Article
| Title: | On the distribution of the total number of generators of $h$-free and $h$-full elements in an Abelian monoid (English) |
| Author: | Das, Sourabhashis |
| Author: | Kuo, Wentang |
| Author: | Liu, Yu-Ru |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 1 |
| Year: | 2026 |
| Pages: | 303-333 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let ${\mathfrak m}$ be an element of an Abelian monoid, and let $\Omega ({\mathfrak m})$ denote the total number of prime elements (counted with multiplicity) generating ${\mathfrak m}$. We investigate the distribution of $\Omega ({\mathfrak m})$ over the subsets of $h$-free and $h$-full elements, obtaining moment estimates and establishing its normal order within these subsets. This extends the authors' previous work (see S. Das et al., 2025c) on $\omega ({\mathfrak m})$, where multiplicities of prime elements were not considered. In particular, we develop new identities involving sums over prime elements, which play a central role in the analysis. Several applications are presented, including ideals in number fields, effective divisors in global function fields, and effective zero-cycles on geometrically irreducible projective varieties. (English) |
| Keyword: | omega function |
| Keyword: | Abelian monoid |
| Keyword: | the first moment |
| Keyword: | the second moment |
| Keyword: | $h$-free element |
| Keyword: | $h$-full element |
| MSC: | 11K65 |
| MSC: | 11N80 |
| MSC: | 20M32 |
| DOI: | 10.21136/CMJ.2026.0365-25 |
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| Date available: | 2026-03-13T09:34:54Z |
| Last updated: | 2026-03-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153574 |
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| Reference: | [1] Das, S., Elma, E., Kuo, W., Liu, Y.-R.: On the number of irreducible factors with a given multiplicity in function fields.Finite Fields Appl. 92 (2023), Article ID 102281, 22 pages. Zbl 1556.11112, MR 4633691, 10.1016/j.ffa.2023.102281 |
| Reference: | [2] Das, S., Kuo, W., Liu, Y.-R.: Distribution of $\omega(n)$ over $h$-free and $h$-full numbers.Int. J. Number Theory 21 (2025), 715-737. Zbl 08005207, MR 4875888, 10.1142/S1793042125500368 |
| Reference: | [3] Das, S., Kuo, W., Liu, Y.-R.: Generalizations of Erdős-Kac theorem with applications.Available at https://arxiv.org/abs/2506.02432 (2025), 38 pages. 10.48550/arXiv.2506.02432 |
| Reference: | [4] Das, S., Kuo, W., Liu, Y.-R.: On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid.J. Théor. Nombres Bordeaux 37 (2025,), 1041-1081. Zbl 8147531, MR 4997381, 10.5802/jtnb.1351 |
| Reference: | [5] Das, S., Kuo, W., Liu, Y.-R.: On the number of prime factors with a given multiplicity over $h$-free and $h$-full numbers.J. Number Theory 267 (2025), 176-201. Zbl 07939413, MR 4801523, 10.1016/j.jnt.2024.08.007 |
| Reference: | [6] Elboim, D., Gorodetsky, O.: Multiplicative arithmetic functions and the generalized Ewens measure.Isr. J. Math. 262 (2024), 143-189. Zbl 1562.11128, MR 4803424, 10.1007/s11856-024-2609-x |
| Reference: | [7] Erdős, P., Kac, M.: The Gaussian law of errors in the theory of additive number theoretic functions.Am. J. Math. 62 (1940), 738-742. Zbl 0024.10203, MR 0002374, 10.2307/2371483 |
| Reference: | [8] Hardy, G. H., Ramanujan, S.: The normal number of prime factors of a number $n$.Quart. J. Math. 48 (1917), 76-92 \99999JFM99999 46.0262.03. MR 2280878 |
| Reference: | [9] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers.Oxford University Press, Oxford (2008). Zbl 1159.11001, MR 2445243 |
| Reference: | [10] Jakimczuk, R., Lalín, M.: The number of prime factors on average in certain integer sequences.J. Integer Seq. 25 (2022), Article ID 22.2.3, 15 pages. Zbl 1492.11140, MR 4390042 |
| Reference: | [11] Knopfmacher, J.: Abstract Analytic Number Theory.Dover Books on Advanced Mathematics. Dover, New York (1990). Zbl 0743.11002, MR 1068138 |
| Reference: | [12] Lalín, M., Zhang, Z.: The number of prime factors in $h$-free and $h$-full polynomials over function fields.Publ. Math. Debr. 104 (2024), 377-421. Zbl 07857962, MR 4744694, 10.5486/PMD.2024.9694 |
| Reference: | [13] Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale.Chelsea Publishing, New York (1949), German. Zbl 0045.32202, MR 0031002 |
| Reference: | [14] Liu, Y.-R.: A generalization of the Erdős-Kac theorem and its applications.Can. Math. Bull. 47 (2004), 589-606. Zbl 1078.11058, MR 2099756, 10.4153/CMB-2004-057-4 |
| Reference: | [15] Liu, Y.-R.: A generalization of the Turán theorem and its applications.Can. Math. Bull. 47 (2004), 573-588. Zbl 1078.11057, MR 2099755, 10.4153/CMB-2004-056-7 |
| Reference: | [16] Lorenzini, D.: An Invitation to Arithmetic Geometry.Graduate Studies in Mathematics 9. AMS, Providence (1996). Zbl 0847.14013, MR 1376367, 10.1090/gsm/009 |
| Reference: | [17] Rosen, M.: Number Theory in Function Fields.Graduate Texts in Mathematics 210. Springer, New York (2002). Zbl 1043.11079, MR 1876657, 10.1007/978-1-4757-6046-0 |
| Reference: | [18] Saidak, F.: An elementary proof of a theorem of Delange.C. R. Math. Acad. Sci., Soc. R. Can. 24 (2002), 144-151. Zbl 1085.11048, MR 1940553 |
| Reference: | [19] Turán, P.: On a theorem of Hardy and Ramanujan.J. Lond. Math. Soc. 9 (1934), 274-276. Zbl 0010.10401, MR 1574877, 10.1112/jlms/s1-9.4.274 |
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