A note on the two variable Artin's conjecture (Q6556217)

In [J. Number Theory 85, No. 2, 291--304 (2000; Zbl 0966.11042)], \textit{P. Moree} and \textit{P. Stevenhagen} proposed a two-variable Artin conjecture. They proved that the set of primes \(p\) for which a given \(b\) is contained in the residue class \(a \bmod p\) has positive density \(\delta(a,b)\) when \(a\) and \(b\) are multiplicatively independent. In [J. Number Theory 194, 8--29 (2019; Zbl 1437.11141)], \textit{M. R. Murty} et al. showed that the number of such primes \(p \le x\) is \(\gg x/\log^2 x\). In the present paper, this lower bound is improved using ideas from \textit{D. R. Heath-Brown} [Q. J. Math., Oxf. II. Ser. 37, 27--38 (1986; Zbl 0586.10025)]: if \(W = \{m_1, m_2, m_3\}\) is a set of three multiplicatively independent integers such that none of \(m_1\), \(m_2\), \(m_3\), \(-3m_1m_2\), \(-3m_2m_3\), \(-3m_3m_1\), \(m_1m_2m_3\) is a perfect square, then there is at least one pair of distinct elements \(a, b \in W\) such that \N\[\N \# \{p \le x: b \bmod p \in \langle a \bmod p \rangle \} \gg x \log \log x/\log^2 x. \N\]