| Title:
|
On the value distribution of a class of arithmetic functions (English) |
| Author:
|
Nowak, Werner Georg |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
1 |
| Year:
|
1996 |
| Pages:
|
117-134 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
arithmetic functions |
| Keyword:
|
value distribution |
| Keyword:
|
finite Abelian groups |
| MSC:
|
11K65 |
| MSC:
|
11N37 |
| MSC:
|
11N45 |
| MSC:
|
11N64 |
| idZBL:
|
Zbl 0854.11050 |
| idMR:
|
MR1396164 |
| . |
| Date available:
|
2009-01-08T18:22:25Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118816 |
| . |
| Reference:
|
[1] De Koninck J.M., Ivić A.: Topics in arithmetical functions.North Holland Publ. Co., Amsterdam-New York-Oxford, 1980. MR 0589545 |
| Reference:
|
[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 |
| Reference:
|
[3] Ivić A.: The Riemann zeta-function.New York-Chichester, J. Wiley & Sons, 1985. MR 0792089 |
| Reference:
|
[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 |
| Reference:
|
[5] Krätzel E.: Lattice Points.Kluwer Acad. Publ., Dordrecht-Boston-London, 1988. MR 0998378 |
| Reference:
|
[6] Krätzel E.: The distribution of values of $a(n)$.Arch. Math. 57 (1991), 47-52. Zbl 0701.11039, MR 1111114 |
| Reference:
|
[7] Krätzel E., Wolke D.: Über die Anzahl der abelschen Gruppen gegebener Ordnung.Analysis 14 (1994), 257-266. MR 1302542 |
| Reference:
|
[8] Kühleitner M.: Comparing the number of Abelian groups and of semisimple rings of a given order.Math. Slovaca, to appear. MR 1390704 |
| Reference:
|
[9] Nowak W.G.: Sums of reciprocals of general divisor functions and the Selberg divisor problem.Abh. Math. Sem. Hamburg 61 (1991), 163-173. Zbl 0739.11037, MR 1138281 |
| Reference:
|
[10] Nowak W.G.: On the average number of finite Abelian groups of a given order.Ann. sc. math. Québec 15 (1991), 193-202. Zbl 0749.11043, MR 1151477 |
| Reference:
|
[11] Rieger G.J.: Zum Teilerproblem von Atle Selberg.Math. Nachr. 30 (1965), 181-192. MR 0190105 |
| Reference:
|
[12] Selberg A.: Note on a paper by L.G. Sathe.J. Indian Math. Soc. 18 (1954), 83-87. Zbl 0057.28502, MR 0067143 |
| Reference:
|
[13] Wolke D.: Über die zahlentheoretische Funktion $ømega(n)$.Acta Arith. 55 (1990), 323-331. Zbl 0705.11051, MR 1069186 |
| Reference:
|
[14] Wolke D.: On a problem of A. Rényi.Monatsh. Math. 111 (1991), 323-330. Zbl 0742.11045, MR 1116950 |
| Reference:
|
[15] Wu J.: Sur un probléme de Rényi.Monatsh. Math. 117 (1994), 303-322. Zbl 0818.11037, MR 1279119 |
| . |
