MATHEMATICS OF COMPUTATION Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0025-5718(XX)0000-0

DENSITY OF CUBIC FIELD DISCRIMINANTS

DAVID P. ROBERTS

Abstract. In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density of cubic field discriminants. Our refinement is plausible theoretically and agrees very well with computational data.

1. Introduction

Let a_{n} be the number of isomorphism classes of abelian cubic fields with discriminant n. Let b_{n} be the number of isomorphism classes of non-abelian cubic fields with discriminant n. The numbers a_{n} are very well understood. The numbers b_{n} have been the subject of extensive theoretical and computational study for at least sixty years, but are less well understood. The object of this note is to contribute to the study of these b_{n}, by bringing together the theoretical and computational literature.

For α ∈ {−, +}, define

g_{α}(x) = ∑_{n=1}^{x} b_{αn}. (1.1)

Put C_{−} = 3 and C_{+} = 1. The main theoretical result is due to Davenport and Heilbronn [8]:

g_{α}(x) ∼ C_{α} \frac{1}{12ζ(3)}x. (1.2)

The numbers b_{αn} have been computed for larger and larger n by many authors. The most recent results are due to Belabas [3], who introduced a vastly improved method and worked through the cutoff x = 10^{11}.

It has been discussed by several authors that the “match” between computational evaluation of g_{α}(x) and the asymptotic formula (1.2) is not very good, as illustrated by Tables 4.1, 4.2, and 6.1 below. For example, the paper [9] dealt with the case α = −; here is an excerpt from pages 322-324, with some trivial notational changes:

If, however, the reader were to plot the density versus x, he would be somewhat astonished to see that this density is increasing so slowly that his first impression would be that it will not make it to Davenport-Heilbronn limit. Thus it remains a challenging problem, assuming that the D-H limit is not in error, to explain the origin of this very slow convergence. This problem was indicated by Shanks in [13],[1], and on the real side in [13],[10], where the problem is further aggravated

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 11N56, 11R16.

by even slower convergence. To date, and to our knowledge, no good quantitative explanation of this phenomenon has been given.

In this paper we conjecture a quantitative explanation of this slow-convergence phenomenon.

2. The abelian case

It is best to briefly review the situation in the abelian case. One has the formula

∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = −1 + (1 + \frac{2}{81^{s}} ) ∏_{p≡1 (6)} (1 + \frac{2}{p^{2s}} ) . (2.1)

Put

f(x) = ∑_{n=1}^{x} a_{n}. (2.2)

Cohn [4] showed that the function (2.1) continues analytically to Re(s) > 1/4, except for a simple pole at s = 1/2. He computed the residue of this pole. This gave him more than enough information to conclude that

f(x) ∼ (\frac{11\sqrt{3}}{36π} ∏_{p≡1 (6)} \frac{(p + 2)(p − 1)}{p(p + 1)}) x^{1/2}. (2.3)

Define h_{−}(x) = g_{−}(x) and

h_{+}(x) = g_{+}(x) + \frac{1}{3}f(x). (2.4)

From (1.2) and (2.3) one sees that (1.2) would remain true with g_{α} replaced by h_{α}. We henceforth focus more on h_{α}, as it appears naturally in (3.2) and (3.3).

3. The basic conjecture

Belabas proved [3, (8) and (20)] that, for either α, the difference between the sides of (1.2) is o(x/(log x)^{2}). From his extensive computational data he speculated [3, page 1232] that these differences are o(x/(log x)^{β}) for arbitrarily large β. We believe a stronger statement. Recall that C_{−} = 3 and C_{+} = 1. Put K_{−} = \sqrt{3} and K_{+} = 1.

Conjecture 3.1. For each α,

h_{α}(x) = C_{α} \frac{1}{12ζ(3)}x + K_{α} \frac{4ζ(1/3)}{5Γ(2/3)^{3}ζ(5/3)}x^{5/6} + o(x^{5/6}). (3.1)

Numerically, (3.1) works out to

h_{−}(x) = (0.2079768431 …)x − (0.2557983756 …)x^{5/6} + o(x^{5/6})
h_{+}(x) = (0.0693256143 …)x − (0.1476852610 …)x^{5/6} + o(x^{5/6})

Note that the coefficient of x^{5/6} is negative in each case, the negative sign coming from ζ(1/3) < 0. This negativity corresponds to the fact that computationally one has always found fewer fields than the Davenport-Heilbronn asymptotic would suggest.

The heuristic we are about to sketch is too crude to suggest any particular error term in (3.1). The o(x^{5/6}) we have written there corresponds to the weakest conjecture in which our conjectured new term actually plays a role. However, Tables 4.1 and 4.2 below suggest that the error term should be reducible at least most of the way to o(x^{1/2}).

We are led to our conjecture theoretically as follows. Define a Dirichlet series

(3.2)  ξ_{α}(s) = ∑_{k∈K_{α}} \frac{1}{| Aut(k)||D_{k}|^{s}} .

Here K_{α} is the set of isomorphism classes of cubic fields k with discriminant of sign α. Also, D_{k} is the discriminant of k while Aut(k) is the automorphism group of k; so | Aut(k)| is 3 or 1 according to whether k is abelian or not, the former case occuring only for α = +.

For S a finite set of primes, define a weighted version of (3.2)

ξ_{α,S}(s) = ∑_{k∈K_{α}} \frac{η_{k,S}(s)}{| Aut(k)||D_{K}|^{s}} . (3.3)

Here the weights η_{k,S}(s) are given on page 122 of [6]. These weights are themselves Dirichlet series ∑ c_{m}(k, S)m^{−s} with integral coefficients c_{m}(k, S) decreasing to the Kronecker symbol δ_{m1} as S grows. Thus ξ_{α,S} → ξ_{α} in the sense of formal Dirichlet series.

The point of introducing the approximations ξ_{α,S} is that one understands them analytically better than one understands ξ_{α} itself. From Davenport’s earlier work [7] one knows that the rightmost pole of ξ_{α,∅} is at s = 1. The analytic behavior of ξ_{α,∅} was investigated more explicitly by Shintani [14], who found the next pole at s = 5/6. More generally, it is proved in [5, Prop. 6.2], that each ξ_{α,S} continues to the whole s-plane, meromorphically with only poles at s = 1 and s = 5/6. These poles are simple, their residues C_{α}r_{1,S} and K_{α}r_{5/6,S} being deducible from the first two equations in [5, Thm. 6.2i]. In the case s = 1, one has to set b_{α} equal to zero in the corresponding equation; this b_{α} corresponds to quadratic fields, included in ξ_{α,S} in [5] and [6], but not included here.

From the formulas in [5, Thm. 6.2i], one sees that the numbers r_{1,S} are positive, and decreasing as S grows. Similarly, the numbers r_{5/6,S} are negative and increasing as S grows. Let r_{1} and r_{5/6} be their respective limits. One does not have enough control at present to rigorously apply any “poles-control-growth” theorems. However the general formalism suggests that h_{α}(x) grows as C_{α}r_{1}x + K_{α}(6/5)r_{5/6}x^{5/6}. This is the source of our Conjecture 3.1. The constant r_{5/6} is actually nicer than we had initially expected from its source as the limit of the r_{5/6,S}; special zeta values arise because of the factorization of the right side of (5.3) below.

To follow our heuristic in a more detailed way, one should first understand it in the more refined context of Section 5 below. We recommend looking next at Prop. 2.1 and the discussion around it in [6]; this treats in more classical language the special case of [5, Thm 6.2i] we need, except it does not repeat the formula for r_{5/6,S}.

4. Numerical evidence for the basic conjecture

Our conjecture for α = − compares with Belabas’s data [3, page 1232] for negative-discriminant cubic fields as in Table 4.1. Here and below, x = 10^{j}. Also H_{α}(x) denotes the right side of (1.2) while H_{α}^{∗}(x) denotes the first two terms of the right side of (3.1). Clearly H_{−}^{∗} matches the data h_{−} substantially better than H_{−} Table 4.1. Imaginary cubic fields: computation h_{−} and theory H_{−}^{∗}

\begin{matrix}j & h_{−}(x) & \frac{h_{−}(x)}{H_{−}(x)} & \frac{h_{−}(x)}{H_{−}^{∗}(x)} & \frac{h_{−}(x) − H_{−}^{∗}(x)}{x^{1/2}} \\ 2 & 7 & 0.337 & 0.7843510 & -0.1925 \\ 3 & 127 & 0.611 & 0.9993210 & -0.0027 \\ 4 & 1,520 & 0.731 & 0.9943300 & -0.0867 \\ 5 & 17,041 & 0.819 & 0.9998781 & -0.0066 \\ 6 & 182,417 & 0.877 & 1.0001096 & 0.0200 \\ 7 & 1,905,514 & 0.916 & 1.0000100 & 0.0060 \\ 8 & 19,609,185 & 0.943 & 0.9999394 & -0.1188 \\ 9 & 199,884,780 & 0.961 & 0.9999850 & -0.0951 \\ 10 & 2,024,660,098 & 0.974 & 1.0000009 & 0.0176 \\ 11 & 20,422,230,540 & 0.982 & 1.0000003 & 0.0218 \end{matrix}

does. The poor match of H_{−} to h_{−} in the range 10^{5} ≤ x ≤ 10^{6} formed the basis of the quote in Section 1.

For α = +, we start from the values of g_{+}(x) + f(x) listed on [3, page 1232]. We use (2.1) to compute f(x) and thereby deduce h_{+}(x) = g_{+}(x) + (1/3)f(x). Our

Table 4.2. Real cubic fields: computation h_{+} and theory H_{+}^{∗}

\begin{matrix}j & g_{+}(x) & f(x) & \frac{h_{+}(x)}{H_{+}(x)} & \frac{h_{+}(x)}{H_{+}^{∗}(x)} & \frac{h_{+}(x)−H_{+}^{∗}(x)}{x^{1/2}} & \frac{g_{+}(x)−H_{+}^{∗}(x)}{x^{1/2}} \\ 2 & 0 & 2 & 0.096 & 8.5889786 & 0.0589 & -0.0078 \\ 3 & 22 & 5 & 0.341 & 1.0461129 & 0.0330 & -0.0197 \\ 4 & 366 & 16 & 0.536 & 0.9900166 & -0.0374 & -0.0908 \\ 5 & 4,753 & 51 & 0.688 & 1.0010833 & 0.0163 & -0.0374 \\ 6 & 54,441 & 159 & 0.786 & 0.9988436 & -0.0631 & -0.1161 \\ 7 & 592,421 & 501 & 0.855 & 0.9999134 & -0.0162 & -0.0690 \\ 8 & 6,246,698 & 1,592 & 0.901 & 1.0000259 & 0.0161 & -0.0369 \\ 9 & 64,654,353 & 5,008 & 0.933 & 1.0000097 & 0.0198 & -0.0330 \\ 10 & 661,432,230 & 15,851 & 0.954 & 0.9999988 & -0.0081 & -0.0609 \\ 11 & 6,715,773,873 & 50,152 & 0.969 & 1.0000002 & 0.0046 & -0.0482 \end{matrix}

hope that the error term in Conjecture 3.1 can be reduced to o(x^{1/2}) would require that the numbers in the last column of Table 4.1 and the next to last column of Table 4.2 tend to zero. The data seems rather borderline in support of o(x^{1/2}), but strongly indicative that our conjecture as stated is true, and moreover that the o(x^{5/6}) can be reduced at least somewhat.

From Cohn’s result (2.3), at most one linear combination g_{+}(x) + uf(x) can agree with H_{+}^{∗}(x) to within o(x^{1/2}). The last two columns of Table 4.2 suggest that u = 1/3 is more likely than u = 0. In fact u = 0.31 … would give the best fit, in a least squares sense, and this agrees well with the theoretically favored u = 1/3.

5. The refined conjecture

The Davenport-Heilbronn theorem comes in a more refined version and so does our conjecture. Let A be the set of symbols {111, 21, 3, 1^{2}1, 1^{3}}. Then for K a cubic field and p a prime, the splitting behavior of p in K determines an element of A. If p is unramified, then it either splits completely (111), splits partially (21), or remains inert (3). If p is ramified, then it is either partially ramified (1^{2}1) or totally ramified (1^{3}).

Up until now, we have been using α to denote a sign, + or −. Henceforth, we use α to denote the following data. First, a finite set support(α) ⊂ {∞, 2, 3, 5, … } of places of Q, containing ∞. Second, a sign α_{∞} ∈ {+, −}, as before. Third, for each prime p ∈ support(α), an element α_{p} ∈ A. Let K_{α} be the set of isomorphism classes of cubic fields meeting the given local condition α_{v} for each v ∈ support(α). One naturally has f_{α}, g_{α}, h_{α}, ξ_{α}, and ξ_{α,S}, all direct generalizations of the notions introduced above in the special case support(α) = {∞}.

Define C_{∞,−} = 3 and C_{∞,+} = 1, as in Section 1 and K_{∞,−} = \sqrt{3} and K_{∞,+} = 1, as in Section 3. Define p-adic analogs as follows:

\begin{matrix}α_{p} & C_{p,α_{p}} & K_{p,α_{p}} \\ 111 & 1/6C_{p} & (1 + p^{−1/3})^{3}/6K_{p} \\ 21 & 1/2C_{p} & (1 + p^{−1/3})(1 + p^{−2/3})/2K_{p} \\ 3 & 1/3C_{p} & (1 + p^{−1})/3K_{p} \\ 1^{2}1 & 1/pC_{p} & (1 + p^{−1/3})^{2}/pK_{p} \\ 1^{3} & 1/p^{2}C_{p} & (1 + p^{−1/3})/p^{2}K_{p} \end{matrix} (5.1)

Here

(5.2)  C_{p} = 1 + p^{−1} + p^{−2}
(5.3)  K_{p} = \frac{(1 − p^{−5/3})(1 + p^{−1})}{1 − p^{−1/3}}.

are normalizing factors, chosen so that the column-sums ∑ C_{p,α_{p}} and ∑ K_{p,α_{p}} are both 1. The values of C_{p,α_{p}} are taken from e.g. [10, page 589] while the K_{p,α_{p}} are taken from [5, Table 5.1]. Note that both the C_{p,α_{p}} and the K_{p,α_{p}} are actually given by procedures uniform in α_{p}. For example, the ratios (K_{p,α_{p}}K_{p})/(C_{p,α_{p}}C_{p}) are all given by replacing a factor f^{e} in α_{p} by a factor (1 + p^{−f/3}), e.g. 1^{2}1 ↦ (1 + p^{−1/3})(1 + p^{−1/3}).

We expect that Conjecture 3.1 holds verbatim for these more general α, with

(5.4)  C_{α} = ∏_{v∈support(α)} C_{v,α_{v}}
(5.5)  K_{α} = ∏_{v∈support(α)} K_{v,α_{v}}.

Our heuristic continues to make sense in this more localized context. In fact the basic references [5] and [6] are in this context.

6. Numerical evidence for the refined conjecture

In this section, we work with twenty different α. All of them have support {∞, p} and α_{∞} = +. We let p run over {2, 3, 5, 7} and let α_{p} run over the five-element set A. For simplicity, we compare computation with theory only at x = 10^{7}.

We get the numbers g_{α}(10^{7}) by renormalizing the first four lines of [10, Table 11]. We get the numbers f_{α}(10^{7}) by a simple procedure based on (2.1). Including these extra 501 fields, with total mass 501/3 = 167, improves the fit slightly, as would be expected from line j = 7 of Table 4.2.

Table 6.1 gives the comparison. This table illustrates how the slowness of the

Table 6.1. Real cubic fields and their p-adic decomposition:

computation h_{α} and theory H_{α}^{∗} (x fixed at 10^{7})

p=2  p=3  p=5  p=7
α_{p} \frac{h_{α}(x)}{H_{α}(x)} \frac{h_{α}(x)}{H_{α}^{∗}(x)} \frac{h_{α}(x)}{H_{α}(x)} \frac{h_{α}(x)}{H_{α}^{∗}(x)} \frac{h_{α}(x)}{H_{α}(x)} \frac{h_{α}(x)}{H_{α}^{∗}(x)} \frac{h_{α}(x)}{H_{α}(x)} \frac{h_{α}(x)}{H_{α}^{∗}(x)}
111 0.706 0.9999 0.721 0.9998 0.734 0.9994 0.740 0.9994
21 0.851 0.9999 0.856 1.0001 0.858 0.9996 0.858 0.9997
3 0.924 1.0001 0.923 0.9999 0.920 1.0002 0.917 1.0008
1^{2}1 0.836 0.9996 0.835 0.9997 0.832 1.0002 0.829 0.9985
1^{3} 0.909 1.0003 0.903 0.9998 0.895 1.0006 0.891 1.0025

convergence h_{α}(x)/H_{α}(x) to 1 seems to depend systematically on α_{p}, the totally split case α_{p} = 111 being the slowest. On the other hand, h_{α}(x)/H_{α}^{∗}(x) is much closer to 1, with no apparent tendency to be above or below.

7. Three concluding remarks

By using the five element set A, we are falling slightly short of the most locally refined conjecture, because there are different ways a prime can be ramified. Thus, for example, if a field K is 1^{3} at p = 3, then there are actually nine possibilities for its 3-adic completion K_{3}; both C_{3,1^{3}} and K_{3,1^{3}} should be split into nine parts, in the proportions indicated in the (p, α_{p}) = (3, 1^{3}) slot on the following chart:

\begin{matrix}α_{p} & p = 2 & p = 3 & p ≥ 5 \\ 1^{2}1 & \frac{1}{4} \frac{1}{4} \frac{1}{8} \frac{1}{8} \frac{1}{8} \frac{1}{8} & \frac{1}{2} \frac{1}{2} & \frac{1}{2} \frac{1}{2} \\ 1^{3} & 1 & \frac{1}{3} \frac{1}{3} \frac{1}{9} \frac{1}{81} \frac{1}{81} \frac{1}{81} \frac{1}{81} \frac{1}{81} \frac{1}{81} & 1 or \frac{1}{3} \frac{1}{3} \frac{1}{3} \end{matrix} (7.1)

Similarly, if a field K is 1^{3} at p ≥ 5, there are either 1 or 3 possibilities for K_{3}, according to whether p ≡ 5, or 1 modulo 6. In general, a p-adic algebra K_{p} of type α_{p} contributes to C_{p,α_{p}} and K_{p,α_{p}} in proportion to 1/(| Aut(K_{p})||D_{K_{p}}|).

Our basic theoretical references [5] and [6] work over arbitrary number fields as well as finite extensions of F_{p}(t) for p ≥ 5. We expect that the analog of our fully refined Conjecture 3.1 holds there too, but at present there is no corresponding published computational data.

The most important remaining problem is to prove our conjecture. The generally pessimistic discussion on [6, page 124] and [2, page 620] suggests to us that the way may be difficult. However, one ingredient of a proof might be the functional equation of ξ_{α,S} with respect to s ↦ 1 − s, studied in [14], [5], and [11]. Another ingredient might be [12, Thm. 3], which concerns growth of arithmetic functions whose associated Dirichlet series satisfy such a functional equation.

References

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[2] K. Belabas, Crible et 3-rang des corps quadratiques, Ann. Inst. Fourier (Grenoble), 46 (1996), 909-949.

[3] , A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237.

[4] H. Cohn, The density of abelian cubic fields, Proc. Amer. Math. Soc. 5 (1954), 476-477.

[5] B. Datsovsky and D. J. Wright, The adelic zeta function associated with the space of binary cubic forms, II: local theory, J. Reine Angew. Math. 367 (1986), 27-75.

[6] , Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988) 116-138.

[7] H. Davenport, On the class number of binary cubic forms, I & II., J. London Math. Soc. 26 (1951), 183-198 (Corrigendum, ibid 27 (1952) 512).

[8] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (II), Proc. Roy. Soc. Lond. A 322 (1971) 405-420

[9] G. W. Fung and H. G. Williams, On the computation of complex cubic fields, with discriminant D ≥ −10^{6}, Math. Comp. 55 (1990), 313-325. Errata Math. Comp. 63 (1994), 433.

[10] P. Llorente and J. Quer, On totally real cubic fields with discriminant d < 10^{7}, Math. Comp. 50 (1988), 581-594.

[11] Y. Ohno, A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms, Amer. J. of Math. 119 (1997), 1083-1094.

[12] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. 100 (1974), 131-170.

[13] D. Shanks, Review of [1], Math. Comp. 30, (1976), 670-673.

[14] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan, 24 (1972), 132-188.

Department of Mathematics, Rutgers University, Piscataway, NJ 08854 Current address: Division of Science and Mathematics, University of Minnesota-Morris, Mor-

ris, MN 56267

E-mail address: roberts@morris.umn.edu
URL: http://cda.morris.umn.edu/~roberts/