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On the function \(a_ p\), \(p^{a_ p(n)}\| n\), \((n>1)\). (English) Zbl 0798.11003

Verf. untersucht arithmetische Eigenschaften der Ordnungsfunktionen \(a_ p\) \((p \in \mathbb P)\), \(\mathbb P\) die Menge aller Primzahlen; sie sind als additive Funktionen durch \(a_ p (q^ k) = k\) falls \(q=p\), 0 falls \(q \neq p\) definiert. Er bestimmt die Konvergenzabszisse der erzeugenden Dirichlet- Reihe \((\sigma = 1)\), den Mittelwert von \(a_ p( = 1/(p-1))\) sowie die natürlichen Dichten der Niveaumengen \(a_ p^{-1} (\{k\})\) und \(\{n:a_ p(n) | n\}\). Die Beweise sind elementar.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
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References:

[1] COOPER C. N., KENNEDY R. E.: Chebyshev’s inequality and natural density. Amer. Math. Monthly 96 (1989), 118-124. · Zbl 0694.10004
[2] FAST H.: Sur la convergence statistique. Coll. Math. 2 (1951), 241-244. · Zbl 0044.33605
[3] HARDY G. H., WRIGHT E. M.: An Introduction to the Theory of Numbers. (3rd edition), Clarendon Press, Oxford, 1954. · Zbl 0058.03301
[4] ŠALÁT T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003
[5] STRAUCH O.: On statistical convergence of bounded sequences. · Zbl 0505.10026
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