Counting \(S_5\)-fields with a power saving error term (Q2879426)

The authors give a sketch of proof for following result: for \(i=0,1,2\) and \(0<X\) let \(N_5^{(i)}(X)\) denote the number of quintic number fields with Galois group \(S_5\), with \(i\) complex places, and the absolute value of their discriminants being bounded by \(X\). Then there are explicitly given constants \(c_i\) such that NEWLINE\[NEWLINE N_5^{(i)}(X) = c_i X + O_\epsilon (X^{\frac {399}{400}+ \epsilon}).NEWLINE\]NEWLINE The proof employs results of \textit{M. Bhargava} [Ann. Math. (2) 172, No. 3, 1559--1591 (2010; Zbl 1220.11139)] counting the number of quintic rings, and the Selberg sieve to obtain the result for quintic fields.