Proceedings of the Japan Academy, Series A, Mathematical Sciences

On a distribution property of the residual order of $a \pmod {p}$

Koji Chinen and Leo Murata

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Abstract

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 $l = 1, 3$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 79, Number 2 (2003), 28-32.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116443604

Digital Object Identifier
doi:10.3792/pjaa.79.28

Mathematical Reviews number (MathSciNet)
MR1960739

Zentralblatt MATH identifier
1071.11054

Subjects
Primary: 11N05: Distribution of primes
Secondary: 11N25: Distribution of integers with specified multiplicative constraints 11R18: Cyclotomic extensions

Keywords
Residual order Artin's conjecture (for primitive roots)

Citation

Chinen, Koji; Murata, Leo. On a distribution property of the residual order of $a \pmod {p}$. Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 2, 28--32. doi:10.3792/pjaa.79.28. https://projecteuclid.org/euclid.pja/1116443604


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References

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