This work introduces the first in-depth study of $h$-free and $h$-full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let $\mathfrak{m}$ be an element of an abelian monoid, with $\omega (\mathfrak{m})$ denoting the number of distinct prime elements generating $\mathfrak{m}$. We study the moments of $\omega (\mathfrak{m})$ over subsets of $h$-free and $h$-full elements, establishing the normal order of $\omega (\mathfrak{m})$ within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.
Ce travail présente la première étude approfondie des éléments $h$-free et $h$-full dans les monoïdes abéliens, offrant une approche unifiée pour comprendre leur rôle dans diverses structures mathématiques. Soit $\mathfrak{m}$ un élément d’un monoïde abélien, avec $\omega (\mathfrak{m})$ désignant le nombre d’éléments premiers distincts engendrant $\mathfrak{m}$. Nous étudions les moments de $\omega (\mathfrak{m})$ sur des sous-ensembles d’éléments $h$-free et $h$-full, établissant l’ordre normal de $\omega (\mathfrak{m})$ au sein de ces sous-ensembles. Nos résultats sont ensuite appliqués aux corps de nombres, aux corps de fonctions globaux et aux variétés projectives géométriquement irréductibles, mettant en évidence la portée étendue de cette approche.
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Keywords: the omega function, abelian monoids, the first moment, the second moment, normal order, $h$-free elements, $h$-full elements
Sourabhashis Das 1 ; Wentang Kuo 1 ; Yu-Ru Liu 1
CC-BY-ND 4.0
@article{JTNB_2025__37_3_1041_0,
author = {Sourabhashis Das and Wentang Kuo and Yu-Ru Liu},
title = {On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {1041--1081},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {3},
doi = {10.5802/jtnb.1351},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1351/}
}
TY - JOUR AU - Sourabhashis Das AU - Wentang Kuo AU - Yu-Ru Liu TI - On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 1041 EP - 1081 VL - 37 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1351/ DO - 10.5802/jtnb.1351 LA - en ID - JTNB_2025__37_3_1041_0 ER -
%0 Journal Article %A Sourabhashis Das %A Wentang Kuo %A Yu-Ru Liu %T On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid %J Journal de théorie des nombres de Bordeaux %D 2025 %P 1041-1081 %V 37 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1351/ %R 10.5802/jtnb.1351 %G en %F JTNB_2025__37_3_1041_0
Sourabhashis Das; Wentang Kuo; Yu-Ru Liu. On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 1041-1081. doi: 10.5802/jtnb.1351
[1] On an additive arithmetic function, Pac. J. Math., Volume 71 (1977) no. 2, pp. 275-294 | MR | DOI
[2] Prime factors with given multiplicity in -free and -full polynomials over function fields, Integers, Volume 25 (2025), A9, 37 pages | Zbl | MR
[3] On the number of irreducible factors with a given multiplicity in function fields, Finite Fields Appl., Volume 92 (2023), 102281, 22 pages | DOI | MR | Zbl
[4] A subset generalization of the Erdős-Kac theorem over number fields with applications (2024) (submitted)
[5] Distribution of over -free and -full numbers, Int. J. Number Theory, Volume 21 (2025) no. 03, pp. 715-737 | DOI | Zbl | MR
[6] On the number of prime factors with a given multiplicity over -free and -full numbers, J. Number Theory, Volume 267 (2025), pp. 176-201 | Zbl | MR
[7] The normal number of prime factors of a number [Quart. J. Math. 48 (1917), 76–92], Collected papers of Srinivasa Ramanujan, AMS Chelsea Publishing, 2000, pp. 262-275 | MR
[8] An introduction to the theory of numbers, Oxford University Press, 2008, xxii+621 pages (revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by Andrew Wiles) | MR | Zbl | DOI
[9] The distribution of powerful integers, Ill. J. Math., Volume 26 (1982) no. 4, pp. 576-590 | MR | Zbl
[10] The number of prime factors on average in certain integer sequences, J. Integer Seq., Volume 25 (2022) no. 2, 22.2.3, 15 pages | MR | Zbl
[11] Sums of and over the -free parts and -full parts of some particular sequences, Integers, Volume 22 (2022), A113, 22 pages | MR | Zbl
[12] Analytic arithmetic of algebraic function fields, Lecture Notes in Pure and Applied Mathematics, 50, Marcel Dekker, 1979, iii+130 pages | MR | Zbl
[13] Abstract analytic number theory, Dover Books on Advanced Mathematics, Dover Publications, 1990, xii+336 pages | MR | Zbl
[14] The number of prime factors in -free and -full polynomials over function fields, Publ. Math. Debr., Volume 104 (2024) no. 3-4, pp. 377-421 | Zbl | MR
[15] Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes, Math. Ann., Volume 56 (1903) no. 4, pp. 645-670 | DOI | MR | Zbl
[16] Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea Publishing, 1949, vii+147 pages | MR | Zbl
[17] A generalization of the Erdős-Kac theorem and its applications, Can. Math. Bull., Volume 47 (2004) no. 4, pp. 589-606 | DOI | MR | Zbl
[18] A generalization of the Turán theorem and its applications, Can. Math. Bull., Volume 47 (2004) no. 4, pp. 573-588 | DOI | MR | Zbl
[19] An invitation to arithmetic geometry, Graduate Studies in Mathematics, 9, American Mathematical Society, 1996, xvi+397 pages | DOI | MR | Zbl
[20] Number theory in function fields, Graduate Texts in Mathematics, 210, Springer, 2002, xii+358 pages | DOI | MR | Zbl
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