On a distribution property of the residual order of \(a \pmod p\). III. (English) Zbl 1102.11002

Chinen, Koji; Murata, Leo

J. Math. Soc. Japan 58, No. 3, 693-720 (2006).

The authors consider the multiplicative order of \(a \pmod p\), \(D_a(p)\), and are interested in the set of primes \(p\) such that \(D_a(p)\equiv j \pmod{q^i}\) with \(q\) a prime and \(i\geq 1\) an arbitrary integer. The work presented is a continuation of [K. Chinen and L. Murata, J. Number Theory 105, No. 1, 60–81 (2004; Zbl 1045.11066) and J. Number Theory 105, No. 1, 82–100 (2004; Zbl 1045.11067)] (where the case \(q^i=4\) was considered) and the present results were announced in [K. Chinen and L. Murata, Proc. Japan Acad., Ser. A Math. Sci. 80, No. 9, 182–186 (2004; Zbl 1073.11062)]. Under the assumption that \(a\geq 2\) is not of the form \(a_0^h\) with \(a_0\) an integer and \(h\geq 2\) they announce, under GRH (the Generalized Riemann Hypothesis), various interesting results. (For the rest of the review I assume the truth of GRH.) They prove that the latter set has a density \(\Delta_a(q^i,j)\). Moreover, one has \(\Delta_a(q^i,j)=\Delta_a(q^{i-1},j)/q\) if \(q\) is an odd prime and \(i\geq 2\). They also give an explicit expression for \(\Delta_a(q,j)\) in case \(q\) is odd.
These results have been established independently and in greater generality by the reviewer [J. Number Theory 117, No. 2, 330–354 (2006; Zbl 1099.11053)]. A survey of the main results obtained by the reviewer in this area can be found in [Electron. Res. Announc. Am. Math. Soc. 12, 121–128 (2006; Zbl 1186.11061)]. The authors express the density \(\Delta_a(q^i,j)\) as a fivefold sum which they then evaluate further; the reviewer, however, expressed it as a twofold sum. Nevertheless, both approaches are rather technical and involve computing Galois theoretic intersection coefficients.

Reviewer: Pieter Moree (Bonn)

MSC:

11A07	Congruences; primitive roots; residue systems
11N05	Distribution of primes
11N25	Distribution of integers with specified multiplicative constraints
11N69	Distribution of integers in special residue classes

Keywords:

multiplicative order; residue class; density

Citations:

Zbl 1045.11066; Zbl 1045.11067; Zbl 1073.11062; Zbl 1099.11053; Zbl 1186.11061

References:

[1]	K. Chinen and L. Murata, On a distribution property of the residual order of \(a \pmod p\), J. Number Theory, 105 (2004), 60-81. · Zbl 1045.11066
[2]	K. Chinen and L. Murata, On a distribution property of the residual order of \(a \pmod p\), II, In: Proceedings of the Conference “New Aspects of Analytic Number Theory”, held at Kyoto University, November 26-30, 2001, RIMS Kokyuroku, 1274 (2002), 62-69. · Zbl 1071.11054
[3]	K. Chinen and L. Murata, On a distribution property of the residual order of \(a \pmod p\) – IV, In: Proceedings of the Conference “Analytic Number Theory and Surrounding Areas” held at Kyoto University, September 29-October 3, 2003, RIMS Kokyuroku, 1384 (2004), 169-174. · Zbl 1071.11054
[4]	H. Hasse, Über die Dichte der Primzahlen \(p\), für die eine vorgegebene ganzrationale Zahl \(a\neq0\) von durch eine vorgegebene Primzahl \(l\neq2\) teilbarer bzw. unteilbarer Ordnung mod \(p\) ist, Math. Ann., 162 (1965), 74-76. · Zbl 0135.10203
[5]	H. Hasse, Über die Dichte der Primzahlen \(p\), für die eine vorgegebene ganzrationale Zahl \(a\neq0\) von gerader bzw. ungerader Ordnung mod \(p\) ist, Math. Ann., 166 (1966), 19-23. · Zbl 0139.27501
[6]	L. Murata, and K. Chinen, On a distribution property of the residual order of \(a \pmod p\) – II, J. Number Theory, 105 (2004), 82-100. · Zbl 1045.11067
[7]	L. Murata, and K. Chinen, On a distribution property of the residual order of \(a \pmod p\), III, In: Proceedings of the Conference “Diophantine Problems and Analytic Number Theory”, held at Kyoto University, October 21-25, 2002, RIMS Kokyuroku, 1319 (2003), 139-147. · Zbl 1071.11054
[8]	R. W. K. Odoni, A conjecture of Krishnamurthy on decimal periods and some allied problems, J. Number Theory, 13 (1981), 303-319. · Zbl 0471.10031
[9]	K. Prachar, Primzahlverteilung, Springer, 1957. · Zbl 0080.25901
[10]	J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci., 54 (1981), 323-401. · Zbl 0496.12011
[11]	K. Wiertelak, On the density of some sets of primes, I, Acta Arith., 34 (1978), 183-196. · Zbl 0373.10029
[12]	K. Wiertelak, On the density of some sets of primes, IV, Acta Arith., 43 (1984), 177-190. · Zbl 0531.10049