IMAGINARY QUADRATIC FIELDS

WITH SMALL ODD CLASS NUMBER

Steven Arno M. L. Robinson Ferrell S. Wheeler

Supercomputing Research Center 17100 Science Drive, Bowie, MD 20715 SRC-TR-93-102

December 1993

Abstract. In this paper we solve the class number m problem for the ten odd values of m between 5 and 23. “Large” discriminants are ruled out using a result of Oesterlé. Methods of Stark and Montgomery-Weinberger are used to treat a portion of the discriminants of “medium” size. In order to eliminate the remaining, smaller discriminants of “medium” size, we introduce new methods which partition the space of minima (of reduced quadratic forms of given negative fundamental discriminant) in a tractable manner. “Small” discriminants are treated by combining the new minima partitions with a computationally intensive sieve.

§1 Introduction

Let −d be the discriminant of an imaginary quadratic field with class number h(−d). As is well known, Gauss [9] conjectured that h(−d) tends to infinity with d. Hence, for fixed m, it was natural to ask for a complete list of negative fundamental disciminants −d such that h(−d) = m. This problem is usually referred to as Gauss’ class number problem or Gauss’ class number m problem.

In 1934, Heilbronn [14] succeeded in proving Gauss’ conjecture, thereby placing the class number problem on firm ground. The following year, Siegel [20] showed that for any ϵ > 0 there exists a constant c_{ϵ} > 0 such that h(−d) > c_{ϵ}d^{1/2−ϵ} as d → ∞. Unfortunately, neither result was effective, and no further progress was made until the 1950’s when Heegner [13] offered a solution for the class number 1 problem based on new ideas from the theory of modular functions. It is interesting to recall that Heegner’s proof was generally discounted until the “gaps” in his argument were explained many years later (see [5], [8], [21], [24]). In the interim period, however, the first accepted proof of the class number 1 problem was given by Stark [23] in 1967. Shortly thereafter, Baker [2, 4] found another proof based on the theory of transcendental numbers. In 1971, Baker [3] and Stark [26] independently resolved the class number 2 problem as well. However, there seemed to be little hope of generalizing these methods to solve higher class number problems.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11R; Secondary 11Y, 11J.

Key words and phrases. Binary quadratic forms, Class numbers, Discriminants, Imaginary quadratic fields.

In 1976, Goldfeld [10] presented a deep and entirely unexpected result which provided the framework for a general attack on the class number problem. He showed that if there exists a Weil curve whose associated L-function has a zero of at least the third order at s = 1, then for any ϵ > 0 there exists an effectively computable constant c_{ϵ} such that

(1.1)  h(−d) > c_{ϵ}(log d)^{1−ϵ}.

Of course, the utility of Goldfeld’s result depended on finding an appropriate elliptic curve. And though one would expect to find a Weil curve with a high order zero at s = 1 based on the celebrated conjecture of Birch and Swinnerton-Dyer [5], technical difficulties kept things on hold for several years. Finally, in 1983 Gross and Zagier [11] were able to show that certain curves must have a zero of at least the third order at s = 1, thereby completing the attack of Goldfeld.

Goldfeld’s proof was later simplified by Oesterlé [19], who provided, among other things, explicit constants for Goldfeld’s theorem. As a result, Oesterlé was able to complete the class number 3 problem as well. The class number 4 problem was solved by the first author [1] who combined new techniques with the well-known methods of Stark [22], Montgomery-Weinberger [18], and Oesterlé [19]. In hindsight, [1] contains a prototype for the partitioning of minima (of reduced quadratic forms of discriminant −d), which plays a crucial role in this paper.

The central concern of this paper is the class number m problem for small, odd m. The aforementioned partitioning of minima enables us to significantly improve on earlier estimates. Our results are summarized in the following theorem.

Theorem 1. For each odd integer m satisfying 5 ≤ m ≤ 23, the class number m problem is solved. For each such m, a complete list of negative fundamental discriminants −d for which h(−d) = m can be found in Appendix A.

Let −d be the discriminant of an imaginary quadratic field with class number h(−d). In Table 1 we present the number of fields satisfying h(−d) = m and the largest such d for each odd m satisfying 5 ≤ m ≤ 23.

Table 1. Upper Bound on d Satisfying h(−d) = m

\begin{matrix}m & # of d & Max. d \\ 5 & 25 & 2683 \\ 7 & 31 & 5923 \\ 9 & 34 & 10627 \\ 11 & 41 & 15667 \\ 13 & 37 & 20563 \\ 15 & 68 & 34483 \\ 17 & 45 & 37123 \\ 19 & 47 & 38707 \\ 21 & 85 & 61483 \\ 23 & 68 & 90787 \end{matrix}

The paper is organized as follows. In §2 we use Oesterlé’s [19] explicit constants to produce a d_{3}(m) such that if d ≥ d_{3}(m), then h(−d) ≠ m. In §3 we provide the theoretical justification for the partitioning of minima mentioned above. In §4 we prove several technical lemmas concerning the auxiliary function

F (x) = \sqrt{x} ∑_{n=1}^{∞} d(n)K_{0}(nx) (x > 0)

which are needed in §5.

We divide §5 into three subsections, each of which rules out a certain range of fundamental discriminants. In §5.1, following the approach of Montgomery-Weinberger [18], we produce a d_{2}(m) such that if d_{2}(m) ≤ d ≤ d_{3}(m), then h(−d) ≠ m. At the end of §5.1 we note that the methods of §5.1, even when pushed to their limits, do not reduce the range of admissible discriminants d to the point where a computationally intensive sieve (like the one introduced in §6) can be used to complete the class number m problem when m > 7. This shows that some new argument is necessary. In §5.2 we exploit the partitioning of minima introduced in §3 to produce a d_{1}(m) such that if d_{1}(m) ≤ d ≤ d_{2}(m), then h(−d) ≠ m. When m is odd and 5 ≤ m ≤ 13, d_{1}(m) is small enough to allow the class number m problem to be completed with the computationally intensive sieve of §6. In §5.3 we use a more sophisticated version of the arguments in §5.2 to produce a d_{0}(m) such that if d_{0}(m) ≤ d ≤ d_{1}(m), then h(−d) ≠ m. When m is odd and 15 ≤ m ≤ 23, d_{1}(m) is small enough to allow the class number m problem to be completed with the computationally intensive sieve of §6. We combine the partitioning of minima with the aforementioned sieve to complete the proof of Theorem 1 in §6.

Notation. Let ℤ, ℤ^{+}, ℚ, ℝ and ℂ denote the ring of integers, the set of positive integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. Let p and q denote pimes in ℤ^{+}. The Kronecker symbol is denoted by either

(\frac{m}{n})

or (m|n), depending on which is more convenient. As is customary, we let ω(n) denote the number of distinct prime divisors of n and d(n) the total number of positive divisors of n. Finally, let e(x) = e^{2πix}.

The number −d denotes a negative fundamental discriminant while χ_{1}(n) = (−d|n), a real primitive character modulo d.

Let

Q_{d} = {Q(x, y) = ax^{2} + bxy + cy^{2} | b^{2} − 4ac = −d, Q is reduced}

denote the set of reduced binary quadratic forms with discriminant −d. The notation ∑_{Q_{d}} denotes a sum over all Q ∈ Q_{d} along with the associated coefficients a, b, and c.

The modified Bessel function of the second kind is given by

(1.2)  K_{ν}(z) = \frac{1}{2} ∫_{1}^{∞} e^{−(z/2)(t+t^{−1})}(t^{ν−1} + t^{−ν−1}) dt  (ℜz > 0, ν ∈ ℂ).

§2 The High Range

As mentioned in §1, Oesterlé [19] provided explicit constants for the c_{ϵ} in Goldfeld’s inequality (1.1). Indeed, if −d is a negative fundamental discriminant with class number h(−d) and

ϑ(d) = ∏_{p|d}^{∗} (1 − \frac{⌊2\sqrt{p}⌋}{p + 1}) ,

where the product is taken over all prime divisors p of d with the exception of the largest prime divisor, then

ϑ(d) log d ≤ Ch(−d)

where C = 55 if (d, 5077) = 1 and C = 7000 otherwise.

In order to evaluate ϑ(d) when h(−d) is odd, recall that the Fundamental Theorem of Genera due to Gauss [9] states that

2^{ω(d)−1} | h(−d).

If h(−d) is odd, then the preceding result implies that d has only one prime divisor. Since −d is a discriminant, we see that −d is either −4, −8, or −p for some odd prime p = 3 (mod 4). Thus, since h(−4) = h(−8) = 1, we have

(2.1)  h(−d) is odd and h(−d) > 1 ⟹ d is prime and d = 3 (mod 4).

Also, when h(−d) is odd, (2.1) implies ϑ(d) = 1. Further, since (d, 5077) = 1, we have

log d ≤ 55h(−d).

Thus, for m ∈ {5, 7, …, 23}, we have h(−d) > m if d ≥ d_{3}(m) where d_{3}(m) is given in Table 2.

Table 2. h(−d) ≠ m for d ≥ d_{3}(m)

\begin{matrix}m & d_{3}(m) \\ 5 & 10^{120} \\ 7 & 10^{168} \\ 9 & 10^{215} \\ 11 & 10^{263} \\ 13 & 10^{311} \\ 15 & 10^{359} \\ 17 & 10^{407} \\ 19 & 10^{454} \\ 21 & 10^{502} \\ 23 & 10^{550} \end{matrix}

If, however, h(−d) is even, then d may be composite. This would allow the possibility that a minimum a of a reduced form could satisfy (a, d) > 1. This would introduce many technical difficulties, the least of which is a smaller value of ϑ(d). For these reasons, we confine our attention to the case where h(−d) is odd.

§3 Minima Results

The coefficient a is referred to as the minimum of the form ax^{2} + bxy + cy^{2} ∈ Q_{d}, while the multiset of minima for a given negative fundamental discriminant −d is denoted

M_{d} = {a | ax^{2} + bxy + cy^{2} ∈ Q_{d}}.

Note, 1 ∈ M_{d} exactly once and is referred to as the principal minimum, since 1 is the minimum of the principal form.

Lemma 1. If a_{1} ∈ M_{d}, a|a_{1} and a ≤ (d/4)^{1/2}, then a ∈ M_{d}.

Proof. Let a_{1}x^{2} + b_{1}xy + c_{1}y^{2} ∈ Q_{d} so that b_{1}^{2} − 4a_{1}c_{1} = −d. It follows at once that the quadratic congruence b^{2} = −d (mod 4a) is solvable. Hence, there exists a b such that b^{2} = −d (mod 4a), with −2a < b ≤ 2a. Further, this range can be sharpened to ensure that −a < b ≤ a by replacing b with b − 2a if a < b ≤ 2a and b with b + 2a if −2a < b ≤ −a. Letting c = (b^{2} + d)/(4a) and observing that c ≥ d/(4a) ≥ a, by hypothesis, shows that the quadratic form Q = ax^{2} + bxy + cy^{2} is reduced. □

Lemma 2. Let a ∈ ℤ^{+} be odd and satisfy a ≤ (d/4)^{1/2}. Then, a ∈ M_{d} if and only if every prime divisor p of a satisfies (−d|p) = 1.

Proof. First, assume that a ∈ M_{d}. Since b^{2} − 4ac = −d, we have p|a implies b^{2} = −d (mod p). Thus, (−d|p) = 1. Conversely, assume that (−d|p) = 1 for each prime divisor p of a. Then the congruence b^{2} = −d (mod p) is solvable, and by Hensel’s lemma, the congruence b^{2} = −d (mod p^{α}) is solvable for all α ∈ ℤ^{+}. Also, since −d = 0 or 1 (mod 4), the congruence b^{2} = −d (mod 4) is solvable. Hence, by the Chinese Remainder Theorem, there exists a b such that the congruence b^{2} = −d (mod 4a) is solvable. The proof can now be completed in the same manner as the proof of Lemma 1. □

It is important to note that Hensel’s lemma precludes the possibility of lifting over the prime 2 as was done for the odd primes. For example, if d ≥ 256 and 4 ∈ M_{d}, then this does not necessarily imply that 8 ∈ M_{d}. However, if d ≥ 256 and 8 ∈ M_{d}, then Lemma 1 does imply 4 ∈ M_{d}.

Lemma 3. Suppose h(−d) > 1 and h(−d) is odd. If a > 1 and ax^{2} + bxy + cy^{2} ∈ Q_{d}, then ax^{2} − bxy + cy^{2} is a distinct member of Q_{d}.

Proof. If ax^{2} + bxy + cy^{2} is a reduced form, then ax^{2} − bxy + cy^{2} will be a distinct reduced form unless (i) b = 0, (ii) b = a, or (iii) a = c. If (i) is true, then 4ac = d, which contradicts (2.1). If (ii) is true, then a^{2} − 4ac = a(a − 4c) = −d. Thus, by (2.1) and the hypothesis that a > 1, we have a = d, which is impossible since a < (d/3)^{1/2}. If (iii) is true, then b^{2} −4a^{2} = (b−2a)(b+2a) = −d. Since b > 0 in case (iii), we have by (2.1) that b−2a = −1, i.e., that a = 1, contradicting our hypothesis. □

Lemma 4. Suppose h(−d) > 1 and h(−d) is odd. Let a ∈ ℤ^{+} be odd and satisfy a ≤ (d/4)^{1/2}. If a ∈ M_{d}, then a appears in M_{d} exactly 2^{ω(a)} times.

Proof. Using the Chinese Remainder Theorem and Lemma 3, the proof is completed in the same manner as the proof of Lemma 2. □

Lemma 5. If a > 1, a ∈ M_{d} and (a, d) = 1, then a > (d/4)^{1/h(−d)}.

Proof. Suppose p|a and p is an odd prime. Then b^{2} = −d (mod p) implies (−d|p) = 1. Also, if 2|a, then b^{2} = −d (mod 8). Hence, −d = 1 (mod 8), so that (−d|2) = 1. Thus, if p is any prime dividing a we have (−d|p) = 1. This implies that p splits in ℚ (\sqrt{−d}), ⟨p⟩ = ℘_{1}℘_{2} with ℘_{1} ≠ ℘_{2}. Thus, ℘_{1}^{h(−d)} = ⟨β⟩ is a principal ideal, and β ∉ ℤ. It follows that p^{h(−d)} = N(℘_{1}^{h(−d)}) = N(⟨β⟩) ≥ d/4. Since a ≥ p, it follows that a ≥ (d/4)^{1/h(−d)}. Note that a > 1 implies h(−d) > 1, so that (d/4)^{1/h(−d)} ∉ ℤ. The lemma follows. □

It follows from (2.1) and Lemma 5 that

(3.1)  (\frac{d}{4})^{1/h(−d)} < a ≤ (\frac{d}{3})^{1/2}  (a ∈ M_{d}\\{1\}, h(−d) > 1 is odd).

To improve on these bounds, we separate the minima to a certain extent using the multiplicative structure of M_{d} developed in Lemmas 1–4.

Assume that h(−d) > 1 and h(−d) is odd. From Lemma 3 we know that every a ∈ M_{d}\{1} appears an even number of times in M_{d}. Also, by (3.1) if d > 2^{h(−d)+2}, then 2 is not a minimum. It then follows by Lemma 1 that a must be odd if a ∈ M_{d}. In the remainder of this section assume that d > 2^{h(−d)+2}. Define the multiset M_{d}^{∗} by

(3.2)  M_{d}^{∗} = {a | ax^{2} + bxy + cy^{2} ∈ Q_{d}, a > 1, b > 0}.

From Lemma 3 we have |M_{d}^{∗}| = (h(−d) − 1)/2.

Definition 1. A partition for d is a list of (h(−d) − 1)/2 pairs of functions (l_{a}(d), u_{a}(d)), each increasing in d, corresponding to the (h(−d) − 1)/2 elements in M_{d}^{∗}.

Definition 2. We say a partition for d covers M_{d} if

l_{a}(d) ≤ a ≤ u_{a}(d)

for every a ∈ M_{d}^{∗}.

We begin with a simple example. Suppose that h(−d) = 5. Let p be the smallest nonprincipal member of M_{d}. It is easy to see from Lemma 2 that if p is composite then d < 16/3. This contradicts the fact that h(−d) > 1, so p must be prime. In past investigations of minima one generally used the result contained in Lemma 5, which states that all minima a, except a = 1, must satisfy a > (d/4)^{1/5}. In particular, the minima could simultaneously be small, each minima lying close to the bound (d/4)^{1/5}. However, using Lemma 2 and Lemma 4, we see that if p is about (d/4)^{1/5} in size, then there are exactly two reduced forms with minimum p and two reduced forms with minimum p^{2}. This excludes the possibility that the minima are simultaneously small. Using similar reasoning it is not hard to see that if h(−d) = 5, then M_{d} is covered by one of the following three partitions.

\begin{matrix}1 & (d/4)^{1/5} ≤ p ≤ (d/4)^{1/4} & (d/4)^{2/5} ≤ p^{2} ≤ (d/4)^{1/2} \\ 2 & (d/4)^{1/4} ≤ p ≤ (d/4)^{1/3} & (d/4)^{1/4} ≤ a ≤ (d/3)^{1/2} \\ 3 & (d/4)^{1/3} ≤ a ≤ (d/3)^{1/2} & (d/4)^{1/3} ≤ a ≤ (d/3)^{1/2} \end{matrix}

For each fixed value of m = 5, 7, …, 23, a set of partitions covering all possible M_{d} is given in Appendix B. In these tables, p and q denote the first and second smallest prime minima in M_{d}, respectively, while a denotes a generic member of M_{d}, v = d/4 and w = d/3. In order to simplify our presentation of a covering partition, we now introduce some additional notation. The notation (3p.), for example, denotes that the inequalities for the first three powers of the prime are trivially inferred. Similarly, the notation (3), for example, means the inequality is to be listed three times. In this notation, the covering partitions for M_{d} when h(−d) = 5 are given by

1 v^{1/5} ≤ p ≤ v^{1/4} (2p.)
2 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2}
3 v^{1/3} ≤ a ≤ w^{1/2} (2).

In order to give a better idea of how the partitions in Appendix B were generated, we will go through the details of partition number 10 for class number 23 (Table B10). The partition is given by

10 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{3/16} ≤ q ≤v^{1/4} (2p.) v^{23/80} ≤ pq ≤ v^{3/8}(2)
v^{31/80} ≤ p^{2}q ≤ v^{1/2} (2) v^{3/8} ≤ a ≤ w^{1/2}.

Assume the smallest nonprincipal minimum p satisfies v^{1/10} ≤ p ≤ v^{1/8}, then p^{2}, p^{3}, p^{4} are all ≤ (d/4)^{1/2}, implying that p, p^{2}, p^{3}, p^{4} ∈ M_{d}, which accounts for 8 minima. If the second smallest minima q satisfies q ≤ (d/4)^{3/16}, then q, q^{2}, pq, pq^{2}, p^{2}q are ≤ (d/4)^{1/2}, implying that they are in M_{d}. But, this accounts for 16 new minima, which would make 25 total. Thus, v^{3/16} ≤ q. Assume q ≤ (d/4)^{1/4}. Now, q, q^{2}, pq, p^{2}q ≤ (d/4)^{1/2}, so we have accounted for 21 minima. If any further minima a satisfy a ≤ (d/4)^{3/8}, then pa ≤ (d/4)^{1/2} would imply that there are greater than 23 minima. Therefore, the remaining two minima must satisfy (d/4)^{3/8} ≤ a ≤ (d/3)^{1/2}. All of the partitions in the tables of Appendix B are computed in this same fashion. By using a disjoint set of assumptions on the first and second largest prime minima p and q, these sets of partitions can be seen to cover all possible M_{d}.

§4 Properties of F

In this section we prove several technical lemmas concerning the function F defined by

(4.1)  F (x) = \sqrt{x} ∑_{n=1}^{∞} d(n)K_{0}(nx)  (x > 0).

These lemmas are needed in §5.2 when dealing with discriminants in the “middle range.” It is well-known [27, p. 202] that if ν is fixed and x is real, then

(4.2)  K_{ν}(x) ∼ (\frac{π}{2x})^{1/2} e^{−x}  (x → ∞).

Letting ν = 0 in (4.2), it follows that the series defining F (x) converges uniformly on compact subsets of (0, ∞). Hence,

(4.3)  F′(x) = \frac{1}{2\sqrt{x}} ∑_{n=1}^{∞} d(n)(2nxK′_{0}(nx) + K_{0}(nx))  (x > 0).

It follows from the well-known identity

(4.4)  K_{1}(x) = −K′_{0}(x)

and (4.2) with ν = 0, 1, that the series in (4.3) also converges uniformly on compact subsets of (0, ∞). Hence, term-by-term differentiation of (4.3) combined with the differential equation

(4.5)  vK′′_{0}(v) + K′_{0}(v) = vK_{0}(v),

with argument v = nx, imply that

(4.6)  F′′(x) = \frac{1}{x^{3/2}} ∑_{n=1}^{∞} d(n)K_{0}(nx) ((nx)^{2} − 1/4)  (x > 0).

On several occasions in §5.2 and §5.3 we will need to compute accurate approximations to F at certain small arguments. For such purposes, the following crude generalization of [18, Lemma 8]) suffices.

Lemma 6. If x > 0, N is a nonnegative integer, and

Δ_{N}(x) = F (x) − \sqrt{x} ∑_{n=1}^{N} d(n)K_{0}(nx),

then

|Δ_{N}(x)| ≤ \frac{2}{\sqrt{x}}(1 + log(N + 1 + 2/x))e^{−(N+1)x/2}

where the empty sum is understood to equal zero when N = 0. Proof. Since x > 0, we see from (4.1) and (1.2) that

0 ≤ x^{−1/2}Δ_{N}(x) ≤ ∑_{n=N+1}^{∞} d(n) ∫_{1}^{∞} e^{−nxt/2} \frac{dt}{t} = ∑_{n=N+1}^{∞} d(n) ∫_{nx/2}^{∞} e^{−u} \frac{du}{u}.

Using partial summation, we have

x^{−1/2}Δ_{N}(x) ≤ ∫_{(N+1)x/2}^{∞} e^{−u}( ∑_{n≤2u/x} d(n))\frac{du}{u}.

For y ≥ 1 we have

∑_{n≤y} d(n) = ∑_{n≤y} ⌊\frac{y}{n}⌋ ≤ y ∑_{n≤y} \frac{1}{n} ≤ y(1 + log y),

so that

|x^{−1/2}Δ_{N}(x)| ≤ ∫_{(N+1)x/2}^{∞} e^{−u} \frac{2u}{x}(1 + log(2u/x))\frac{du}{u}.

An integration by parts gives

|x^{−1/2}Δ_{N}(x)| ≤ \frac{2}{x} ((1 + log(N + 1))e^{−(N+1)x/2} + ∫_{(N+1)x/2}^{∞} e^{−u} \frac{du}{u}) .

Note that the function g, defined by

g(v) = ∫_{v}^{∞} e^{−u} \frac{du}{u} − e^{−v} log (1 + 1/v) ,

is increasing for v > 0, and lim_{v→∞} g(v) = 0. Hence,

∫_{(N+1)x/2}^{∞} e^{−u} \frac{du}{u} ≤ e^{−(N+1)x/2} log (1 + \frac{2}{(N + 1)x}) ,

and the lemma follows. □

On several occasions in Lemma 8, we need to compute crude approximations to F′ at certain small arguments. For these purposes, the following lemma suffices.

Lemma 7. If x > 0, N ∈ ℤ^{+}, and

δ_{N}(x) = F′(x) − \frac{1}{2\sqrt{x}} ∑_{n=1}^{N} d(n)(K_{0}(nx) − 2nxK_{1}(nx)),

then

|δ_{N}(x)| ≤ \frac{2}{\sqrt{x}} \{\frac{2}{x}(1 + log(N + 1 + 2/x)) + \frac{N + 1 − Ne^{−x/2}}{(1 − e^{−x/2})^{2}} \} e^{−(N+1)x/2}.

Proof. From (4.3) and (4.4) we have

δ_{N}(x) = \frac{1}{2\sqrt{x}} ∑_{n=N+1}^{∞} d(n)(K_{0}(nx) − 2nxK_{1}(nx))

= \frac{1}{2x}Δ_{N}(x) − \sqrt{x} ∑_{n=N+1}^{∞} nd(n)K_{1}(nx),

where Δ_{N}(x) is as in Lemma 6. From (1.2), we have

|δ_{N}(x)| ≤ \frac{Δ_{N}(x)}{2x} + \sqrt{x} ∑_{n=N+1}^{∞} nd(n) ∫_{n=1}^{∞} e^{−nxt/2}dt

≤ \frac{Δ_{N}(x)}{2x} + \frac{2}{\sqrt{x}} ∑_{n=N+1}^{∞} ne^{−nx/2} (since d(n) ≤ n)

= \frac{Δ_{N}(x)}{2x} + \frac{2}{\sqrt{x}} (\frac{N}{1 − e^{−x/2}} + \frac{1}{(1 − e^{−x/2})^{2}} ) e^{−(N+1)x/2}.

Lemma 7 then follows from an application of Lemma 6. □

The next lemma will play an important role in §5.2 and §5.3. Lemma 8. F (x) is a strictly decreasing function of x for x > 0. Proof. We shall prove the lemma by showing that F′(x) < 0 for x > 0. It is convenient to break the proof into three cases. For the first case we assume that x > \frac{1}{2}. From (1.2) we have that

2xK′_{0}(x) + K_{0}(x) = −\frac{1}{2} ∫_{1}^{∞} e^{−(x/2)(t+1/t)} (2x(1 + 1/t^{2}) − 2/t) dt.

Since x > \frac{1}{2}, we see that

2xK′_{0}(x) + K_{0}(x) < −\frac{1}{2} ∫_{1}^{∞} e^{−(x/2)(t+1/t)}(1 − 1/t)^{2} dt < 0.

In particular, we have

2nxK′_{0}(nx) + K_{0}(nx) < 0 (n ∈ ℤ^{+}, x > 1/2).

Hence, each term in (4.3) is negative and F′(x) < 0. Thus, Lemma 8 is proved in the case x > \frac{1}{2}.

Next, assume that 1/10 ≤ x ≤ 1/2. Let

G_{k}(x) = \frac{1}{2\sqrt{x}} ∑_{n=1}^{k} d(n)(2nxK′_{0}(nx) + K_{0}(nx)),

the sum of the first k terms in the expression for F′(x) given by (4.3). Note that δ_{k}(x) = F′(x) − G_{k}(x). By considering the first k terms in (4.6) we see that G_{k} is decreasing for x < \frac{1}{2k}. By considering all but the first k terms of (4.6), we have that δ_{k} is increasing for x > \frac{1}{2k+2}. It follows that

(4.7)  G_{k}(x) + δ_{k}(x) ≤ G_{k} (\frac{1}{2k + 2}) + δ_{k} (\frac{1}{2k} )  (\frac{1}{2k + 2} < x < \frac{1}{2k} ) .

Using Lemma 7, values of G_{k} and δ_{k} are listed in Table 3 which are accurate to the number of digits shown. These values combined with (4.7) and the continuity of F′(x) imply that F′(x) < 0 for 1/10 ≤ x ≤ 1/2.

Table 3. F′(x) < 0 for 1/10 ≤ x ≤ 1/2

\begin{matrix}k & G_{k}(\frac{1}{2k}) & δ_{k}(\frac{1}{2k}) \\ 1 & -0.51 & -4.76 \\ 2 & -1.79 & -17.95 \\ 3 & -3.09 & -38.23 \\ 4 & -5.38 & -63.68 \\ 5 & -6.55 & -95.81 \end{matrix}

Finally, assume that 0 < x < 1/10. By partial summation we have

F′(x) = −\frac{1}{2\sqrt{x}} ∫_{x^{−}}^{∞}( ∑_{n≤v/x} d(n))(2vK′′_{0}(v) + 3K′_{0}(v)) dv

because the boundary terms vanish due to (4.4) and (4.2) with v = 0, 1. Using (4.5), (4.4), and the fact that ∑_{n≤v/x} d(n) vanishes for 0 < v < x, we have

(4.8)  F′(x) = \frac{1}{2\sqrt{x}} ∫_{0^{+}}^{∞}( ∑_{n≤v/x} d(n))(K_{1}(v) − 2vK_{0}(v)) dv.

It is well-known [12, p.347] that for y ≥ 1,

∑_{k≤y} \frac{1}{k} = log y + γ + ∫_{y}^{∞} \frac{t − ⌊t⌋}{t^{2}} dt − \frac{y − ⌊y⌋}{y},

where γ is Euler’s constant. Thus for y ≥ 1, we have

|∑_{k≤y} \frac{1}{k} − log y − γ| < \frac{1}{y} .

Using this estimate with y = \sqrt{t} in

∑_{n≤t} d(n) = 2t ∑_{k≤\sqrt{t}} 1/k − 2 ∑_{k≤\sqrt{t}}(t/k − ⌊t/k⌋) − t + ((\sqrt{t})^{2} − ⌊\sqrt{t}⌋^{2})

(an identity valid for t ≥ 1 arising from Dirichlet’s elementary approach to the Dirichlet divisor problem [12, Theorem 320]), it is easy to see that for t ≥ 1,

(4.9)  |∑_{n≤t} d(n) − t log t − (2γ − 1)t| < 4\sqrt{t}.

It is also not difficult to see that (4.9) also holds for 0 < t < 1.

Using the inequalities of (4.9) in (4.8), we have, since K_{0}(x) > 0 and K_{1}(x) > 0 when x > 0,

(4.10) \begin{aligned}F′(x) < \frac{1}{2\sqrt{x}} & ∫_{0}^{∞} ((v/x) log(v/x) + (2γ − 1)v/x + 4(v/x)^{1/2}) K_{1}(v) dv− \\ & \frac{1}{\sqrt{x}} ∫_{0}^{∞} ((v^{2}/x) log(v/x) + (2γ − 1)v^{2}/x − 4(v^{3}/x)^{1/2}) K_{0}(v) dv. \end{aligned}

It is well-known [27, p.388] that if μ and ν are positive real numbers statisfying μ − ν > −1, then

(4.11)  ∫_{0}^{∞} v^{μ}K_{ν}(v) dv = 2^{μ−1}Γ (\frac{μ + ν + 1}{2}) Γ (\frac{μ − ν + 1}{2}) ,

where Γ is the Gamma function. Furthermore, from [16, Theorem 250], it is not difficult to see that differentiation under the integral sign in (4.11) with respect to μ is legitimate. Hence, for positive real μ and ν satisfying μ − ν > −1, we have

(4.12)

∫_{0}^{∞} v^{μ} log vK_{ν}(v)dv =2^{μ−1}Γ (\frac{μ + ν + 1}{2}) Γ (\frac{μ − ν + 1}{2}) ×

(log 2 + \frac{1}{2}ψ (\frac{μ + ν + 1}{2}) + \frac{1}{2}ψ (\frac{μ − ν + 1}{2})) ,

where ψ = Γ′/Γ is the Digamma function.

Using (4.11) and (4.12) to evaluate the integrals in (4.10) yields

F′(x) < \frac{1}{2\sqrt{x}} (\frac{π log x}{2x} + \frac{π log(2/e^{2+γ})}{2x} + Γ^{2}(1/4)\sqrt{2x}) .

Since we are in the case for which 0 < x < 1/10, it follows that F′(x) < 0. Lemma 8 is proved. □

It is interesting to note that the asymptotic behavior of F (x) as x → 0^{+} can be determined using arguments similar to those introduced in the latter part of the proof of Lemma 8. Indeed, it is not difficult to show that

|F (x) − (\frac{π log(1/x)}{2\sqrt{x}} − \frac{π log(2/e^{γ})}{2\sqrt{x}})| ≤ \frac{Γ^{2}(1/4)}{\sqrt{2}} (x > 0).

§5 The Medium Range

Let χ be a real primitive character modulo k for some integer k > 1. In the case where −k is a negative fundamental discriminant we take χ(n) = (−k|n) (see [7, p. 40]). Define

A_{d}(s) = ∑_{Q_{d}} χ(a)a^{−s}

and

P_{k}(s) = ∏_{p|k}(1 − p^{−s}).

In 1965 Stark [22] exploited a formula for the zeta function of a quadratic number field (i.e., ζ(s)L(s, χ_{1})) to show that if a tenth fundamental discriminant −d of class number 1 existed, then d > exp(2.2 10^{7}). Later Stark [25] developed a formula for L(s, χ)L(s, χχ_{1}) analogous to the formula for ζ(s)L(s, χ_{1}). Montgomery and Weinberger [18] exploited this formula to obtain similar results for class numbers 2 and 3. Indeed, if (k, d) = 1, then

(5.1)  (\frac{k\sqrt{d}}{2π})^{s−1/2} Γ(s)L(s, χ)L(s, χχ_{1}) = T_{d}(s) + T_{d}(1 − s) + U_{d}(s)

where

T_{d}(s) = (\frac{k\sqrt{d}}{2π})^{s−1/2} Γ(s)ζ(2s)P_{k}(2s)A_{d}(s),

(5.2)  U_{d}(s) = \frac{4\sqrt{π}}{k} ∑_{Q_{d}} a^{−1/2} ∑_{n=1}^{∞} K_{s−1/2}(\frac{πn\sqrt{d}}{ak})n^{s−1/2}V_{Q}(s, n),

and

V_{Q}(s, n) = ∑_{y|n} y^{1−2s} ℜ\{∑_{j=1}^{k} χ(Q(j, y))e (\frac{jn}{ky}) e (\frac{bn}{2ak} )\}.

Let s_{0} = 1/2 + it_{0}, with t_{0} > 0, be a zero of L(s, χ). Substituting s = s_{0} into (5.1) gives

T_{d}(s_{0}) + T_{d}(\overline{s_{0}}) = −U_{d}(s_{0}).

Applying the Schwartz Reflection Principle to T_{d} implies

2 |T_{d}(s_{0})| cos(arg T_{d}(s_{0})) = −U_{d}(s_{0})

which, in turn, gives

(5.3)  |sin arg(iT_{d}(s_{0}))| = |\frac{U_{d}(s_{0})}{2T_{d}(s_{0})}| .

The method for the middle range consists in showing that this equality is false for large intervals of d under the assumption that h(−d) is some fixed odd integer. Indeed, for fixed k and t_{0}, define the constants

ξ_{1} = t_{0}/2, (5.4)

ξ_{2} = t_{0} log (\frac{k}{2π} ) + arg\{iΓ(1/2 + it_{0})ζ(1 + 2it_{0})P_{k}(1 + 2it_{0})\}, (5.5)

ξ_{3} = 2|Γ(1/2 + it_{0})ζ(1 + 2it_{0})P_{k}(1 + 2it_{0})|. (5.6)

To show (5.3) is false, all we need to show is that

(5.7)  |sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| > \frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|}.

§5.1 The Range d_{2}(m) ≤ d ≤ d_{3}(m)

Lemma 9. Let t ∈ ℝ^{+}. Suppose m = h(−d) is odd and fixed. Then,

(5.8)  |A_{d}(1/2 + it)| ≥ 1 − \frac{m − 1}{(d/4)^{1/(2m)}} .

Furthermore, if d > max{4e^{2m}, 4(m − 1)^{2m}}, then

(5.9)  |arg A_{d}(1/2 + it)| ≤ \frac{t(1 − 1/m) log(d/4)}{(d/4)^{1/(2m)} − (m − 1)}.

Proof. Both (5.8) and (5.9) are trivially true if m = 1, so assume m > 1. Using the lower bound in (3.1) gives

|A_{d}(1/2 + it) − 1| = | ∑_{Q_{d},a≠1} χ(a)a^{−1/2−it}|

≤ ∑_{Q_{d},a≠1} a^{−1/2} (5.10)

≤ \frac{m − 1}{(d/4)^{1/(2m)}} . (5.11)

Now, (5.8) follows from (5.11) and the triangle inequality.

For the remainder of the proof assume the d > max{4e^{2m}, 4(m − 1)^{2m}}. Since d > 4(m − 1)^{2m}, we know from (5.11) that

|A_{d}(1/2 + it) − 1| < 1,

so that A_{d}(1/2 + it) lies in the right half plane. Hence, when we write arg A_{d}(1/2 + it) in (5.9) and below, we can, without loss of generality, assume we are dealing with the principal value of the argument. Let L be the line segment joining 1/2 to 1/2 + it. Then an equation for L is given by ℓ(u) = 1/2 + iu, 0 ≤ u ≤ t. Since d > 4(m − 1)^{2m}, we know by (5.8) that A_{d} doesn’t vanish on L. Furthermore, A_{d}(s) is an entire function of s. It follows [17, p. 218] that

∫_{L} \frac{A′_{d}(z)}{A_{d}(z)}dz = ∫_{0}^{t} \frac{A′_{d}(1/2 + iu)}{A_{d}(1/2 + iu)}i du = log A_{d}(1/2 + iu) |_{0}^{t} .

Evaluating the right-hand side and taking imaginary parts yields

| arg A_{d}(1/2 + it)| = |ℑ ∫_{0}^{t} \frac{A′_{d}(1/2 + iu)}{A_{d}(1/2 + iu)}i du|

≤ t max_{0≤u≤t} |\frac{A′_{d}(1/2 + iu)}{A_{d}(1/2 + iu)}| (5.12)

Note that

|A′_{d}(1/2 + it)| = | ∑_{Q_{d},a≠1} χ(a)a^{−1/2−it} log a|

≤ ∑_{Q_{d},a≠1} a^{−1/2} log a. (5.13)

Now, x^{−1/2} log x is a decreasing function of x for x > e^{2}. Using (3.1) and the hypothesis d > 4e^{2m}, it then follows from (5.13) that

(5.14)  |A′_{d}(1/2 + it)| ≤ \frac{(1 − 1/m) log(d/4)}{(d/4)^{1/(2m)}}.

Using (5.14) and (5.8) in (5.12) yields (5.9), and the lemma is proved. □

Lemma 10. Let t ∈ ℝ. If k > 1 is an odd square-free integer, then

(5.15)  |U_{d}(1/2 + it)| ≤ \frac{4}{d^{1/4}} ∑_{r|k} \frac{3^{ω(r)}2^{ω(k)−ω(r)}}{r^{1/2}} ∑_{a∈M_{d}} F (\frac{π\sqrt{d}r^{2}}{ak} ).

Proof. From the definition of U_{d} in (5.2) we deduce at once that

(5.16)  |U_{d}(1/2 + it)| ≤ \frac{4\sqrt{π}}{k} ∑_{Q_{d}} a^{−1/2} ∑_{n=1}^{∞} K_{0}(\frac{π\sqrt{d}n}{ak})|V_{Q}(1/2 + it, n)|.

Using an argument of Weil [28, his inequality (5)], Montgomery and Weinberger [18, Lemma 7] have shown that

|V_{Q}(1/2 + it, n)| ≤ 2^{ω(k)}k^{1/2} ∑_{y|n} ∏_{p|(y,n/y,k)} \frac{p^{1/2}}{2}

= 2^{ω(k)}k^{1/2} ∑_{r|k} 2^{−ω(r)}r^{1/2} ∑_{{y|n \atop (y,n/y,k)=r}} 1,

because k is square-free. Since (y, n/y, k) = r implies r^{2}|n, we have

|V_{Q}(1/2 + it, n)| ≤ 2^{ω(k)}k^{1/2} ∑_{{r|k \atop r^{2}|n}} 2^{−ω(r)}r^{1/2}d(n).

Inserting this into (5.16) gives

(5.17)  |U_{d}(1/2 + it)| ≤ \frac{4\sqrt{π}}{k^{1/2}} ∑_{a∈M_{d}} a^{−1/2} ∑_{r|k} 2^{ω(k)−ω(r)}r^{1/2} ∑_{{n=1 \atop r^{2}|n}}^{∞} d(n)K_{0}(\frac{π\sqrt{d}n}{ak}).

Let n = r^{2}m and note that d(n) ≤ d(r^{2})d(m) = 3^{ω(r)}d(m) since r is square-free. From (5.17) we have

|U_{d}(1/2 + it)| ≤ \frac{4\sqrt{π}}{k^{1/2}} ∑_{a∈M_{d}} a^{−1/2} ∑_{r|k} 3^{ω(r)}2^{ω(k)−ω(r)}r^{1/2} ∑_{m=1}^{∞} d(m)K_{0}(\frac{π\sqrt{d}r^{2}m}{ak}).

Applying definition (4.1) to the inner sum gives the result. □

Corollary. Let t ∈ ℝ and m = h(−d). If k > 1 is an odd square-free integer, then

|U_{d}(1/2 + it)| ≤ \frac{8k^{1/2} log k}{3^{1/4}π^{1/2}d^{1/4}} (m − 1 + e^{−π(d^{1/2}−\sqrt{3})/(2k)}) ∏_{p|k}(2 + 3p^{−3/2}).

Proof. Replace F in Lemma 10 with the upper bound given in Lemma 6 with N = 0 to give

|U_{d}(1/2 + it)| ≤ \frac{8k^{1/2}}{π^{1/2}d^{1/2}} ∑_{r|k} \frac{3^{ω(r)}2^{ω(k)−ω(r)}}{r^{3/2}} ×

∑_{a∈M_{d}} a^{1/2} (1 + log (1 + \frac{2ak}{πd^{1/2}r^{2}} )) e^{−πd^{1/2}r^{2}/(2ak)}.

In the inner sum, the a = 1 term is treated seperately. When a > 1 we use the inequality a ≤ (d/3)^{1/2}. In both cases, we also use the inequalities r ≥ 1, d ≥ 3, and

(1 + log (1 + \frac{2k}{π\sqrt{3}})) e^{−π\sqrt{3}/(2k)} ≤ log k (k ≥ 2)

to finish the proof of the corollary. □

For the remainder of §5.1, we assume there exists a discriminant −d satisfying the following properties:

(I) h(−d) = m, where m ∈ {5, 7, 9, …, 23} is fixed; (II) d_{2}(m) ≤ d ≤ 10^{850}, where d_{2}(m) is given in Table 4.

Our goal is to show that (5.7) is true for d with a suitable choice of k and s_{0}. For this purpose, we need a small zero, s_{0} = 1/2 + it_{0}, of L(s, χ). Weinberger [29] has computed several such zeroes, each corresponding to a different value of k. In this section, we use

k = 115147 and t_{0} = 0.003157614

where the absolute error in t_{0} is less than 10^{−8}, but we only make use of the first 4 significant places. From (5.4)-(5.6) we then have

ξ_{1} = 0.001579, ξ_{2} = 0.02875, ξ_{3} = 555.8,

where these approximations are accurate to the number of places shown.

Since m ≤ 23 and d ≥ d_{2}(m) ≥ 10^{63} by Table 4, the right-hand side of (5.8) is positive. Hence, letting t = t_{0} in Lemma 9, we see from (5.8) that |A_{d}(s_{0})| doesn’t vanish. Thus, using (5.8) and the corollary to Lemma 10 with t = t_{0}, we have

\frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ R_{2}(d),

where

R_{2}(d) = \frac{8k^{1/2} log k (m − 1 + e^{−π(d^{1/2}−\sqrt{3})/(2k)}) ∏_{p|k}(2 + 3p^{−3/2})}{ξ_{3}3^{1/4}π^{1/2}d^{1/4}(1 − (m − 1)/(d/4)^{1/(2m)})}.

Clearly R_{2}(d) is decreasing in d so that

(5.18)  \frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ R_{2}(d_{2}(m))

since d ≥ d_{2}(m). Upper bounds for R_{2}(d_{2}(m)) are given in Table 4.

Since 5 ≤ m ≤ 23 we have max{4e^{2m}, 4(m − 1)^{2m}} = 4(m − 1)^{2m} < 10^{63} < d_{2}(m). Hence, all of the hypotheses of Lemma 9 hold for t = t_{0} and m ∈ {5, 7, …, 23}. We deduce from (5.9) that

| arg A_{d}(s_{0})| ≤ α_{2}(d),

where

α_{2}(d) = \frac{t(1 − 1/m) log(d/4)}{(d/4)^{1/(2m)} − (m − 1)}.

It is easy to see that α_{2}(d) is decreasing for d ≥ 4e, so we certainly have

(5.19)  | arg A_{d}(s_{0})| ≤ α_{2}(d_{2}(m))

since d ≥ d_{2}(m). Upper bounds for α_{2}(d_{2}(m)) are given in Table 4.

Let β_{2}(m) be defined by

β_{2}(m) = ξ_{1} log d_{2}(m) + ξ_{2} − α_{2}(d_{2}(m))

and γ_{2}(m) be defined by

γ_{2}(m) = ξ_{1} log 10^{850} + ξ_{2} + α_{2}(d_{2}(m)).

Lower bounds for β_{2}(m) and upper bounds for γ_{2}(m) are given in Table 4. Using Table 4, the fact that d_{2}(m) ≤ d, and (5.19), we have

0 < β_{2}(m) ≤ ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}).

Using the fact that d ≤ 10^{850}, (5.19), and Table 4, we have

ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}) ≤ γ_{2}(m) < π.

Hence,

(5.20)  |sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| ≥ min{| sin β_{2}(m)|, | sin γ_{2}(m)|}.

Lower bounds for | sin β_{2}(m)| and | sin γ_{2}(m)| are given in Table 4.

From (5.20) and Table 4 we deduce that

|sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| > R_{2}(d_{2}(m)).

In light of (5.18), (5.7) follows immediately. Since (5.3) is false, we conclude that h(−d) ≠ m for m ∈ {5, 7, …, 23} and d_{2}(m) ≤ d ≤ 10^{850}. Note that from Table 2 in §2, we have certainly covered the range d_{2}(m) ≤ d ≤ d_{3}(m).

Table 4. h(−d) ≠ m for d_{2}(m) ≤ d ≤ 10^{850}

\begin{matrix}m & d_{2}(m) & R_{2}(d_{2}(m)) & α_{2}(d_{2}(m)) & β_{2}(m) & | sin β_{2}(m)| & γ_{2}(m) & | sin γ_{2}(m)| \\ 5 & 10^{65} & 2.2 10^{−14} & 1.4 10^{−7} & 0.264 & 0.26 & 3.121 & 0.020 \\ 7 & 10^{65} & 3.3 10^{−14} & 1.1 10^{−5} & 0.264 & 0.26 & 3.121 & 0.020 \\ 9 & 10^{66} & 2.5 10^{−14} & 9.9 10^{−5} & 0.268 & 0.26 & 3.121 & 0.020 \\ 11 & 10^{68} & 9.9 10^{−15} & 3.9 10^{−4} & 0.275 & 0.27 & 3.121 & 0.020 \\ 13 & 10^{73} & 6.8 10^{−16} & 8.2 10^{−4} & 0.293 & 0.28 & 3.121 & 0.020 \\ 15 & 10^{76} & 1.5 10^{−16} & 1.7 10^{−3} & 0.303 & 0.29 & 3.122 & 0.019 \\ 17 & 10^{79} & 3.1 10^{−17} & 2.9 10^{−3} & 0.312 & 0.30 & 3.124 & 0.017 \\ 19 & 10^{79} & 3.8 10^{−17} & 5.6 10^{−3} & 0.310 & 0.30 & 3.126 & 0.015 \\ 21 & 10^{79} & 4.8 10^{−17} & 1.1 10^{−2} & 0.304 & 0.29 & 3.132 & 0.009 \\ 23 & 10^{79} & 6.8 10^{−17} & 2.0 10^{−2} & 0.295 & 0.29 & 3.141 & 5.9 10^{−4} \end{matrix}

The d_{2}(m) listed in Table 4 are far too large to allow the completion of the class number m problem using the computationally intensive sieve in §6 to investigate the range d < d_{2}(m). Pushing the proceeding arguments to their limit, it is possible to produce a d_{2}^{∗}(m) (< d_{2}(m)) such that if d_{2}^{∗}(m) < d < d_{3}(m), then h(−d) ≠ m. Approximate values of d_{2}^{∗}(m) are given in Table 5. Note, however, that for odd m > 9, d_{2}^{∗}(m) is also far too large to allow one to complete the class number problem by using a computationally intensive sieve. Some type of further argument is necessary. In §5.2 and §5.3, the idea of partitioning minima is introduced, which allows us to reduce the range of admissible discriminants so that a computationally intensive sieve can be used. It turns out that our new arguments are powerful enough to rule out the range d_{2}^{∗}(m) < d < d_{2}(m), obviating the need to push the arguments of this subsection to their tedious limits.

Table 5. Lower Bounds on d Obtained Without Partition Estimates

\begin{matrix}m & d_{2}^{∗}(m) & 4(m − 1)^{2m} \\ 5 & 4.5 10^{12} & 4.2 10^{6} \\ 7 & 4.4 10^{14} & 3.1 10^{11} \\ 9 & 2.2 10^{18} & 7.2 10^{16} \\ 11 & 7.0 10^{23} & 4.0 10^{22} \\ 13 & 8.7 10^{29} & 4.6 10^{28} \\ 15 & 2.1 10^{36} & 9.7 10^{34} \\ 17 & 8.4 10^{42} & 3.5 10^{41} \\ 19 & 5.3 10^{49} & 2.0 10^{48} \\ 21 & 5.0 10^{56} & 1.8 10^{55} \\ 23 & 6.8 10^{63} & 2.3 10^{62} \end{matrix}

The last column of Table 5 underscores the fact that the methods in this subsection do not suffice when m > 9. This column arises from the fact that the denominator on the right-hand side of (5.9) must be positive. In other words, d > 4(m − 1)^{2m}. Note that when m > 9, the last column of Table 5 precludes the use of the computationally intensive sieve in §6, no matter how sharp we make the other estimates in this subsection.

§5.2 The Range d_{1}(m) ≤ d ≤ d_{2}(m)

Lemma 11. Let t ∈ ℝ^{+}. Assume that m = h(−d) is odd and P covers M_{d}. Then,

(5.21)  |A_{d}(1/2 + it)| ≥ 1 − 2 ∑_{a∈M_{d}^{∗}} l_{a}(d)^{−1/2}.

Furthermore, if the right-hand side of (5.21) is positive, then

(5.22)  |arg A_{d}(1/2 + it)| ≤ \frac{2t ∑_{a∈M_{d}^{∗}} f(l_{a}(d))}{1 − 2 ∑_{a∈M_{d}^{∗}} l_{a}(d)^{−1/2}}

where f is the nonincreasing function defined by

f(x) = \{ \begin{matrix}2/e, & for 0 ≤ x ≤ e^{2} \\ x^{−1/2} log x, & for x > e^{2}. \end{matrix}

Proof. The proof is identical to the proof of Lemma 9 except that inequality (3.1) is replaced with the partition inequalities l_{a}(d) ≤ a ≤ u_{a}(d). □

Lemma 12. Let t ∈ ℝ and m = h(−d) > 1. Suppose that m is odd and P covers M_{d}. If k is an odd prime, then

|U_{d}(1/2+it)| ≤ \frac{24m(π\sqrt{3}k + 2)}{d^{1/4}(π\sqrt{3})^{3/2}k^{2}}e^{−π\sqrt{3}k/2}+\frac{16\sqrt{k}(π\sqrt{d} + 2k)}{π^{3/2}d}e^{−π\sqrt{d}/(2k)}+\frac{16}{d^{1/4}} ∑_{a∈M_{d}^{∗}} F (\frac{π\sqrt{d}}{u_{a}(d)k} )

Proof. We use Lemma 10. In the outer sum of (5.15), r = 1 or r = k. By Lemma 8, F is decreasing, so we can bound the inner summands from above by using upper bounds on a. When r = k we use the general upper bound a ≤ (d/3)^{1/2} of (3.1). When r = 1 and a > 1 we use the partition inequality a ≤ u_{a}(d). Thus, we have

|U_{d}(1/2 + it)| ≤ \frac{12m}{d^{1/4}k^{1/2}} F (π\sqrt{3}k) + \frac{8}{d^{1/4}} F (\frac{π\sqrt{d}}{k}) + \frac{16}{d^{1/4}} ∑_{a∈M_{d}^{∗}} F (\frac{π\sqrt{d}}{u_{a}(d)k} ).

Using Lemma 6 with N = 0 and x = π\sqrt{3}k and the inequality log(1 + x) < x (valid for x > 0), we have

|F (π\sqrt{3}k)| ≤ \frac{2}{\sqrt{π}3^{1/4}\sqrt{k}} (1 + \frac{2}{π\sqrt{3}k} )e^{−π\sqrt{3}k/2}.

Similarly, for x = π\sqrt{d}/k, we obtain

|F (π\sqrt{d}/k)| ≤ \frac{2\sqrt{k}}{\sqrt{π}d^{1/4}} (1 + \frac{2k}{π\sqrt{d}})e^{−π\sqrt{d}/(2k)}.

Lemma 12 follows easily. □

For the remainder of §5.2, we assume there exists a discriminant −d satisfying the following properties:

(I) h(−d) = m, where m ∈ {5, 7, 9, …, 23} is fixed; (II) d_{1}(m) ≤ d ≤ d_{2}(m), where d_{1}(m) is given in Table 6 and d_{2}(m) is given in Table 4.

Our goal is to show that (5.7) is true for d with a suitable choice of k and s_{0}. In a manner similar to §5.1, we let s_{0} = 1/2 + it_{0} and use

k = 17923 and t_{0} = 0.030985799.

Weinberger [29] has shown that the error in t_{0} is less than 10^{−8} but we only use the first 5 significant digits. From (5.4)-(5.6) we then have

ξ_{1} = 0.01549, ξ_{2} = 0.2216, ξ_{3} = 57.1,

where all approximations are accurate to the number of places shown.

As in §3, let M_{d} denote the multiset of minima for −d. Turning to Appendix B, note that each entry in Table B\frac{m−3}{2} (e.g., when m = 15, refer to Table B6) consists of (m − 1)/2 pairs of functions (l_{a}(d), u_{a}(d)), each increasing in d, corresponding to the (m − 1)/2 elements in M_{d}^{∗}. In other words, each entry of Table B\frac{m−3}{2} is a partition for d according to Definition 1 in §3. Since d ≥ d_{1}(m) ≥ 2^{m+2} from Table 6, the arguments at the end of §3 show that there is some partition for d in Table B\frac{m−3}{2} which covers M_{d} in the sense of Definition 2 in §3. We henceforth denote this partition by P_{1}.

Applying the first part of Lemma 11 with t = t_{0} and P = P_{1}, we have

(5.23)  |A_{d}(s_{0})| ≥ 1 − 2 ∑_{(l_{a},u_{a})∈P_{1}} l_{a}(d))^{−1/2} ≥ min_{P} (1 − 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d)^{−1/2}) ,

where the minimum is henceforth understood to be over all partitions P for d occurring in Table B\frac{m−3}{2}. Since l_{a}(d) is increasing in d,

(5.24)  min_{P} (1 − 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d)^{−1/2}) ≥ min_{P} (1 − 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d_{1}(m))^{−1/2}) > 0,

with the positivity following from direct computation.

From (5.23), (5.24), and Lemma 12 with t = t_{0}, k = 17923, and P = P_{1}, we have

\frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ R_{1}(d, P_{1}),

where we define

R_{1}(d, P) = \frac{\frac{24m(π\sqrt{3}k+2)}{d^{1/4}(π\sqrt{3})^{3/2}k^{2}} e^{−π\sqrt{3}k/2} + ^{\frac{16(π\sqrt{d}+2k)}{π^{3/2}d}}e^{−π\sqrt{d}/2k} + \frac{16}{d^{1/4}} ∑_{(l_{a},u_{a})∈P} F (\frac{π\sqrt{d}}{u_{a}(d)k} )}{ξ_{3}(1 − 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d)^{−1/2})}.

Thus from (5.23) and (5.24) we have

\frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ max_{P} R_{1}(d, P),

where the maximum is henceforth understood to be over all partitions P for d occurring in Table B\frac{m−3}{2}.

Note that for each u_{a} occurring in each of the partitions for d in Table B\frac{m−3}{2}, \sqrt{d}/u_{a}(d) is a non-decreasing function of d. It follows from Lemma 8 that F (π\sqrt{d}/(u_{a}(d)k)) is a nonincreasing function of d. Therefore, the numerator of R_{1}(d, P) is decreasing in d for any partition P for d appearing in Table B\frac{m−3}{2}. Thus, with the aid of (5.24), we have

(5.25)  \frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ max_{P} R_{1}(d_{1}(m), P)

since d ≥ d_{1}(m). Upper bounds for max_{P} R_{1}(d_{1}(m), P) can be found in Table 6. In order to produce these approximations, we need estimates for the functions F evaluated at small positive arguments. Such estimates are easily obtained by applying Lemma 6 with suitably large N.

Note that from (5.23) and (5.24) we know the right-hand side of (5.21) is positive for all partitions for d appearing in Table B\frac{m−3}{2}. Applying the second part of Lemma 11 with t = t_{0} and P = P_{1}, we obtain

| arg A_{d}(s_{0})| ≤ α_{1}(d, P_{1}),

where we define

α_{1}(d, P) = \frac{2t_{0} ∑_{(l_{a},u_{a})∈P} f(l_{a}(d))}{1 − 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d)^{−1/2}} .

It follows from the rightmost inequality in (5.23) and (5.24) that

(5.26)  | arg A_{d}(s_{0})| ≤ max_{P} α_{1}(d, P),

where the maximum is over all partitions P for d occurring in Table B\frac{m−3}{2}.

Now α_{1}(d, P) is decreasing in d for any fixed partition P for d appearing in Table B\frac{m−3}{2}. Hence,

(5.27)  | arg A_{d}(s_{0})| ≤ max_{P} α_{1}(d_{1}(m), P)

since d ≥ d_{1}(m). Table 6 contains upper bounds for max_{P} α_{1}(d_{1}(m), P).

Let β_{1}(m) be defined by

β_{1}(m) = ξ_{1} log d_{1}(m) + ξ_{2} − max_{P} α_{1}(d_{1}(m), P)

and γ_{1}(m) be defined by

γ_{1}(m) = ξ_{1} log d_{2}(m) + ξ_{2} + max_{P} α_{1}(10^{37}, P).

Lower bounds for β_{1}(m) and upper bounds for γ_{1}(m) are given in Table 6. Using Table 6, the fact that d_{1}(m) ≤ d and (5.27), we have

0 < β_{1}(m) ≤ ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}).

In the other direction, we claim that

ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}) ≤ γ_{1}(m) < π.

Note that from Table 4 and Table 6 we have

d_{1}(m) < 10^{37} < d_{2}(m).

If 10^{37} < d, our claim follows from the fact that d ≤ d_{2}(m), (5.26) coupled with the fact that α_{1}(d, P) is decreasing in d for any fixed partition P for d appearing in Table B\frac{m−3}{2}, and Table 6. When d ≤ 10^{37}, note that by (5.27) we have

ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}) ≤ ξ_{1} log(10^{37}) + ξ_{2} + max_{P} α_{1}(d_{1}(m), P).

With the aid of Table 6, direct calculation shows that

ξ_{1} log(10^{37}) + ξ_{2} + max_{P} α_{1}(d_{1}(m), P) < γ_{1}(m).

Thus our claim is also true when d ≤ 10^{37}. Hence,

|sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| ≥ min{| sin β_{1}(m)|, | sin γ_{1}(m)|}.

From the preceding inequality and Table 6, we conclude that

|sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| > max_{P} R_{1}(d_{1}(m), P).

In light of (5.25), (5.7) follows immediately. Since (5.3) is false, we conclude that h(−d) ≠ m for m ∈ {5, 7, …, 23} and d_{1}(m) ≤ d ≤ d_{2}(m).

Table 6. h(−d) ≠ m for d_{1}(m) ≤ d ≤ d_{2}(m)
m d_{1}(m) max_{P} R_{1}(d_{1}(m)) max_{P} α_{1}(d_{1}(m), P) max_{P} α_{1}(10^{37}, P) β_{1}(m) γ_{1}(m)
5 3.6 10^{11}  0.555  0.041  2.39 10^{−4}  0.592 2.542
7 2.0 10^{12}  0.547  0.072  1.89 10^{−3}  0.588 2.543
9 7.9 10^{12}  0.519  0.134  5.70 10^{−3}  0.547 2.583
11 4.6 10^{13}  0.444  0.245  1.15 10^{−2}  0.463 2.660
13 4.9 10^{14}  0.291  0.448  1.89 10^{−2}  0.297 2.846
15 1.9 10^{16}  0.148  0.643  2.77 10^{−2}  0.158 2.961
17 1.2 10^{18}  0.062  0.794  3.82 10^{−2}  0.072 3.079
19 9.2 10^{19}  0.024  0.906  5.06 10^{−2}  0.027 3.091
21 7.7 10^{21}  0.009  0.991  6.58 10^{−2}  0.010 3.107
23 6.6 10^{23}  0.004  1.065  8.47 10^{−2}  0.005 3.125

For m ∈ {5, 7, 9, 11, 13}, the d_{1}(m) in Table 6 is small enough to allow the class number m problem to be completed by the computationally intensive sieve of §6. However, for m ∈ {15, 17, 19, 21, 23}, we have to resort to further refinements in §5.3.

§5.3 The Range d_{0}(m) ≤ d ≤ d_{1}(m)

Lemma 13. Let t ∈ ℝ^{+}. Assume that m = h(−d) is odd and P covers M_{d}. Let p be the smallest nonprincipal minimum in M_{d}. If ℓ is a nonnegative integer such that the corresponding set S_{ℓ} = {p, p^{2}, …, p^{ℓ}} ⊆ M_{d}^{∗} (where S_{0} is understood to be the empty set), then

(5.28)  |A_{d}(1/2 + it)| ≥ |1 − 2\frac{1 − λ_{p}(t)^{ℓ+1}}{1 − λ_{p}(t)} | − 2 ∑_{a∈M_{d}^{∗}−S_{ℓ}} l_{a}(d)^{−1/2},

where λ_{p}(t) = χ(p)p^{−1/2−it}. Furthermore, if ℜA_{d}(1/2 + iτ) > 0 for 0 ≤ τ ≤ t and the right-hand side of (5.28) is positive, then

(5.29) |arg A_{d}(1/2 + it)| ≤ \frac{2t (log p |∑_{j=1}^{ℓ} jλ_{p}^{j}(t)| + ∑_{a∈M_{d}^{∗}−S_{ℓ}} f(l_{a}(d)))}{|1 − 2(1 − λ_{p}^{ℓ+1}(t))/(1 − λ_{p}(t))| − 2 ∑_{a∈M_{d}^{∗}−S_{ℓ}} l_{a}(d)^{−1/2}} ,

where f is as defined in Lemma 11.

Proof. The proof uses arguments similar to those in the proof of Lemma 9. □

For the remainder of §5.3, we assume there exists a discriminant −d satisfying the following properties:

(I) h(−d) = m, where m ∈ {9, 11, …, 23} is fixed; (II) d_{0}(m) ≤ d ≤ d_{1}(m), where d_{0}(m) is given in Table 7 and d_{1}(m) is given in Table 6.

Our goal is to show that (5.7) is true for d with a suitable choice of k and s_{0}. As in §5.2, let s_{0} = 1/2 + it_{0} and use

k = 17923 and t_{0} = 0.030986.

From (5.4)-(5.6) we then have

ξ_{1} = 0.01549, ξ_{2} = 0.2216, ξ_{3} = 57.1,

where all approximations are accurate to the number of places shown.

For each partition P for d in Table \frac{m−3}{2}, define ℓ_{P} to be the number of powers of the smallest minima in M_{d}^{∗} that can be shown to appear in P using the arguments of §3, if the number of such powers exceeds 2. Otherwise, set ℓ_{P} = 0. Denoting the smallest minima in M_{d}^{∗} by p, let S_{ℓ_{P}} = {p, p^{2}, …, p^{ℓ_{P}} }. Finally, let P^{∗} denote the set of (l_{a}(d), u_{a}(d) in P for which a ∉ S_{ℓ_{P}}.

Since d ≥ d_{0}(m) > 2^{m+2} by Table 7, the arguments at the end of §3 show there is some partition in Table B\frac{m−3}{2} which covers M_{d} in the sense of Definition 2 in §3. We henceforth denote this partiton by P_{0}. Applying the first part of Lemma 13 with t = t_{0}, P = P_{0}, ℓ = ℓ_{P_{0}}, and letting λ_{p} = λ_{p}(t_{0}), we have

(5.30)  \begin{aligned}|A_{d}(s_{0})| & ≥ |1 − 2\frac{1 − λ_{p}^{ℓ_{P_{0}} +1}}{1 − λ_{p}} | − 2 ∑_{(l_{a},u_{a})∈P_{0}^{∗}} l_{a}(d)^{−1/2} \\ & ≥ min_{P} (|1 − 2\frac{1 − λ_{p}^{ℓ_{P} +1}}{1 − λ_{p}} | − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d)^{−1/2}) , \end{aligned}

where the minimum is henceforth understood to be over all partitions P for d appearing in Table B\frac{m−3}{2}. Letting λ_{q} = χ(q)q^{−1/2−it_{0}} and noting that l_{a}(d) and u_{a}(d) are increasing in d, we have

min_{P} (|1 − 2\frac{1 − λ_{p}^{ℓ_{P} +1}}{1 − λ_{p}} | − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d)^{−1/2}) ≥

(5.31) min_{P} min_{q prime, (l_{p},u_{p})∈P^{lp(d_{0}(m))≤q≤up(d_{1}(m))}} (|1 − 2\frac{1 − λ_{q}^{ℓ_{P} +1}}{1 − λ_{q}} | − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d_{0}(m))^{−1/2}) > 0,

with the positivity following from a direct computation. From (5.30), (5.31), and Lemma 12 with t = t_{0}, k = 17923, and P = P_{0}, we have

\frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ R_{0}(d, P_{0}),

where we define

R_{0}(d, P) = \frac{\frac{24m(π\sqrt{3}k+2)}{d^{1/4}(π\sqrt{3})^{3/2}k^{2}} e^{−π\sqrt{3}k/2} + ^{\frac{16(π\sqrt{d}+2k)}{π^{3/2}d}}e^{−π\sqrt{d}/2k} + \frac{16}{d^{1/4}} ∑_{(l_{a},u_{a})∈P} F (\frac{π\sqrt{d}}{u_{a}(d)k} )}{ξ_{3} ( min_{q prime, (l_{p},u_{p})∈P^{lp(d_{0}(m))≤q≤up(d_{1}(m))}}(|1 − 2(1 − λ_{q}^{ℓ_{P} +1})/(1 − λ_{q})| − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d_{0}(m))^{−1/2}))}.

Thus from (5.30) and (5.31) we have

\frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ max_{P} R_{0}(d, P),

where the maximum is henceforth understood to be over all partitions P for d occurring in Table B\frac{m−3}{2}.

Note that for each u_{a} occurring in each of the partitions for d in Table B\frac{m−3}{2}, \sqrt{d}/u_{a}(d) is a non-decreasing function of d. Let P be any of the partitions for d appearing in Table B\frac{m−3}{2}. It follows from Lemma 8 that F (π\sqrt{d}/(u_{a}(d)k)) is a non-increasing function of d. Therefore, the numerator of R_{0}(d, P) is decreasing in d for any partition P for d appearing in Table B\frac{m−3}{2}. Hence with the aid of (5.31) we have

(5.32)  \frac{|U_{d}(s_{0})|}{ξ_{3}|A_{d}(s_{0})|} ≤ max_{P} R_{0}(d_{0}(m), P).

Upper bounds for max_{P} R_{0}(d_{0}(m), P) can be found in Table 7. In order to produce these approximations, we need estimates for the functions F evaluated at small positive arguments. Such estimates are easily obtained by applying Lemma 6 with suitably large N.

Note that from (5.30) and (5.31) we know the right-hand side of (5.28) is positive for all partitions for d appearing in Table B\frac{m−3}{2}. Suppose that 0 ≤ τ ≤ t_{0}. If ℓ_{P_{0}} = 0, then we have

|A_{d}(1/2 + iτ) − 1| ≤ 2 ∑_{(l_{a},u_{a})∈P_{0}} l_{a}(d)^{−1/2} ≤ max_{P} 2 ∑_{(l_{a},u_{a})∈P} l_{a}(d_{0}(m))^{−1/2} < 1,

by a direct calculation. Hence, ℜA_{d}(1/2 + iτ) > 0 when ℓ_{P_{0}} = 0. On the other hand, suppose ℓ_{P_{0}} ≠ 0. It follows from Table B\frac{m−3}{2} and Table 6 that the smallest minima p ∈ M_{d}^{∗} satisfies p ≤ (d_{1}(m)/4)^{1/6} ≤ 7406. Hence, 0 < 2t_{0} log p < π, and it is not difficult to see that

ℜA_{d}(1/2 + iτ) ≥ 1 − 2 ∑_{j=1}^{ℓ_{P_{0}}} \frac{1}{p^{j/2}} + \frac{2 + 2 cos(2t_{0} log p)}{p} − 2 ∑_{(l_{a},u_{a})∈P_{0}^{∗}} l_{a}(d_{0}(m))^{−1/2} ≥

min_{P} min_{q prime, (l_{p},u_{p})∈P^{lp(d_{0}(m))≤q≤up(d_{1}(m))}} (1 − 2 ∑_{j=1}^{ℓ_{P}} \frac{1}{q^{j/2}} + \frac{2 + 2 cos(2t_{0} log q)}{q} − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d_{0}(m))^{−1/2}) .

It then follows by direct computation that ℜA_{d}(1/2 + iτ) > 0 for 0 ≤ τ ≤ t_{0}. Hence, we may apply the second part of Lemma 13 with t = t_{0}, P = P_{0}, and ℓ = ℓ_{P_{0}} to obtain

| arg A_{d}(s_{0})| ≤ \frac{2t_{0} (log p |∑_{j=1}^{ℓ_{P_{0}}} jλ_{p}^{j}| + ∑_{(l_{a},u_{a})∈P_{0}^{∗}} f(l_{a}(d)))}{|1 − 2(1 − λ_{p}^{ℓ_{P_{0}} +1})/(1 − λ_{p})| − 2 ∑_{(l_{a},u_{a})∈P_{0}^{∗}} l_{a}(d)^{−1/2}} .

Hence, since l_{a}(d) and u_{a}(d) are increasing in d, we have

| arg A_{d}(s_{0})| ≤ α_{0}(d, P_{0}),

where we define

α_{0}(d, P) = max_{q prime, (l_{p},u_{p})∈P^{lp(d_{0}(m))≤q≤up(d_{1}(m))}} (\frac{2t_{0} (log q |∑_{j=1}^{ℓ_{P}} jλ_{q}^{j}| + ∑_{(l_{a},u_{a})∈P^{∗}} f(l_{a}(d)))}{|1 − 2(1 − λ_{q}^{ℓ_{P} +1})/(1 − λ_{q})| − 2 ∑_{(l_{a},u_{a})∈P^{∗}} l_{a}(d)^{−1/2}} ) .

It follows from (5.30) and (5.31) that

| arg A_{d}(s_{0})| ≤ max_{P} α_{0}(d, P).

Since f is nonincreasing and l_{p}(d) is increasing in d, we know α_{0}(d, P) is decreasing in d for any fixed partition P for d appearing in Table B\frac{m−3}{2}. Hence,

(5.33)  | arg A_{d}(s_{0})| ≤ max_{P} α_{0}(d_{0}(m), P)

since d ≥ d_{0}(m). Table 7 contains upper bounds for max_{P} α_{0}(d_{0}(m), P).

Let β_{0}(m) be defined by

β_{0}(m) = ξ_{1} log d_{0}(m) + ξ_{2} − max_{P} α_{0}(d_{0}(m), P)

and γ_{0}(m) be defined by

γ_{0}(m) = ξ_{1} log d_{1}(m) + ξ_{2} + max_{P} α_{0}(d_{0}(m), P).

Lower bounds for β_{0}(m) and upper bounds for γ_{0}(m) are given in Table 7. Using Table 7, the fact that d_{0}(m) ≤ d, and (5.33), we have

0 < β_{0}(m) ≤ ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}).

Using the fact that d ≤ d_{1}(m), (5.33), and Table 7, we have

ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}) ≤ γ_{0}(m) < π.

Hence,

|sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| ≥ min{| sin β_{0}(m)|, | sin γ_{0}(m)|}.

From the preceding inequality and Table 7, we conclude that

|sin(ξ_{1} log d + ξ_{2} + arg A_{d}(s_{0}))| > max_{P} R_{0}(d_{0}(m), P).

In light of (5.32), (5.7) follows immediately. Since (5.3) is false, we conclude that h(−d) ≠ m for m ∈ {9, 11, …, 23} and d_{0}(m) ≤ d ≤ d_{1}(m).

Table 7. h(−d) ≠ m for d_{0}(m) ≤ d ≤ d_{1}(m)

\begin{matrix}m & d_{0}(m) & max_{P} R_{0}(d_{0}(m)) & max_{P} α_{0}(d_{0}(m), P) & β_{0}(m) & γ_{0}(m) \\ 9 & 6.4 10^{12} & 0.553 & 0.089 & 0.589 & 0.771 \\ 11 & 2.2 10^{13} & 0.555 & 0.100 & 0.597 & 0.810 \\ 13 & 4.2 10^{13} & 0.556 & 0.116 & 0.591 & 0.862 \\ 15 & 9.4 10^{13} & 0.557 & 0.128 & 0.591 & 0.931 \\ 17 & 1.9 10^{14} & 0.556 & 0.138 & 0.592 & 1.005 \\ 19 & 3.5 10^{14} & 0.555 & 0.149 & 0.591 & 1.083 \\ 21 & 6.5 10^{14} & 0.542 & 0.173 & 0.576 & 1.176 \\ 23 & 10.6 10^{14} & 0.548 & 0.177 & 0.580 & 1.249 \end{matrix}

A comparison of Tables 6 and 7 shows that the gap between d_{0}(m) and d_{1}(m) is increasing rapidly as m increases from 9 to 23.

§6 The Low Range

In this section, we complete the proof of Theorem 1, the statement of which appears in §1. Using the results of §2 and §5 (see Tables 6 and 7 in particular), it suffices to find all negative fundamental discriminants −d with h(−d) ∈ {5, 7, …, 23} such that d ≤ 1.1 10^{15}.

We first consider the case where

(6.1)  7.5 10^{6} ≤ d ≤ 1.1 10^{15}.

Assume there is a negative fundamental discriminant −d with h(−d) ∈ {5, 7, …, 23}. Using results from §3, we see that

(i) (−d|p) = −1 for all odd primes p ≤ (d/4)^{1/23}.

(ii) (−d|p) = −1 for all odd primes p ≤ (d/4)^{1/6} with at most one exception.

(iii) (−d|p) = −1 for all odd primes p ≤ (d/4)^{1/4} with at most two exceptions.

Of course, (1.3) implies (−d|p) ≠ 0 for odd primes p < d. Item (i) follows from Lemmas 1 and 5. If two odd primes p, q ≤ (d/4)^{1/6} satisfy (−d|p) = 1 and (−d|q) = 1, then p, p^{2}, p^{3}, q, q^{2}, q^{3}, pq, p^{2}q, pq^{2} ∈ M_{d}, which implies h(−d) ≥ 25 by Lemmas 1 and 4. Thus, item (ii) is true. Lastly, if three odd primes p, q, r ≤ (d/4)^{1/4} satisfy (−d|p) = (−d|q) = (−d|r) = 1, then Lemmas 1 and 4 give p, p^{2}, q, q^{2}, r, r^{2}, pq, pr, qr ∈ M_{d}, which implies h(−d) ≥ 25. Thus, item (iii) is true.

Even with (i), (ii) and (iii), it would be infeasible to check all prime d in the range (6.1). Instead, we build up a substantially smaller set of possible d by using the Chinese Remainder Theorem on the residue requirements implicit in (i), (ii) and (iii) for a set of small odd primes along with the congruence d = 3 (mod 4). The requirement that (−d|p) = −1 (or (−d|p) = 1) implies that d is in one of (p − 1)/2 residue classes mod p. To apply this method we divide (6.1) into 3 shorter intervals.

We now restrict our attention to the subcase where 2.65 10^{13} ≤ d ≤ 1.1 10^{15}. The upper bounds in (i), (ii) and (iii) applied to 2.65 10^{13} are 3.6, 137, and 1604, respectively. Consider the set of moduli

{3, 4, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43}

with one residue mod 4 and (p − 1)/2 residues for the odd primes p. Applying the Chinese Remainder Theorem results in 20575296000 residues modulo m = 1137631420145220. Among the 11 odd primes p > 3 there is at most one satisfying (−d|p) = 1. Thus, there are 12 cases to consider resulting in 2.5 10^{11} possible residue classes mod m in which d must fall. Since m > 1.1 10^{15}, all d in the given range are being considered. These values were generated and checked using (ii) and (iii) on a Cray 2 supercomputer (using C) in 19.8 hours. A total of 43 additional primes were needed, and the following two values passed the tests: 625483863652783 and 652681014370807. For each of these two values of d, the list {(−d|p_{i})}_{i=2}^{56} contains only −1 except in exactly two positions. The rarity of such numbers is a good check for the accuracy of the code.

Next, consider the subcase where 4.32 10^{10} ≤ d ≤ 2.66 10^{13}. As above, the upper bounds in (i), (ii) and (iii) applied to 4.32 10^{10} are 2.7, 47, and 322, respectively, and we consider the set of 12 moduli

{3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 41, 47}

with one residue mod 4 and (p − 1)/2 residues for the odd primes p. Applying the Chinese Remainder Theorem results in 983664000 residues modulo m = 26653797550380. As above, there 12 possible cases, and since m > 2.66 10^{13} all possible d in this range are considered. The number of such d is 1.2 10^{10}. Checking these values of d took 1.2 hours on a Cray 2. An additional 38 primes were used, and the only value that passed the tests was 7647157072003.

Now consider the final subcase where 7.5 10^{6} ≤ d ≤ 4.35 10^{10}. The bounds (ii) and (iii) applied to 7.5 10^{6} are 11 and 37, respectively. We use the set of 10 moduli {3, 4, 5, 7, 13, 17, 19, 23, 29, 37}. The modulus for the Chinese Rmainder Theorem is m = 43523390820, which satisfies m > 4.35 10^{10}. The number of residues in one case is 7185024, but here there are 43 cases. In reality, we used the fact that 3 ≤ (d/4)^{1/12} and 5 ≤ (d/4)^{1/8} to reduce the number of cases to 37. The resulting 93405312 possible values of d produced were eliminated in 7 minutes on the Cray 2.

Finally, there remains the case where d ≤ 7.5 10^{6}. For each d in this range, we computed the class number by counting the number of reduced forms of discriminant −d. In other words, we searched for integers a, b, and c with 0 < a < (d/3)^{1/2} and c = (b^{2} − d)/(4a) such that either −a < b ≤ a < c or 0 ≤ b ≤ a = c. This straightforward approach required just under two hours on the Cray-2, rendering further optimization unnecessary. A complete listing of the negative fundamental discriminants for class numbers 5, 7, …, 23 is given in Appendix 1, completing the proof of Theorem 1. It is worth noting that Buell [6] had previously computed class numbers of imaginary quadratic number fields for d ≤ 4 10^{6}, and our results agree perfectly with his in this range. Futhermore, in recent unpublished work, Buell has independently verified our results up to 7.5 10^{6}, using our method of separating minima that was introduced in §3.

Acknowledgement

All of our numerical computations and explorations were done with the aid of Mathematica on a SPARCstation, with the exception of the C code written for the Cray-2 routines discussed in §6.

References

1. S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 40 (1992), 321–334.

2. A. Baker, Linear forms in the logarithms of algebraic numbers. I, Mathematika 13 (1966), 204–216.

3. A. Baker, Imaginary quadratic fields with class number 2, Ann. of Math. 94 (1971), 139–152.

4. A. Baker, Transcendental Number Thoery, Cambridge Univ. Press, New York, 1975.

5. B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108.

6. D. A. Buell, Class groups of quadratic fields. II, Math. Comp. 48 (1987).

7. H. Davenport, Multiplicative Number Theory, 2nd ed., Graduate Texts in Math. 74, Springer, New York, 1980.

8. M. Deuring, Imaginäre quadratische Zahl Körper mit der Klassenzahl Eins, Invent. Math. 5 (1968), 169–179.

9. C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966.

10. D. M. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa 3 (1976), 623–663.

11. B. Gross and D. Zagier, Points de Heegner et derivées de fonctions L, C. R. Acad. Sci. Paris 297 (1983), 85–87.

12. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford University Press, London, 1968.

13. K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.

14. H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. Oxford Ser. 25 (1934), 150–160.

15. A. E. Ingham, The Distribution of Prime Numbers, Cambridge Tract in Mathematics No. 30, Cambridge University Press, Cambridge, 1990.

16. H. Kestelman, Modern Theories of Integration, Dover Publications, New York, 1960.

17. N. Levinson and R. M. Redheffer, Complex Variables, Holden-Day, Inc., San Francisco, 1970.

18. H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1974), 529–542.

19. J. Oesterlé, Nombres de classes des corps quadratiques imaginaires, Sém. Bourbaki, 1983–1984, exp. 631.

20. C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1936), 83–86.

21. C. L. Siegel, Zum Bewiese des Starkschen Satzes, Invent. Math. 5 (1968), 180–191.

22. H. M. Stark, On complex quadratic number fields with class number equal to one, Trans. Amer. Math. Soc. 122 (1966), 112–119.

23. H. M. Stark, A complete determination of the complex quadratic fields of class number one, Michigan Math. J. 14 (1967), 1–27.

24. H. M. Stark, On the “gap” in a theorem of Heegner, J. Number Theory 1 (1969), 16–27.

25. H. M. Stark, L-functions and character sums for quadratic forms (II), Acta Arith. 15 (1969), 307–317.

26. H. M. Stark, On complex quadratic fields with class number two, Math. Comp. 29 (1975), 289–302.

27. G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, Cambridge University Press, London, 1966.

28. A. Weil, On some exponential sums, Proc. N. A. S. 34 (1948), 204–220.

29. P. J. Weinberger, On small zeros of Dirichlet L-functions, Math. Comp. 29 (1975), 319–328.

28 ARNO, ROBINSON AND WHEELER

Appendix A

Negative Fundamental Discriminants

for Class Numbers 5, 7, …, 23. Table A1. Values of d with h(−d) = 5

47 79 103 127 131 179 227 347 443 523 571 619 683 691 739 787 947 1051 1123 1723 1747 1867 2203 2347 2683 Table A2. Values of d with h(−d) = 7

71 151 223 251 463 467 487 587 811 827 859 1163 1171 1483 1523 1627 1787 1987 2011 2083 2179 2251 2467 2707 3019 3067 3187 3907 4603 5107 5923 Table A3. Values of d with h(−d) = 9

199 367 419 491 563 823 1087 1187 1291 1423 1579 2003 2803 3163 3259 3307 3547 3643 4027 4243 4363 4483 4723 4987 5443 6043 6427 6763 6883 7723 8563 8803 9067 10627 Table A4. Values of d with h(−d) = 11

167  271  659  967 1283 1303 1307 1459 1531 1699 2027 2267 2539 2731 2851 2971 3203 3347 3499 3739 3931 4051 5179 5683 6163 6547 7027 7507 7603 7867 8443 9283 9403 9643 9787 10987 13003 13267 14107 14683 15667

Table A5. Values of d with h(−d) = 13

191  263  607  631  727 1019 1451 1499 1667 1907 2131 2143 2371 2659 2963 3083 3691 4003 4507 4643 5347 5419 5779 6619 7243 7963 9547 9739 11467 11587 11827 11923 12043 14347 15787 16963 20563

Table A6. Values of d with h(−d) = 15

239  439  751  971 1259 1327 1427 1567 1619 2243 2647 2699 2843 3331 3571 3803 4099 4219 5003 5227 5323 5563 5827 5987 6067 6091 6211 6571 7219 7459 7547 8467 8707 8779 9043 9907 10243 10267 10459 10651 10723 11083 11971 12163 12763 13147 13963 14323 14827 14851 15187 15643 15907 16603 16843 17467 17923 18043 18523 19387 19867 20707 22003 26203 27883 29947 32323 34483

Table A7. Values of d with h(−d) = 17

383 991 1091 1571 1663 1783 2531 3323 3947 4339 4447 4547 4651 5483 6203 6379 6451 6827 6907 7883 8539 8731 9883 11251 11443 12907 13627 14083 14779 14947 16699 17827 18307 19963 21067 23563 24907 25243 26083 26107 27763 31627 33427 36523 37123

Table A8. Values of d with h(−d) = 19

311  359  919 1063 1543 1831 2099 2339 2459 3343 3463 3467 3607 4019 4139 4327 5059 5147 5527 5659 6803 8419 8923 8971 9619 10891 11299 15091 15331 16363 16747 17011 17299 17539 17683 19507 21187 21211 21283 23203 24763 26227 27043 29803 31123 37507 38707

Table A9. Values of d with h(−d) = 21

431  503  743  863 1931 2503 2579 2767 2819 3011 3371 4283 4523 4691 5011 5647 5851 5867 6323 6691 7907 8059 8123 8171 8243 8387 8627 8747 9091 9187 9811 9859 10067 10771 11731 12107 12547 13171 13291 13339 13723 14419 14563 15427 16339 16987 17107 17707 17971 18427 18979 19483 19531 19819 20947 21379 22027 22483 22963 23227 23827 25603 26683 27427 28387 28723 28867 31963 32803 34147 34963 35323 36067 36187 39043 40483 44683 46027 49603 51283 52627 55603 58963 59467 61483

647 1039 1103 1279 1447 1471 1811 1979 2411 2671 3491 3539 3847 3923 4211 4783 5387 5507 5531 6563 6659 6703 7043 9587 9931 10867 10883 12203 12739 13099 13187 15307 15451 16267 17203 17851 18379 20323 20443 20899 21019 21163 22171 22531 24043 25147 25579 25939 26251 26947 27283 28843 30187 31147 31267 32467 34843 35107 37003 40627 40867 41203 42667 43003 45427 45523 47947 90787

Appendix B Covering Partitions for Class Numbers 5, 7, 9, …, 23.

The following tables give a set of partitions covering all possible multisets M_{d} under the assumption that h(−d) = m for m = 5, 7, …, 23. Let p and q denote the first and second smallest prime minima in M_{d}, respectively, and let a denote a generic member of M_{d}. Let v = d/4 and w = d/3. The notation (np.) implies that inequalities for the first n powers of the prime are to be inferred. The notation (n) simply means the inequality is to be listed a total of n times.

Table B1. Partitions for Class Number 5

1 v^{1/5} ≤ p ≤ v^{1/4} (2p.)
2 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2}
3 v^{1/3} ≤ a ≤ w^{1/2} (2)

Table B2. Partitions for Class Number 7

1 v^{1/7} ≤ p ≤ v^{1/6} (3p.)
2 v^{1/6} ≤ p ≤ v^{1/4} (2p.) v^{1/4} ≤ a ≤ w^{1/2}
3 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (2)
4 v^{1/3} ≤ a ≤ w^{1/2} (3)

Table B3. Partitions for Class Number 9

1 v^{1/9} ≤ p ≤ v^{1/8} (4p.)
2 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/3} ≤ a ≤ w^{1/2}
3 v^{1/6} ≤ p ≤ v^{1/4} (2p.) v^{1/4} ≤ a ≤ w^{1/2} (2)
4 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (3)
5 v^{1/3} ≤ a ≤ w^{1/2} (4)

Table B4. Partitions for Class Number 11

1 v^{1/11} ≤ p ≤ v^{1/10} (5p.)
2 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{3/8} ≤ a ≤ w^{1/2}
3 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (2)
4 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ a ≤ w^{1/2} (3)
5 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (4)
6 v^{1/3} ≤ a ≤ w^{1/2} (5)

32 ARNO, ROBINSON AND WHEELER

Table B5. Partitions for Class Number 13

1 v^{1/13} ≤ p ≤ v^{1/12} (6p.)
2 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{2/5} ≤ a ≤ w^{1/2}
3 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{3/8} ≤ a ≤ w^{1/2} (2)
4 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2)
5 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (3)
6 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)
7 v^{1/6} ≤ p ≤ v^{1/4} (2p.) v^{1/4} ≤ q ≤ v^{1/3} v^{1/4} ≤ a ≤ w^{1/2} (3)
8 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (4)
9 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (5)
10 v^{1/3} ≤ a ≤ w^{1/2} (6)

Table B6. Partitions for Class Number 15

1 v^{1/15} ≤ p ≤ v^{1/14} (7p.)
2 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{5/12} ≤ a ≤ w^{1/2}
3 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{2/5} ≤ a ≤ w^{1/2} (2)
4 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{1/4} ≤ a ≤ w^{1/2} (3)
5 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{7/24} ≤ pq ≤ v^{5/12} (2)
6 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2) v^{1/3} ≤ a ≤ w^{1/2}
7 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (4)
8 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)  v^{1/4} ≤ a ≤ w^{1/2}
9 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ q ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (4)
10 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (5)
11 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (6)
12 v^{1/3} ≤ a ≤ w^{1/2} (7)

Table B7. Partitions for Class Number 17

1 v^{1/17} ≤ p ≤ v^{1/16} (8p.)
2 v^{1/16} ≤ p ≤ v^{1/14} (7p.) v^{3/7} ≤ a ≤ w^{1/2}
3 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{5/12} ≤ a ≤ w^{1/2} (2)
4 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{3/10} ≤ a ≤ w^{1/2} (3)
5 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{1/4} ≤ q ≤ v^{3/8}  v^{7/20} ≤ pq ≤ v^{1/2} (2) v^{3/8} ≤ a ≤ w^{1/2}
6 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{3/8} ≤ a ≤ w^{1/2} (4)
7 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{7/24} ≤ pq ≤ v^{5/12} (2) v^{1/3} ≤ a ≤ w^{1/2}
8 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2) v^{1/3} ≤ a ≤ w^{1/2} (2)
9 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (5)
10 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)  v^{1/4} ≤ a ≤ w^{1/2} (2)
11 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ q ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (5)
12 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (6)
13 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (7)
14 v^{1/3} ≤ a ≤ w^{1/2} (8)

Table B8. Partitions for Class Number 19

1 v^{1/19} ≤ p ≤ v^{1/18} (9p.)
2 v^{1/18} ≤ p ≤ v^{1/16} (8p.) v^{7/16} ≤ a ≤ w^{1/2}
3 v^{1/16} ≤ p ≤ v^{1/14} (7p.) v^{3/7} ≤ a ≤ w^{1/2} (2)
4 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{1/3} ≤ a ≤ w^{1/2} (3)
5 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{3/10} ≤ a ≤ w^{1/2} (4)
6 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{1/4} ≤ q ≤ v^{3/8}  v^{7/20} ≤ pq ≤ v^{1/2} (2) v^{3/8} ≤ a ≤ w^{1/2} (2)
7 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{3/8} ≤ a ≤ w^{1/2} (5)
8 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{7/24} ≤ pq ≤ v^{5/12} (2) v^{1/3} ≤ a ≤ w^{1/2} (2)
9 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2) v^{1/4} ≤ a ≤ w^{1/2} (3)
10 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (6)
11 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)  v^{1/4} ≤ a ≤ w^{1/2} (3)
12 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ q ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (6)
13 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (7)
14 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (8)
15 v^{1/3} ≤ a ≤ w^{1/2} (9)

Table B9. Partitions for Class Number 21

1 v^{1/21} ≤ p ≤ v^{1/20} (10p.)
2 v^{1/20} ≤ p ≤ v^{1/18} (9p.) v^{4/9} ≤ a ≤ w^{1/2}
3 v^{1/18} ≤ p ≤ v^{1/16} (8p.) v^{7/16} ≤ a ≤ w^{1/2} (2)
4 v^{1/16} ≤ p ≤ v^{1/14} (7p.) v^{5/14} ≤ a ≤ w^{1/2} (3)
5 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{1/3} ≤ a ≤ w^{1/2} (4)
6 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{1/4} ≤ q ≤ v^{3/10} v^{1/3} ≤ pq ≤ v^{2/5} (2) v^{5/12} ≤ p^{2}q ≤ v^{1/2} (2)
7 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{3/10} ≤ a ≤ w^{1/2} (5)
8 v^{1/10} ≤ p ≤ v^{1/8} (4p.)  v^{3/16} ≤ q ≤ v^{1/4} (2p.) v^{23/80} ≤ pq ≤ v^{3/8} (2) v^{31/80} ≤ p^{2}q ≤ v^{1/2} (2)
9 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{1/4} ≤ q ≤ v^{3/8} v^{7/20} ≤ pq ≤ v^{1/2} (2) v^{1/4} ≤ a ≤ w^{1/2} (3)
10 v^{1/10} ≤ p ≤ v^{1/8} (4p.)  v^{3/8} ≤ a ≤ w^{1/2} (6)
11 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{7/24} ≤ pq ≤ v^{5/12} (2) v^{1/4} ≤ a ≤ w^{1/2} (3)
12 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2) v^{1/4} ≤ a ≤ w^{1/2} (4)
13 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (7)
14 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)  v^{1/4} ≤ a ≤ w^{1/2} (4)
15 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ q ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (7)
16 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (8)
17 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (9)
18 v^{1/3} ≤ a ≤ w^{1/2} (10)

34 ARNO, ROBINSON AND WHEELER

Table B10. Partitions for Class Number 23

1 v^{1/23} ≤ p ≤ v^{1/22} (11p.)
2 v^{1/22} ≤ p ≤ v^{1/20} (10p.) v^{9/20} ≤ a ≤ w^{1/2}
3 v^{1/20} ≤ p ≤ v^{1/18} (9p.) v^{4/9} ≤ a ≤ w^{1/2} (2)
4 v^{1/18} ≤ p ≤ v^{1/16} (8p.) v^{3/8} ≤ a ≤ w^{1/2} (3)
5 v^{1/16} ≤ p ≤ v^{1/14} (7p.) v^{5/14} ≤ a ≤ w^{1/2} (4)
6 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{1/4} ≤ q ≤ v^{1/3}  v^{9/28} ≤ pq ≤ v^{5/12} (2) v^{11/28} ≤ p^{2}q ≤ v^{1/2} (2)
7 v^{1/14} ≤ p ≤ v^{1/12} (6p.) v^{1/3} ≤ a ≤ w^{1/2} (5)
8 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{1/5} ≤ q ≤ v^{3/10}  v^{17/60} ≤ pq ≤ v^{2/5} (2) v^{11/30} ≤ p^{2}q ≤ v^{1/2} (2) v^{2/5} ≤ a ≤ w^{1/2}
9 v^{1/12} ≤ p ≤ v^{1/10} (5p.) v^{3/10} ≤ a ≤ w^{1/2} (6)
10 v^{1/10} ≤ p ≤ v^{1/8} (4p.)  v^{3/16} ≤ q ≤ v^{1/4} (2p.) v^{23/80} ≤ pq ≤ v^{3/8} (2) v^{31/80} ≤ p^{2}q ≤ v^{1/2} (2) v^{3/8} ≤ a ≤ w^{1/2}
11 v^{1/10} ≤ p ≤ v^{1/8} (4p.) v^{1/4} ≤ q ≤ v^{3/8} v^{7/20} ≤ pq ≤ v^{1/2} (2) v^{1/4} ≤ a ≤ w^{1/2} (4)
12 v^{1/10} ≤ p ≤ v^{1/8} (4p.)  v^{3/8} ≤ a ≤ w^{1/2} (7)
13 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{7/24} ≤ pq ≤ v^{5/12} (2) v^{1/4} ≤ a ≤ w^{1/2} (4)
14 v^{1/8} ≤ p ≤ v^{1/6} (3p.) v^{1/4} ≤ q ≤ v^{1/3} v^{3/8} ≤ pq ≤ v^{1/2} (2) v^{1/4} ≤ a ≤ w^{1/2} (5)
15 v^{1/8} ≤ p ≤ v^{1/6} (3p.)  v^{1/3} ≤ a ≤ w^{1/2} (8)
16 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/6} ≤ q ≤ v^{1/4} (2p.) v^{1/3} ≤ pq ≤ v^{1/2} (2)  v^{1/4} ≤ a ≤ w^{1/2} (5)
17 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/4} ≤ q ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (8)
18 v^{1/6} ≤ p ≤ v^{1/4} (2p.)  v^{1/3} ≤ a ≤ w^{1/2} (9)
19 v^{1/4} ≤ p ≤ v^{1/3}  v^{1/4} ≤ a ≤ w^{1/2} (10)
20 v^{1/3} ≤ a ≤ w^{1/2} (11)

Supercomputing Research Center, 17100 Science Drive, Bowie, MD 20715 E-mail address: arno@super.org, robinson@super.org, wheeler@super.org