Artin prime producing quadratics (Q2479808)

Let \(g\) be a non--square integer, different from \(-1\). If \(p\) is a prime such that \(g\) is primitive root modulo \(p\), then \(p\) is called an `Artin prime\'\ (with respect to \(g\)). The paper studies integral quadratic polynomials generating many Artin primes with respect to some integer. In 1957, W.R.~Griffith thought that \(10\) is a primitive root modulo any prime of the form \(10n^2+7\), and few years later \textit{D. H.~Lehmer} [Scripta Math. 26, 117--119 (1963; Zbl 0114.26206)] proved that all the primes \(p < 10^7\) of the form \(326n^2+3\) are Artin primes with respect to \(326\) with probability \(1\). To be precise, for any primitive (quadratic or not) polynomial \(f(X) \in {\mathbb Z} [X]\) and any integer \(g\) as above, the author lets \((p_j)_{j\geq 1}\) be the sequence of all the primes not dividing \(g\) extracted by the sequence \((f(i))_{i\geq 0}\); then, he considers the largest integer \(r\) (when it exists) such that \(g\) is a primitive root modulo \(p_j\), for all \(j = 1,\ldots,r\), and defines \(c_g(f)\) as the number of distinct primes in \(\{p_1,\ldots,p_r\}\) (in case \(g\) is a primitive root modulo \(p_j\) for all \(j\geq 1\), he lets \(c_g(f) = \infty\)). The main problem one can think of is to determine whether there exist \(g\) and \(f\) such that \(c_g(f)=\infty\), and, if this is the case, to find some explicit examples. For the two cases mentioned above, it is easy nowadays to compute that \(c_{10}(10X^2+7) = 16\) and \(c_{326}(326X^2+3) = 206\). Indeed, at the end of the paper the author is led to conjecture that \(c_g(f) < \infty\) for any quadratic polynomial \(f\) and any integer \(g\). Therefore, one would like to find \(g\) and \(f\) such that \(c_g(f)\) is as large as possible. Under some well--known and generally accepted conjectures, the author gives many heuristic arguments and rigorous results related to such a problem. Also, he points out the connection with quadratic polynomials producing primes, with the splitting of primes \(p=f(n)\) in quadratic fields, and with the value of some \(L\)--functions at \(s=2\). Hence, the author gives an heuristic algorithm to find some integer \(g\) and some quadratic polynomial \(f\) with \(c_g(f)\) large; this algorithm was used by Y.~Gallot to find an explicit example with \(c_g(f)=31082\), which is reported in this paper. The main final result is that, under Dickson's conjecture, for any fixed \(g\) and any fixed positive integer \(m\), there exists a quadratic polynomial \(f\) such \(c_g(f) = m\). The author then conjectures that this holds true unconditionally.