The density of discriminants of quintic rings and fields
Let \(N_5^{(i)}(\xi,\eta)\) denote the number of quintic algebraic number fields \(K\), up to isomorphism, having \(5-2i\) real embeddings and satisfying \(\xi<\text{Disc}(K)<\eta\). The author proves that \[ \lim_{X\to\infty}\frac{N_5^{(0)}(0,X)}{X}=\frac{1}{240}\prod_p (1+p^{-2}-p^{-4}-p^{-5}); \] \[ \lim_{X\to\infty}\frac{N_5^{(1)}(-X,0)}{X}=\frac{1}{24}\prod_p (1+p^{-2}-p^{-4}-p^{-5}); \] \[ \lim_{X\to\infty}\frac{N_5^{(2)}(0,X)}{X}=\frac{1}{16}\prod_p (1+p^{-2}-p^{-4}-p^{-5}). \] Let \(M_5^{(i)}(\xi,\eta)\) denote the number of isomorphism classes of orders \({\mathcal O}\) in quintic fields having \(5-2i\) real embeddings and satisfying \(\xi<\text{Disc}\,{\mathcal O}<\eta\). Then there exists a positive constant \(\alpha\) such that \[ \lim_{X\to\infty}\frac{M_5^{(0)}(0,X)}{X}=\frac{\alpha}{240}; \] \[ \lim_{X\to\infty}\frac{M_5^{(1)}(-X,0)}{X}=\frac{\alpha}{24}; \] \[ \lim_{X\to\infty}\frac{M_5^{(2)}(0,X)}{X}=\frac{\alpha}{16}. \]
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