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Keywords:
arithmetic functions; value distribution; finite Abelian groups
arithmetic functions; value distribution; finite Abelian groups
Summary:
This article deals with the value distribution of multiplicative prime-in\-de\-pendent arithmetic functions $(\alpha (n))$ with $\alpha (n)=1$ if $n$ is $N$-free ($N\ge2$ a fixed integer), $\alpha (n)>1$ else, and $\alpha (2^n)\to\infty$. An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zeta-function. Applications to the enumerative functions of Abelian groups and of semisimple rings of given finite order are discussed.
References:
[1] De Koninck J.M., Ivić A.: Topics in arithmetical functions. North Holland Publ. Co., Amsterdam-New York-Oxford, 1980. MR 0589545
[2] Ivić A.: The distribution of values of the enumerating function of non-isomorphic Abelian groups of finite order. Arch. Math. 30 (1978), 374-379. MR 0498444
[3] Ivić A.: The Riemann zeta-function. New York-Chichester, J. Wiley & Sons, 1985. MR 0792089
[4] Krätzel E.: Die Werteverteilung der Anzahl der nicht-isomorphen abelschen Gruppen endlicher Ordnung und ein verwandtes zahlentheoretisches Problem. Publ. Inst. Math. Belgrade (45) 31 (1982), 93-101. MR 0710948
[5] Krätzel E.: Lattice Points. Kluwer Acad. Publ., Dordrecht-Boston-London, 1988. MR 0998378
[6] Krätzel E.: The distribution of values of $a(n)$. Arch. Math. 57 (1991), 47-52. MR 1111114 | Zbl 0701.11039
[7] Krätzel E., Wolke D.: Über die Anzahl der abelschen Gruppen gegebener Ordnung. Analysis 14 (1994), 257-266. MR 1302542
[8] Kühleitner M.: Comparing the number of Abelian groups and of semisimple rings of a given order. Math. Slovaca, to appear. MR 1390704
[9] Nowak W.G.: Sums of reciprocals of general divisor functions and the Selberg divisor problem. Abh. Math. Sem. Hamburg 61 (1991), 163-173. MR 1138281 | Zbl 0739.11037
[10] Nowak W.G.: On the average number of finite Abelian groups of a given order. Ann. sc. math. Québec 15 (1991), 193-202. MR 1151477 | Zbl 0749.11043
[11] Rieger G.J.: Zum Teilerproblem von Atle Selberg. Math. Nachr. 30 (1965), 181-192. MR 0190105
[12] Selberg A.: Note on a paper by L.G. Sathe. J. Indian Math. Soc. 18 (1954), 83-87. MR 0067143 | Zbl 0057.28502
[13] Wolke D.: Über die zahlentheoretische Funktion $ømega(n)$. Acta Arith. 55 (1990), 323-331. MR 1069186 | Zbl 0705.11051
[14] Wolke D.: On a problem of A. Rényi. Monatsh. Math. 111 (1991), 323-330. MR 1116950 | Zbl 0742.11045
[15] Wu J.: Sur un probléme de Rényi. Monatsh. Math. 117 (1994), 303-322. MR 1279119 | Zbl 0818.11037
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