On the least primitive root in number fields (Q625870)

Let \(K\) be an algebraic number field, and for a prime ideal \(P\) let \(g(P)\) be the primitive root mod \(P\) of minimal norm. The author shows, assuming GRH for Hecke zeta-functions of \(K\), that for any \(K\) one has \(N(g(P))\ll_K n^4(1+\log n)^4\log^2(N(P))\), where \(n\) denotes the number of distinct prime factors of \(N(P)-1\). This improves upon an earlier result of \textit{Y. H. Wang} and \textit{C. Bauer} [Acta Arith. 115, 269--285 (2004; Zbl 1056.11047)]. The proof is based on a generalization of Perron's formula established by \textit{W. Grotz} [Acta Arith. 38, 69--95 (1980; Zbl 0363.12011)].