On a conjecture of Rodier on primitive roots
Let \(N\geq3\) be an integer. A sequence \(a_1,a_2,\dots\) of integers is said to be weakly uniformly distributed \(\bmod\;N\) (WUD \(\bmod\;N\)) if it contains infinitely many elements prime to \(N\) and the condition \((r,N)=(s,N)=1\) implies asymptotic equality of \(\#\{j\leq t:\;a_j\equiv r\pmod N\}\) and \(\#\{j\leq t:\;a_j\equiv s\pmod N\}\) for \(t\) tending to infinity. Let also for squarefree \(g\neq 0,\pm1\) \(P_g\) be the sequence of all primes having \(g\) for its primitive root. The author shows, assuming GRH, that this sequence is WUD \(\bmod N\) if and only if either \(g\equiv 1\pmod 4\) and \(N\) is a power of 2 or \(g\equiv 2 \pmod 4\) and \(N=4\). Details are given only in the case \(g=2\), the general case being treated in a preprint (MPI Bonn, preprint 5/1997). The proof in this case relies on the methods used by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)], \textit{H. W. Lenstra, jun.} [Invent. Math. 42, 201-224 (1977; Zbl 0362.12012)] and \textit{G. Cooke} and \textit{P. J. Weinberger} [Commun. Algebra 3, 481-524 (1975; Zbl 0315.12001)].
scientific article; zbMATH DE number 1114170