### Distribution of orders in number fields

Nathan Kaplan, Jake Marcinek, Ramin Takloo-Bighash
2015 Research in the Mathematical Sciences
In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields. Remark 2. Note that by Theorem 1.5 of
more » ... Theorem 1.5 of [10], the zeta function η K (s) has an analytic continuation to a domain of the form s > α − with α > 0 the abscissa of convergence and > 0. Remark 3. A byproduct of our methods, stated as Corollary 4 and Corollary 5 in section 'The proof of Theorems 1 and 2', is an improvement of the upper bounds obtained by Brakenhoff [3], Theorem 5.1 and Theorem 8.1. Remark 4. It would be interesting to obtain information about the constant C K . For the cubic case, the results of [6] stated below give precise values for C K . Corollary 1 of Nakagawa [24] gives the value of C K in terms of certain Euler products, but it is not clear if these Euler products have any conceptual meaning. For higher degree extensions, we know nothing about the constants C K . We define an order O in O L to be a subring with identity of O L which is of Z-rank d. Again, we set η L (s) = O order 1 [ O L : O] s . As usual, knowing the analytic properties of η L (s) via Tauberian arguments, e.g., Theorem 9, gives us information about the function N L (B) := |{O ⊂ O L ; Oan order, [ O L : O] ≤ B}| . Our methods give an asymptotic formula for N L (B) whenever [ L : Q] ≤ 5. Let us explain the simplest possible case. For d ∈ N, we set N d (B) := N Q d (B). Given k ∈ N, we define f d (k) to be the number of orders in Z d of index equal to k. Clearly, N d (B) = k≤B f d (k). It is easy to see that the function f d (k) is multiplicative, i.e., if k 1 , k 2 are coprime integers then f d (k 1 k This is the prototype of the problem that we study in this paper: Despite its innocent appearance, this is a very difficult problem, and prior to our work, the only cases for which an asymptotic formula is known for N d (B) are d = 2, 3, 4 [19]. Here, we obtain an asymptotic formula for N 5 (B), and give non-trivial bounds for N d (B) when d > 5. Definition 1. Let d, k ∈ N. We define a < Z d (k) to be number of subrings S of Z d , not necessarily with identity, such that [ Z d : S] = k.