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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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Pieter Moree
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

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.

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