The distribution of the multiplicative index of algebraic numbers over residue classes

Pieter Moree; Antonella Perucca; Pietro Sgobba

doi:10.1007/s12188-024-00276-2

The distribution of the multiplicative index of algebraic numbers over residue classes

Pieter Moree, Antonella Perucca, Pietro Sgobba^*

^*Corresponding author for this work

Department of Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let K be a number field and G a finitely generated torsion-free subgroup of K^×. Given a prime p of K we denote by ind_p(G) the index of the subgroup (Gmodp) of the multiplicative group of the residue field at p. Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.

Original language	English
Pages (from-to)	1-17
Number of pages	17
Journal	Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
Volume	94
Issue number	1
DOIs	https://doi.org/10.1007/s12188-024-00276-2
Publication status	Published - Apr 2024

Keywords

Multiplicative index and order
Natural density
Primary: 11R45
Primes in arithmetic progression
Reductions of algebraic numbers
Secondary: 11A07, 11R44

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10.1007/s12188-024-00276-2

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Moree, P., Perucca, A., & Sgobba, P. (2024). The distribution of the multiplicative index of algebraic numbers over residue classes. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 94(1), 1-17. https://doi.org/10.1007/s12188-024-00276-2

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KW - Secondary: 11A07, 11R44

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The distribution of the multiplicative index of algebraic numbers over residue classes

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