For a given finitely generated multiplicative subgroup of the rationals which possiblycontain negative numbers, we derive, subject to GRH, formulas for the densities of primes forwhich the index of the reduction group has a given value. We completely classify the cases of rankone, torsion groups for which the density vanishes and the the set of primes for which the indexof the reduction group has a given value, is finite. For higher rank groups we propose some partialresults. Finally, we present some computations comparing the approximated density computedwith primes up to1010and those predicted by the Riemann Hypothesis
Pappalardi, F., Abdullah, H., Mustafa, A.A. (2021). DENSITY OF THE “QUASI r-RANK ARTIN PROBLEM”. FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI, 65(1), 73-93 [10.7169/FACM/1902].
DENSITY OF THE “QUASI r-RANK ARTIN PROBLEM”
Francesco Pappalardi
Membro del Collaboration Group
;Andam Ali MustafaMembro del Collaboration Group
2021-01-01
Abstract
For a given finitely generated multiplicative subgroup of the rationals which possiblycontain negative numbers, we derive, subject to GRH, formulas for the densities of primes forwhich the index of the reduction group has a given value. We completely classify the cases of rankone, torsion groups for which the density vanishes and the the set of primes for which the indexof the reduction group has a given value, is finite. For higher rank groups we propose some partialresults. Finally, we present some computations comparing the approximated density computedwith primes up to1010and those predicted by the Riemann HypothesisI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
