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Über verallgemeinerte Momente additiver Funktionen. (On generalized moments of additive functions). (German) Zbl 0606.10043
Characterizations of additive functions f are given, for which \[ \| \phi \circ | f| \|:=\limsup_{x\to \infty}(1/x)\sum_{n\leq x}\phi ( | f(n)|) \] is bounded, where \(\phi: {\mathbb{R}}^+\to {\mathbb{R}}^+\) is monotone and (1) \(\phi (x+y) \ll \phi (x)+\phi(y)\) \((x,y\geq 0)\) or (2) \(\phi (x)=c^ x\) \((x\in {\mathbb{R}})\). (A typical example is \(\phi (x)=x^{\alpha}\) \((\alpha >0)\) for \(x\geq 0.)\)
The main result is the following theorem. Let \(f: {\mathbb{N}}\to {\mathbb{R}}\) be additive and \(\phi(y)\uparrow \infty\) as \(y\to \infty\). Further, assume that (1) (or (2)) holds. Then \(\| \phi \circ | f| \| <\infty\) if and only if the series \[ \sum_{p,\quad | f(p)| >1}p^{-1},\quad \sum_{p,\quad | f(p)| \leq 1}| f(p)|^ 2 p^{-1}, \] \[ \sum_{p}\sum_{m\geq 1,\quad | f(p^ m)| >1}\phi (| f(p^ m)|) p^{-m} \] converge and \[ \sum_{p\leq x,\quad | f(p)| \leq 1}f(p) p^{-1} = O(1)\text{ as } x\to \infty. \]

MSC:
11K65 Arithmetic functions in probabilistic number theory
11N37 Asymptotic results on arithmetic functions
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[1] Elliott, P. D. T. A.: High power analogous to the Turán–Kubilius inequality, and an application to number theory. Can. J. Math.32, 893–907 (1980). · Zbl 0444.10051 · doi:10.4153/CJM-1980-068-0
[2] Elliott, P. D. T. A.: On additive functions whose limiting distribution possess a finite mean and variance. Pac. J. Math.48, 47–55 (1973). · Zbl 0271.10047
[3] Elliott, P. D. T. A.: Mean value theorems for multiplicative functions bounded in meana-power,a>1. J. Australian Math. Soc. (Series A)29, 177–205 (1980). · Zbl 0426.10050 · doi:10.1017/S1446788700021182
[4] Elliott, P. D. T. A.: Probabilistic Number Theory I, II. New York-Heidelberg-Berlin: Springer. 1979, 1980. · Zbl 0431.10029
[5] Erdös, P.: On the distribution of additive functions. Ann. Math.47, 1–20 (1946). · Zbl 0061.07902 · doi:10.2307/1969031
[6] Hildebrand, A., Spilker, J.: Charakterisierung der additiven, fastperiodischen Funktionen. Manuscripta Math.32, 213–230 (1980). · Zbl 0446.10041 · doi:10.1007/BF01299602
[7] Indlekofer, K.-H.: Cesáro means of additive functions. Analysis,6, 1–24 (1986) (Preprint 1980).
[8] Indlekofer, K.-H.: Properties of uniformly summable multiplicative functions. Periodica Math. Hung.17, 143–161 (1986) (Preprint 1980). · Zbl 0594.10037 · doi:10.1007/BF01849323
[9] Indlekofer, K.-H.: A mean-value theorem for multiplicative functions. Math. Z.172, 255–271 (1980). · Zbl 0426.10049 · doi:10.1007/BF01215089
[10] Indlekofer, K.-H.: Limiting distributions and mean-values of multiplicative arithmetical functions. J. reine angew. Math.328, 116–127 (1981). · Zbl 0455.10036 · doi:10.1515/crll.1981.328.116
[11] Indlekofer, K.-H.: Gleichgradige Summierbarkeit bei verallgemeinerten Momenten additiver Funktionen. Preprint 1986.
[12] Kubilius, J.: Probabilistic Methods in the Theory of Numbers. Translations of Mathematical Monographs. 11. Providence, Rhode Island: Amer. Math. Soc. 1964. · Zbl 0133.30203
[13] Rusza, I. Z.: Generalized moments of additive functions. J. Number Theory18, 27–33 (1984). · Zbl 0524.10042 · doi:10.1016/0022-314X(84)90039-8
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