TY - JOUR
T1 - The distribution of the multiplicative index of algebraic numbers over residue classes
AU - Moree, Pieter
AU - Perucca, Antonella
AU - Sgobba, Pietro
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024
Y1 - 2024
N2 - 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 indp(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.
AB - 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 indp(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.
KW - Multiplicative index and order
KW - Natural density
KW - Primary: 11R45
KW - Primes in arithmetic progression
KW - Reductions of algebraic numbers
KW - Secondary: 11A07, 11R44
UR - http://www.scopus.com/inward/record.url?scp=85189872171&partnerID=8YFLogxK
U2 - 10.1007/s12188-024-00276-2
DO - 10.1007/s12188-024-00276-2
M3 - Article
AN - SCOPUS:85189872171
SN - 0025-5858
JO - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
JF - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
ER -