On a distribution property of the residual order of \(a\pmod p\) with a quadratic residue condition (Q1943247)

For a given integer \(a\) and an odd prime \(p\) coprime to \(a\), let \(D_a(p)\) denote the multiplicative order of \(a(\text{mod~}p)\). The first author and Leo Murata proved that under the Generalized Riemann Hypothesis (GRH), the set of primes \(p\) such that \(D_a(p)\) is in a prescribed residue class has a natural density, gave explicit formulae for these densities in some simpler cases and obtained many further results. The reviewer by a different method obtained similar results in a series of papers, see his survey [\textit{P. Moree}, Electron. Res. Announc. Am. Math. Soc. 12, 121--128 (2006; Zbl 1186.11061)]. None of these papers of the reviewer is mentioned in the references. In the present paper the authors are interested in the set \(S_{a,b}(k,l)\) of primes \(p\) coprime to \(ab\) such that \(D_a(p)\equiv l(\text{mod~}k)\) and in addition \(b\) is a square modulo \(p\). By quadratic reciprocity it is enough to consider the set of primes \(p\) such that \(D_a(p)\equiv l(\text{mod~}k)\) and in addition \(p\) is in arbitrary prescribed arithmetic progression. The reviewer has shown that this set, under GRH, has a density and gave an expression for it as double sum (Eqn. (14) in [\textit{P. Moree}, J. Number Theory 117, No. 2, 330--354 (2006; Zbl 1099.11053)]). The authors only determine the natural density of the set \(S_{a,b}(k,l)\) in the special cases \((k,l)=(2,j),(q,0),(4,l)\), where \(j=0,1\), \(q\) is an odd prime and \(l=0,2\). It is easy to see that for \(p\) coprime to \(2a\), \(D_a(p)\) being even is equivalent with \(p\) dividing \(a^m+1\) for some \(m\geq 1\). Using this observation and quadratic reciprocity it is not difficult to derive Theorem 1 (concerning \((k,l)=(2,0)\) from the main result of the reviewer and \textit{B. Sury} [Int. J. Number Theory 5, No. 4, 641--665 (2009; Zbl 1190.11052)], but with error term improved from \(O(x(\log x \log \log x)^{-1})\) to \(O(x \log^{-7/6}x)\). The easiest case is where \(a=b=2\). Then one considers the set of odd primes \(p\) for which \(D_2(p)\) is even and \((2/p)=1\). It is easy to see that this equals the set of primes \(p\equiv 1(\text{mod~}8)\) that divide \(2^m+1\) for some \(m\geq 1\). The density of the latter set of primes was computed by Moree and Sury [ibid.] to be \(5/24\) in agreement with the result of the authors.