Open Access
Feb. 2003 On a distribution property of the residual order of $a \pmod {p}$
Koji Chinen, Leo Murata
Proc. Japan Acad. Ser. A Math. Sci. 79(2): 28-32 (Feb. 2003). DOI: 10.3792/pjaa.79.28


Let $a$ be a positive integer which is not a perfect $h$-th power with $h \geq 2$, and $Q_a(x; k,l)$ be the set of primes $p \leq x$ such that the residual order of $a$ in $\mathbf{Z} / p\mathbf{Z}^{\times}$ is congruent to $l \bmod{k}$. It seems that no one has ever considered the density of $Q_a(x; k,l)$ for $l \ne 0$ when $k \geq 3$. In this article, the natural densities of $Q_a(x; 4,l)$ ($l = 0, 1, 2, 3$) are considered. When $l = 0, 2$, calculations of $\sharp Q_a(x; 4,l)$ are simple, and we can get these natural densities unconditionally. On the contrary, the distribution properties of $Q_a(x; 4,l)$ for $l = 1, 3$ are rather complicated. Under the assumption of Generalized Riemann Hypothesis, we determine completely the natural densities of $\sharp Q_a(x; 4,l)$ for $l = 1, 3$.


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Koji Chinen. Leo Murata. "On a distribution property of the residual order of $a \pmod {p}$." Proc. Japan Acad. Ser. A Math. Sci. 79 (2) 28 - 32, Feb. 2003.


Published: Feb. 2003
First available in Project Euclid: 18 May 2005

zbMATH: 1071.11054
MathSciNet: MR1960739
Digital Object Identifier: 10.3792/pjaa.79.28

Primary: 11N05
Secondary: 11N25 , 11R18

Keywords: Artin's conjecture (for primitive roots) , residual order

Rights: Copyright © 2003 The Japan Academy

Vol.79 • No. 2 • Feb. 2003
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