On a class number formula of Hurwitz

In a little-known paper Hurwitz gave an inﬁnite series representation of the class number for positive deﬁnite binary quadratic forms. In this paper we give a similar formula in the indeﬁnite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.


Introduction
Adolf Hurwitz made a number of important and influential contributions to the theory of binary quadratic forms. Yet his paper [Hur1] on an infinite series representation of the class number in the positive definite case, which appeared in the Dirichlet-volume of Crelle's Journal of 1905, has been essentially ignored. About the only references to this paper we found in the literature are in Dickson's book [Di,p.167] and the more recent paper [Sc]. Perhaps one reason for this neglect is that Hurwitz gives a rather general treatment of certain projective integrals which, when applied in this special case, tends to obscure the basic mechanism behind the proof. Our main object here is to establish an indefinite version of Hurwitz's formula and give a direct and uniform treatment of both cases. We also want to clarify the relation between these formulas and the much better known class number formulas of Dirichlet.
In another largely ignored paper [Hur2], published after his death, Hurwitz further developed his method and applied it to get a formula for the class number of integral positive definite ternary quadratic forms. His general method deserves to be better known and we plan to give some different applications of it in future work. Duke  2 Dirichlet's formulas Before stating the Hurwitz formula and the indefinite version we will first set notation and recall Dirichlet's formulas. A standard reference is Landau's book [La]. For convenience we will generally use the notation from [DIT]. Consider the real binary quadratic form which is absolutely convergent for Re s > 1. Let w −3 = 3, w −4 = 2 and w d = 1 for d < −4.
Theorem 1 (Dirichlet). For any fundamental d < 0 (2.4) Let L(s, χ d ) = n≥1 χ d (n)n −s be the Dirichlet L-series with χ d (·) the Kronecker symbol. Then by counting solutions to the quadratic congruence implicit in (2.2) we have and so π 2 6 res s=1 Z d (s) = L(1, χ d ). This leads to the usual finite versions of Dirichlet's formulas: when d < 0 we have that

Hurwitz's formula and the indefinite case
Hurwitz's formula gives the class number when d < 0 in terms of an absolutely convergent analogue of the divergent series Z d (1) from (2.2). A nice feature is that approximations increase monotonically to their limit.
Theorem 2 (Hurwitz). For d < 0 a fundamental discriminant Each term in the sum is positive and the sum converges.
In fact, this holds for any negative discriminant d when 1 w d h(d) is replaced by the Hurwitz class number H(−d), although Hurwitz only stated the formula for even d. The proofs immediately extend to include all negative discriminants. The convergence of this series is rather slow. Hurwitz also gave the equivalent formulation (3.1) where r, s are positive.
Actually Hurwitz gave a whole series of formulas for h(d) consisting of sums of series of the same shape that individually converge faster. The nicest one is given by (3.2) We will prove this and give others in §8 below. Turning now to the indefinite case, we will show the following.
where reduced means reduced in the sense of Zagier, meaning that a, c > 0 and b > a + c.
The sum over reduced forms is finite and each term in the infinite sum is positive, the sum being convergent.
Note that it is no longer true that each individual factor (b + 2a), b, (b + 2c) is positive but their product is positive. As in the positive definite case, approximations increase monotonically as more terms are taken.
Consider the example d = 5. Taking a ≤ 100 and |b| ≤ 100 in the infinite sum yields the approximation 0.961098 to the correct value while taking a ≤ 500 and |b| ≤ 500 gives 0.962282. Similarly to (3.2), we can derive faster converging series at the expense of more complicated formulas. Here is the next case, to be proven at the end of §8.
As another example, Since the rational numbers given by the finite sums give lower bounds for L(1, it is obviously of interest (and no doubt extremely difficult) to estimate them from below. Numerically, the values of R j (d) for j = 1, 2 seem to account for at least a constant proportion of the value of L(1, χ d ), with the constant being larger for R 2 than for R 1 .
As will become clear, it is possible to give formulas of the same shape as those in (3.3) and (3.5) for the residue at s = 1 of an ideal class zeta function (and hence for log d ) by suitably restricting both the finite and infinite sums over [a, b, c]. Note that Kohnen and Zagier in [KZ,p. 223] gave the values of such zeta functions at s = 1 − k for k ∈ {2, 3, 4, 5, 7} as sums over reduced forms of certain polynomials in a, b, c.

Eisenstein series
It is instructive to sketch in some detail proofs of (2.3) and (2.4) of Theorem 1 that are prototypes for our proofs of Theorems 2 and 3. Define for τ ∈ H, the upper half-plane, and for Re(s) > 1 the Eisenstein series Here Γ ∞ consists of the translations by integers in Γ. Let ∆ be the (positive) hyperbolic Laplacian. Since ∆Imτ = s(s − 1)Imτ and ∆ commutes with the usual linear fractional action τ → gτ , it follows that E(τ, s) is an eigenfunction of −∆ : It is a crucial result that E(τ, s) has a meromorphic continuation in s to C and that it has a simple pole at s = 1 with residue that is constant. In fact with respect to the usual invariant measure dµ(τ ).
To each real positive definite binary quadratic form Q = [a, b, c] of discriminant d associate the point Note that for τ → gτ the usual linear fractional action for g ∈ PSL(2, R) Then it is straightforward to check using (2.1) that and (2.3) of Theorem 1 follows from this and (4.1). The case d > 0 is more involved. Let S Q be the oriented semi-circle defined by oriented counterclockwise if a > 0 and clockwise if a < 0. Given z ∈ S Q let C Q be the directed arc on S Q from z to the image of z under the canonical generator g Q of the isotropy subgroup of Q, which is given by where t, u were defined in Theorem 1. We want to show that 1) . Then the second formula of Theorem 1 follows by (4.1) and the fact that (4.6) But (see e.g. the proof of Lemma 7 in [DIT]) upon using the substitution u = tan θ 2 for which sin θ = 2u 1+u 2 and dθ = 2 1+u 2 du. This gives (4.5) hence (2.4) of Theorem 1.
(5.4) Suppose that for some constant C > 1 we have that Im τ ≤ C. Separating the k = 0 and k = 1 terms from the sum in (5.3) shows that for such τ with − 1 This is trivial for a = 0 and follows from |aτ + b| ≥ |Im τ | otherwise. The sum (5.4) contains at most finitely many terms where Im gτ > C. Also we may assume that − 1 2 ≤ Re gτ ≤ 1 2 . Thus by (5.5) and (5.6) the sum in (5.4) is majorized by a constant multiple of (c,d)=1 where the constant depends only on C. The claimed convergence follows from our assumption on s 1 , s 2 , s 3 . It is plain that this assumption can be weakened in various ways and we will still have convergence.
In analogy with (4.1) we have the following.

Proof of Theorem 3
We turn now to the proof of Theorem 3, again assuming Proposition 1. As before, the case d > 0 is harder. Similarly to (4.7) we have The factor of 2 in (6.1) is due to the fact that the sum is restricted to a > 0. As before let we get that Making the substitution u = tan θ 2 so that cos θ = 1−u 2 1+u 2 , sin θ = 2u 1+u 2 and dθ = 2 1+u 2 du yields
Proof. The integral can be evaluated easily using partial fractions.
We must split the evaluation of I Q = I Q (1, 1, 1) in (6.2) into four cases depending on the signs of AA and BB . Denote by I ±,± Q the corresponding value of I Q according to these signs. We have .
(6.4) By (6.1), (6.4), Proposition 1 and (4.6) we have All sums are over a, b, c with a > 0 and satisfying b 2 − 4ac = d. The positivity of the terms and the convergence of the infinite series follows from that of the Poincaré series and (6.1). The finite sum R 1 (d) may be simplified.
Lemma 3. For R 1 (d) in (6.6) and all discriminants d > 0 we have the identity the sum being over all Zagier reduced forms of discriminant d, which is a finite sum.
Proof. Since a > 0, elementary calculations give that, (2a − b) 2 < d if and only if a + c < b and that b 2 > d if and only if c > 0. Hence for the first sum in (6.6) we have which is the second sum in (6.6).
which finishes the proof.
Theorem 3 now follows from (6.5) since the conditions in the infinite sum in (6.5) a > 0, b 2 > d and (2a + b) 2 > d are equivalent to a, c, a + b + c > 0.

Point-pair invariant
In this section we will prove Proposition 1 and hence finish the proofs of Theorems 2 and 3. Instead of following Hurwitz we will obtain it as a simple consequence of Selberg's theory [Se] of point-pair invariants.
Recall from [Se] that an eigenfunction φ of the Laplacian ∆ is also an eigenfunction of the invariant integral operator Therefore Γ\H K(τ, z)dµ(z) = c is constant since 1 is an eigenfunction of ∆ and it is easy to compute that c = π. Proposition 1 follows from the next lemma.
Proof. Let F 1 = {z = x + iy : 0 ≤ x ≤ 1 2 and |z − 1| ≥ 1} be the hyperbolic triangle with vertices at 0, e πi/3 and ∞. Note that F 1 is obtained from the standard closed fundamental domain F for Γ by mapping the left-hand half of F to the hyperbolic triangle with corners at 0, i and e πi/3 by using the inversion S = ( 0 −1 1 0 ) . Thus F 1 is also a closed fundamental domain for Γ. Let where T = ± ( 1 1 0 1 ). Any τ ∈ H can be expressed uniquely as τ = τ Q for a positive definite real Q with disc(Q) = 1. For this Q we have by (4.3). A straightforward calculation using (4.2), (7.1) and (7.2) when z = x + iy and Q = [a, b, c] has disc(Q) = 1 gives k(τ Q , z) = y 3 c + bx + a(x 2 + y 2 ) −3 .
This completes the proofs of Theorems 2 and 3.

Proof of Formula (3.5)
The constant linear combinations of Poincaré series we get can be used in the indefinite case as well. Thus in order to prove formula (3.5) we apply the following analog of Lemma 2.