The least prime in a conjugacy class (Q2707863)

The conjugacy classes \(C\) of the title arise when \(K\) is a number field, \(L\) is a finite Galois extension of \(K\), and \(\mathfrak q\) is a prime ideal of \(L\) which divides a prime ideal \(\mathfrak p\) of \(K\) which is unramified in \(L\). When \(({\mathfrak q},L/K)\) is a Frobenius automorphism in \(G=\text{ Gal(L/K)}\) the set \(\text{ Fr}_{\mathfrak p}=\text{ Fr}_{\mathfrak p}(L/K)=\{({\mathfrak q},L/K):{\mathfrak q}|{\mathfrak p}\}\) forms a full conjugacy class in \(G\). NEWLINENEWLINENEWLINELet \(\pi_C(x,L/K)\) denote the number of such \(\mathfrak p\) of degree one with \(\text{ Fr}_{\mathfrak p}=C\) such that the rational prime \({\mathbb N}_{K/Q}\mathfrak p\) does not exceed \(x\), and let \(P(C,L/K)\) denote the least value of \(x\) for which this number is positive. Further, denote \(P(L/K) = \max_C P(C,L/K)\). The author discusses and formulates a number of conjectures relating to these numbers \(P\), and establishes some results on the connections and implications between these conjectures.NEWLINENEWLINENEWLINE\textit{J. C. Lagarias} and \textit{A. M. Odlyzko} [Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975, 409-464 (1977; Zbl 0362.12011)] conjectured the following generalisation of Linnik's Theorem: there is an absolute constant \(c_1>0\) such that \(P(L/K) \ll (\log d_L)^{c_1}\), in which \(d_L\) denotes the absolute value of the discriminant of \(L/\mathbb Q\). With \textit{H. L. Montgomery} [Invent. Math. 54, 271-296 (1979; Zbl 0401.12014)] they showed unconditionally that \(P(L/K) \ll d_L^{c_2}\) for some absolute constant \(c_2\), and that on the GRH (the Riemann Hypothesis for the Dedekind zeta-function \(\zeta_L(s)\)) one would obtain \(P(L/K) \ll \log^2 d_L\). The author conjectures that there are positive absolute constants \(a,b\) such that \(P(L/K) \ll d_L^{an_K/n_L}(\log d_L)^b\). Here \(n_K=[K:Q]\) and \(n_L=[L:Q]\). On the basis of the GRH he conjectures \(P(C,L/K) \ll |C|^{-1}\log^2 d_L\), an estimate that decreases as \(|C|\) increases, as one might expect. NEWLINENEWLINENEWLINEThe Artin Holomorphy conjecture states that in the factorisation of the Dedekind zeta-function as a product of factors \(L(s,\chi)^{\chi(1)}\) the individual \(L\)-functions are holomorphic. On the basis of this conjecture and the GRH the author proves a result slightly weaker than his conjecture: NEWLINE\[NEWLINEP(C,L/K) \ll |C|^{-1}n_K^2(n\log n +\log d_L)^2,NEWLINE\]NEWLINE where \(n=n_L/n_K = [L:K]\). NEWLINENEWLINENEWLINEThe author also discusses the relationship between these and other conjectures with hypotheses about the zeros of the Dedekind zeta-function and of Artin \(L\)-functions. Further, he examines some properties of \(P(C,L/K)\) that arise when one considers subgroups or quotients of \(G\).