CHAPTER 0 Introduction Let K be a number field of degree n over Q. We denote by OK and DK the ring of integers in K and the discriminant of K, respectively. We call a subring of K with 1 an order of K if it is a free Z-module of rank n. We are concerned with the asymptotic behaviour of the number of orders of K with index less than X as X tends to oo. In other words, we study the Dirichlet series o where O runs over all orders of K. The definition of T]K(S) is analogous to that of the Dedekind zeta function CK(S) of K. If n = 2, then TJK{S) is nothing but the Riemann zeta function C(5)- Let n 3 and consider the sums (0.1) Yl \DK\~SVK(2S) and ^ \DK\-Sm(2s), DK0 DK0 where K runs over all cubic fields with positive or negative discriminant. It is known that there is a bijection between the set of equivalence classes of integral binary cubic forms and the set of orders of cubic fields (cf. [2]). Using this bijection, we see that the sums (0.1) are essentially equal to the zeta functions associated to the prehomogeneous vector space of binary cubic forms studied by Shintani [6]. Then it follows from a result of Datskovsky and Wright [1] that ^ W = ^ C ( 2 - ) C ( 3 - 1 ) . The purpose of this paper is to prove an analogous formula for T]K(S) when K is a quartic field. Let K b e a quartic field. We denote by K and G the normal closure of K over Q and the Galois group of K over Q, respectively. Then the Galois group G is one of the following five groups: the symmetric group S4 of degree 4, the alternating group A\ of degree 4, the dihedral group D4 of order 8, the cyclic group C4 of order 4 or the Klein four group V4. If a prime number p factorizes in K as p = p^1 -pegg with Npi = pf*, then we say that p is of type j \ 1 fg9 in K. To give an explicit expression for T]K{S), we introduce the following thirteen polynomials of two variables t and X with coefficients in Z. Received by the editor October 24, 1993, and in revised form April 10, 1995 1
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