A conditional proof of Artin's conjecture for primitive roots (Q2746252)

In the interesting paper under review a conditional proof of Artin's conjecture for primitive roots \(a\bmod p\) is given. In 1927 E. Artin conjectured that any integer \(a\), neither \(\pm 1\) nor a perfect square, is a primitive root modulo \(p\) for infinitely many primes \(p\). Moreover, if \(N_a(x)\) counts the number of such primes \(p\leq x\), then it is expected that \(N_a(x)\sim A(a)x/\log x\) as \(x\to\infty\), where \(A(a)\) is a constant depending on \(a\). \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048); Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics 70, Cambridge University Press (1976; Zbl 0327.10044)] proved Artin's conjecture and an asymptotic formula for \(N_a(x)\) subject to the truth of the Generalized Riemann Hypothesis for Dedekind zeta functions; he also remarked that \(2\) is a primitive root \(\bmod p\) for infinitely many primes \(p\) if the zeta functions over \({\mathbb K}={\mathbb Q}(\sqrt[k]{2},\sqrt[k]{1})\), where \(k\) is a square-free integer, have no zeros with real part \(>1-{1\over 2}e^{-1}-\delta\), where \(\delta>0\). NEWLINENEWLINENEWLINEThe author proves that \(2\) is a primitive root \(\bmod p\) for a positive proportion of primes \(p\), i.e., \(N_2(x)\gg x/\log x\), if the Dedekind zeta function \(\zeta_{\mathbb K}(s)\) is zero free in \(\Re s>1-{1\over 2} \exp(-{12A\over 5})-\delta\), where \(A=\prod_q(1-{1\over q(q-1)})\), \(q\) prime. Since \(A<5/12\), this improves Hooley's result slightly. The proof relies mainly on a variant of the theorem of Bombieri-Vinogradov for ramified primes in Galois extensions due to \textit{M. R. Murty} and \textit{V. K. Murty} [CMS Conf. Proc. 7, 243-272 (1987; Zbl 0619.10039)]. The author considers only the particular case \(a=2\), but the general case can be treated similarly.