(\sum_{r\geq 0}\sum_{0\leq e<g} |f(eg^r)|^2\) are convergent [according to Delange’s theorem the convergence of these two series is necessary and sufficient for real-valued \(g\)-additive functions to have a limit distribution], (d) \(\|f\|_q<\infty\), and the mean-values \(M(f)\) exists, (e) \(\operatorname{Re}(f)\) and \(\operatorname{Im}(f)\) do have limit distributions, are equivalent.
A similar criterion is given for membership in \(\mathcal A^u\) resp. \(\mathcal D^u\).
The proof makes use, for example, of a theorem of H. Delange [Acta Arith. 21, 285-298 (1972; Zbl 0219.10062)], of the mean-value theorem of Delange-Wirsing-Halász, of a Lemma of Elliott, and of a generalized Turán-Kubilius inequality for complex-valued \(g\)-additive functions.
Reviewer: Wolfgang Schwarz (Frankfurt / Main)
MSC:
| 11K65 | Arithmetic functions in probabilistic number theory |
| 11N69 | Distribution of integers in special residue classes |
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
