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Electronic Research Announcements
ISSN 1079-6762

On the distribution of the order over residue classes


Author: Pieter Moree
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 121-128
MSC (2000): Primary 11N37, 11R45; Secondary 11N69
Published electronically: August 18, 2006
MathSciNet review: 2263073
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Abstract: For a fixed rational number $ g\not\in \{-1,0,1\}$ and integers $ a$ and $ d$ we consider the set $ N_g(a,d)$ of primes $ p$ such that the order of $ g$ modulo $ p$ is congruent to $ a\,({\rm mod~}d)$. Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set $ N_g(a,d)$ has a natural density $ \delta_g(a,d)$. Arithmetical properties of $ \delta_g(a,d)$ are described, and $ \delta_g(a,d)$ is compared with $ \delta(a,d)$: the average density of elements in a field of prime characteristic having order congruent to $ a\,({\rm mod~}d)$. It transpires that $ \delta_g(a,d)$ has a strong tendency to be equal to $ \delta(a,d)$, or at least to be close to it.


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Additional Information

Pieter Moree
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
Email: moree@mpim-bonn.mpg.de

DOI: http://dx.doi.org/10.1090/S1079-6762-06-00168-5
PII: S 1079-6762(06)00168-5
Received by editor(s): February 5, 2006
Published electronically: August 18, 2006
Communicated by: Brian Conrey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.