## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

#### 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

#### 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

#### References

• Chinen, K., and Murata, L.: On a distribution property of the residual orders of $a$ (mod $p$). Analytic Number Theory –- Expectations for the 21st Century –-. RIMS Kokyuroku, 1219, 245–255 (2001). (In Japanese).
• Chinen, K., and Murata, L.: On a distribution property of the residual orders of $a$ (mod $p$). Proceedings of the Conference AC2001 (Algebra and Computation) held at the Tokyo Metropolitan Univ. (2001). (In Japanese). (Published electronically in ftp://tnt.math.metro-u.ac.jp/pub/ac/2001).
• Chinen, K., and Murata, L.: On a distribution property of the residual orders of $a$ (mod $p$), II. New Aspects of Analytic Number Theory. RIMS Kokyuroku, 1274, 62–69 (2002). (In Japanese).
• Hasse, H.: Über die Dichte der Primzahlen $p$, für die eine vorgegebene ganzrationale Zahl $a\ne0$ von durch eine vorgegebene Primzahl $l\ne2$ teilbarer bzw. unteibarer Ordnung mod $p$ ist. Math. Ann., 162, 74–76 (1965).
• Hasse, H.: Über die Dichte der Primzahlen $p$, für die eine vorgegebene ganzrationale Zahl $a\ne0$ von gerader bzw. ungerader Ordnung mod $p$ ist. Math. Ann., 166, 19–23 (1966).
• Hooley, C.: On Artin's conjecture. J. Reine Angew. Math., 225, 209–220 (1967).
• Lagarias, J. C., and Odlyzko, A. M.: Effective versions of the Chebotarev density theorem. Algebraic Number Fields (Durham, 1975). Academic Press, London, pp. 409–464 (1977).
• Lenstra Jr., H. W.: On Artin's conjecture and Euclid's algorithm in global fields. Invent. Math., 42, 201–224 (1977).
• Murata, L.: A problem analogous to Artin's conjecture for primitive roots and its applications. Arch. Math., 57, 555–565 (1991).
• Odoni, R. W. K.: A conjecture of Krishnamurthy on decimal periods and some allied problems. J. Number Theory, 13, 303–319 (1981).