ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS
| Autor: | Andrea Ferraguti, Giacomo Micheli |
|---|---|
| Přispěvatelé: | Ferraguti, Andrea, Micheli, Giacomo |
| Rok vydání: | 2015 |
| Předmět: |
Discrete mathematics
Mertens-Cesàro theorem algebraic integers Mathematics::Number Theory natural density General Mathematics 010102 general mathematics number field Field (mathematics) 010103 numerical & computational mathematics Algebraic number field algebraic integer number fields 01 natural sciences zeta function Riemann zeta function Arithmetic zeta function symbols.namesake Quadratic integer symbols Natural density Settore MAT/03 - Geometria 0101 mathematics Mathematics |
| Zdroj: | Bulletin of the Australian Mathematical Society. 93:199-210 |
| ISSN: | 1755-1633 0004-9727 |
| DOI: | 10.1017/s0004972715001288 |
| Popis: | Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$. |
| Databáze: | OpenAIRE |
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