On the distribution of cyclic number fields of prime degree (Q2907092)

Let \(N_{C_{p}}(X)\) denote the number of \(C_{p}\) Galois extensions of \(Q\) with absolute discriminant \(\leq X\). The work of \textit{D. J. Wright} [Proc. Lond. Math. Soc., III. Ser. 58, No. 1, 17--50 (1989; Zbl 0628.12006)] implies that when \(p\) is prime, we have NEWLINE\[NEWLINE N_{C_{p}}(X) = c(p)X^{1/(p-1)} + O(X^{1/p})NEWLINE\]NEWLINE for some positive \(c(p)\). NEWLINENEWLINEIn this paper, the authors, reduce the secondary error term to \(O(X^{1/2(p-1)})\). Moreover, under the Generalized Riemann Hypothesis, they obtain the following stronger result NEWLINE\[NEWLINE N_{C_{p}}(X) = c(p)X^{1/(p-1)} + X^{1/3(p-1)}R_{p}(\log\, X) + O(X^{\frac{1}{4(p-1)}+\epsilon})NEWLINE\]NEWLINE where \(R_{p}(X) \in Q[X]\) is polynomial of degree \(\lfloor p(p-2)/3\rfloor - 1\). This confirms an idea of \textit{H. Cohen, F. Diaz y Diaz} and \textit{M. Olivier} [J. Théor. Nombres Bordx. 18, No. 3, 573--593 (2006; Zbl 1193.11109)] in the case of \(C_{3}\) extensions.