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On a disctribution property of the residual order of \(a\pmod p\). (English) Zbl 1071.11054

Let \(a\) be a rational number and let \(Q_a(x,k,l)\) be the set of primes \(p\leq x\) such that the (residual) order of \(a(\text{ mod~}p)\) is congruent to \(l(\text{ mod~}k)\). In case \(a\) is a positive integer which is not a perfect \(h\)th-power with \(h\geq 2\) and \(k=4\) the authors announce various results regarding \(Q_a(x,k,l)\) and sketch an outline of the proofs. Meanwhile these results and complete proofs have appeared in [J. Number Theory 105, 60–81 (2004; Zbl 1045.11066) and J. Number Theory 105, 82–100 (2004; Zbl 1045.11067)].
Of the two main results Theorem 1.1 is not new (see Zbl 1045.11066 for details). D. Zagier (personal communication) was the first to find a much shorter reformulation of Theorem 1.2. In the reviewer’s approach such a compact formulation results directly. Moreover, he extended Theorem 1.2 to arbitrary rational \(a\) and with smaller error term [J. Number Theory 114, 238–271 (2005; Zbl 1099.11052)].

MSC:

11N05 Distribution of primes
11N25 Distribution of integers with specified multiplicative constraints
11R18 Cyclotomic extensions
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References:

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