On a distribution property of the residual order of <i>a</i> (mod <i>p</i>) — III

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  • Chinen Koji
    Department of Mathematics, Faculty of Engineering, Osaka Institute of Technology
  • Murata Leo
    Department of Mathematics, Faculty of Economics, Meiji Gakuin University

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Other Title
  • On a distribution property of the residual order of a (mod p)(3)
  • On a distribution property of the residual order of a (mod p)- III
  • On a distribution property of the residual order of $a \pmod {p}$
  • On a distribution property of the residual order of a(modp)—II
  • On a distribution property of the residual order of a(modp)
  • On a Distribution Property of the Residual Order of a (mod p)— IV

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Let a be a positive integer which is not a perfect b-th power with b≥2, q be a prime number and Qa(x;qi,j) be the set of primes p≤x such that the residual order of a (mod p) in (Z/pZ)× is congruent to j modulo qi. In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of Qa(x;qi,j) for i≥3 if q=2, i≥1 if q is an odd prime, and for an arbitrary nonzero integer j (the main results of this paper are announced without proof in [3], [7] and [2]).

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