Approximations to π via the Dedekind eta function

J.M. Borwein Math Dept., Simon Fraser Univ., Burnaby, B.C, V5A 1S6 Canada F.G. Garvan^{∗}Math. Dept., University of Florida, Gainesville, FL 32611 March 27, 1996

Contents

1 Introduction 88 2 The function α(r) 89 3 The function α_{p} 90 4 The Symbolic Search for Iterations 91

4.1 Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Modular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Iteration Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 The Quadratic and Quartic Iterations 96 6 The Cubic 100 7 The Septic 102 8 Nonic Iterations 106 9 A Sixteenth Order Iteration 108 10 Appendix: Modular Forms 109 11 Appendix: Class Number 110 Abstract. Arguably the most efficient algorithm currently known for the extended precision calculation of π is a quartic iteration due to J.M. and P.B. Borwein. In their paper, the Borwein’s show how this iteration and others are intimately connected to the work of Ramanujan. This connection is shown utilizing their alpha-function which is defined in terms of theta-functions. They are able to find p-th order iterations based on this function using modular equations for the theta-functions. In this paper we construct an infinite family of functions α_{p}. Each α_{p} gives rise to a p-th order iteration. For p = 4 we obtain a quartic iteration due to the Borweins but not the one that comes from the alpha-function. For p = 3 we obtain a cubic iteration due to the Borweins that does not come from the alpha-function. For p = 7 we find a septic iteration that is analogous to the cubic iteration. For p = 9 we obtain a nonic (ninth order) iteration that does not seem to come from iterating the cubic twice. Our method depends on using the computer and a symbolic algebra package to find and solve certain modular equations.

∗The author was supported in part by NSF Grant DMS-9208813.

1 Introduction

In Bailey, Borwein, and Borwein’s paper [9] an overview of a method is given for constructing series and algorithms for finding rapid approximations for π. Although Ramanujan did not know these algorithms many of the key ingredients are in his notebooks [14]. The algorithms depend crucially on the solvable forms of certain modular equations for the theta-functions due to Ramanujan. In [9] two algorithms are given – one quartic and one quintic algorithm. In a related paper [6] a septic algorithm is sketched. In [5] a general method is given for constructing p-th order algorithms. These algorithms involve defining a sequence {α^{(n)}}_{n=1}^{∞} recursively and for which α^{(n)} converges to 1/π to high order. In general, for us, p-th order convergence of a sequence {α^{(n)}}_{n=1}^{∞} to α^{(∞)} means that α^{(n)} tends to α^{(∞)} and that

|α^{(n+1)} − α^{(∞)}| ≤ C|α^{(n)} − α^{(∞)}|^{p}  (1.1)

for some constant C > 0. The proof of p-th order convergence depends crucially on identifying α^{(n)} as the value of a certain function α(·), which can be defined in terms of elliptic integrals or equivalently in terms of theta-functions.

Since [9] was first written, Borwein and Borwein [7] found an amazing cubic algorithm. See also [8]. This algorithm comes from a certain hypergeometric analog of elliptic integrals that was studied by Ramanujan. In this paper, we make an attempt to unify some of these results and find new algorithms. Instead of a fixed function α(·), we define an infinite family of functions α_{p}(·) for p > 1. Our goal is to construct for each p, a p-th order iteration which converges to 1/π, using the function α_{p}(·).

In Section 2 we briefly describe the Borwein and Borwein α-function^{1} In Section 3 we define α_{p}(·) (for each p > 1), in terms of Dedekind’s^{2} eta

Approximations to π via the Dedekind eta function 89

function. We find that α_{p}(·) satisfies a nice modular transformation property, and a nice p-th order modular equation. In Section 4 we show how the results in Section 3 may be used to construct p-th order iterations which converge to 1/π. The method is illustrated with some maple sessions. In Section 5 we give a brief overview of how our method relates to known quadratic, cubic and quartic Borwein and Borwein iterations. Details are given how the cases p = 2, 4 relate to the quadratic and quartic algorithms. In Section 6 we show how the case p = 3 gives the Borwein and Borwein cubic algorithm.

1It is very easy to compute α for r > 0. One uses the functional equation α(1/r) = ^{\frac{\sqrt{(}r)−α(r)}{r}} for r < 1 and a truncation of the theta expansion [2.4]. Two terms suffice to obtain 10 digits on r > 1.

2URL for biographical information on Dedekind: http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Dedekind.html.

Our main goal in this paper is to somehow mimic the Borwein and Borwein cubic algorithm and obtain analogous higher order algorithms. In Section 7 we obtain an explicit solvable septic iteration which converges to 1/π. In Section 8 we obtain an explicit solvable nonic (ninth order) iteration which converges to 1/π. This nonic iteration does not appear to come from iterating the cubic twice.

Given the organic nature of this document, we hope, in a later version of this paper, to provide more complete details and improvements of the septic and nonic algorithms. In a later version we will also include some mixed order algorithms. For instance, although the case p = 2 leads naturally to a quadratic iteration we may instead use it to construct a new cubic iteration.

2 The function α(r)

Before we can define the alpha-function we need the following classical theta functions:

θ_{2}(q) := ∑_{n=−∞}^{∞} q^{(n+ \frac{1}{2} )^{2}},  (2.1)

θ_{3}(q) := ∑_{n=−∞}^{∞} q^{n^{2}},  (2.2)

θ_{4}(q) := ∑_{n=−∞}^{∞}(−1)^{n}q^{n^{2}}.  (2.3)

The alpha-function can be defined as

α(r) := \frac{\frac{1}{π} − 4\sqrt{r}q \frac{θ̇_{4}}{θ_{4}}}{θ_{3}^{4}} (where q := exp(−π\sqrt{r})).  (2.4)

As r tends to infinity we see that q tends to zero so that we have

lim_{r→∞} α(r) = \frac{1}{π} .  (2.5)

In [9, Theorem 3, p. 215] Borwein, Borwein and Bailey are able to express α(p^{2}r) in terms of α(r) and various theta functions. Utilizing p-th order modular equations for the theta functions, they then are able to construct p-th order iterations that converge to \frac{1}{π}. In the next section we show how to construct p-th order iterations in a different way. Instead of a single alpha-function we construct an infinite family of α_{p}.

3 The function α_{p}

In this section we construct an infinite family of functions α_{p}, where p is any integer greater than 1. Theoretically it is possible to find an update to any order; ie. an equation relating α_{p}(N^{2}r) with α_{p}(r). We will find that this relation is particularly nice when N = p. This will give rise to p-th order iterations with a nice form. Our functions are constructed from the Dedekind eta function instead of the theta functions.

Let q := exp(2πiτ) (with ℑτ > 0). As usual the Dedekind eta function is defined as

η(τ) := exp(πiτ/12) ∏_{n=1}^{∞}(1 − exp(2πinτ))  (3.1)

= q^{1/24} ∏_{n=1}^{∞}(1 − q^{n})  (3.2)

Then

η(−1/τ) = \sqrt{\frac{τ}{i}} η(τ).  (3.3)

See [13, p. 121] for a proof. Now for p > 1 (a positive integer) we define

B_{p}(r) := \frac{η^{p}(τ)}{η(pτ)},  C_{p}(r) := \frac{η^{p}(pτ)}{η(τ)}.  (3.4)

where τ = i\sqrt{r}/\sqrt{p} and q = exp(−2π\sqrt{r}/\sqrt{p}). It should be noted that the functions B_{3} and C_{3} occured naturally in the Borwein-Borwein cubic iteration [7], [8]. Define

α_{p}(r) := \frac{(\frac{1}{π} − q \frac{8\sqrt{r}}{(p−1)\sqrt{p}} \frac{Ḃ}{B} )}{A_{p}(r)},  (3.5)

where

A_{p}(r) := q (\frac{24}{p^{2} − 1}) \{\frac{Ċ}{C} − \frac{Ḃ}{B} \} .  (3.6)

Here Ḃ = \frac{dB}{dq}. From (3.2) we have

A_{p}(r) = 1 + O(q),  (3.7)

and

A_{p}(1/r) = rA_{p}(r),  (3.8)

which follows from (3.3). The definition of α_{p} was chosen so that it had a form analogous to that of (2.4) and that it satisfied a transformation like (3.9) below. Using (3.3) and (3.8) it is not hard to show that

α_{p}(1/r) = \frac{\frac{(p+1)}{3\sqrt{p}} \sqrt{r} − α_{p}(r)}{r}.  (3.9)

Substituting r = 1 gives

α_{p}(1) = \frac{p + 1}{6\sqrt{p}},  (3.10)

Since q → 0 as r → ∞ we see that

lim_{r→∞} α_{p}(r) = \frac{1}{π} .  (3.11)

Theorem 3.1 Let N, p ≥ 1 be fixed. We have

α_{p}(N^{2}r) = α_{p}(r)m_{N,p}(r) + \sqrt{r}ϵ_{N,p}(r),  (3.12)

where

ϵ_{N,p} = \frac{p + 1}{3\sqrt{p}} \{\frac{q \frac{Ḃ}{B} − Nq^{N} \frac{Ḃ}{B} (q^{N})}{q^{N} \frac{Ċ}{C} (q^{N}) − q^{N} \frac{Ḃ}{B} (q^{N})}\} ,  (3.13)

and

m_{N,p} = \frac{A_{p}(r)}{A_{p}(N^{2}r)}.  (3.14)

Further

A_{p} = \frac{1}{p − 1} [pP (q^{p}) − P (q)] ,  (3.15)

where

P (q) := 1 − 24 ∑_{n=1}^{∞} \frac{nq^{n}}{1 − q^{n}} = 24q \frac{η̇}{η} .  (3.16)

Proof 3.1 The statement (3.12) follows easily from (3.5) and (3.6). Equation (3.15) follows easily from the product expansion (3.2) and the definition in (3.4).

When N = p, the function ϵ_{N,p} has a nice form

ϵ_{p,p}(r) = \frac{\sqrt{p}}{3} (1 − m_{p,p}(r)) ,  (3.17)

so that

α_{p}(p^{2}r) = α_{p}(r)m_{p,p}(r) + \frac{\sqrt{rp}}{3} (1 − m_{p,p}(r)) .  (3.18)

The proof of (3.17) follows easily from (3.2), (3.4) and (3.15). From (3.8) and (3.14) we have

m_{p,p}(1/p) = p.  (3.19)

By using (3.18) and (3.19) we find that (3.9) and (3.18) give rise to two equations involving α_{p}(p) and α_{p}(1/p). These equations may be solved easily to yield

α_{p}(p) = α_{p}(1/p) = \frac{1}{3}  (Not a bad starting point for 1/π).  (3.20)

4 The Symbolic Search for Iterations

In this section we show how a computer algebra package like maple can be used to generate p-th order iterations that converge to 1/π from the α_{p}. For a fixed initial value r_{0} and a fixed p we define the sequence {α_{n}}_{n=1}^{∞} by

α_{n} := α_{p} (r_{0} p^{2n}) .  (4.1)

Of course, as it stands, although it is clear that α_{n} converges to 1/π to p-th order, it is not very practical. The idea is to write the sequence recursively. Therefore there are two problems to solve:

1. Find initial values α_{p}(r_{0}).

2. Get α_{n} in terms of α_{n−1}. Table 1 contains some functions we defined in maple and that we used to find and prove initial values and iterations.

Table 1: maple functions

maple function Description
etaq(q,k,t)  q-series expansion of ∏_{j=1}^{∞}(1 − q^{kj}) to order O (q^{t})
etamake(f,t)  η-product expansion of q-series f to order O (q^{t})
findhom(L,n,t) Finds a set of homogeneous relations of degree n

satisfied by the set L of q-series to order O (q^{t})

Our function etaq utilizes the expansion due to Euler:

∏_{j=1}^{∞}(1 − q^{kj}) = ∑_{n=−∞}^{∞}(−1)^{n}q^{3kn(n−1)/2}  (4.2)

We are then able to effectively compute q-series expansions of B_{p}, C_{p}, A_{p}. We have written maple procedures to compute all the necessary functions.

4.1 Initial values

We illustrate a search for initial values with a maple^{3} session. The file funcs contains all the necessary maple functions.

> read funcs:
> readlib(lattice):
> Digits := 30:
> alpha(7,1);

.503952630678969636286022048314

> x:=":
> minpoly(x,2);

−16 + 63 X^{2}

> alpha(7,1/7);

.333333333333333333333333333334

> x:=alpha(2,3);

x := .323697301100981642294746628421

3See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple1.mpl for the associated Maple form interface.

> minpoly(x,4);

−49 + 528 X^{2} − 576 X^{4}

> solve(");

\frac{\sqrt{6}}{4} + \frac{\sqrt{3}}{6}, \frac{\sqrt{6}}{4} − \frac{\sqrt{3}}{6}, −\frac{\sqrt{6}}{4} + \frac{\sqrt{3}}{6}, −\frac{\sqrt{6}}{4} − \frac{\sqrt{3}}{6}

> evalf(alpha(2,3)-(3*sqrt(6)-2*sqrt(3))/12,100);

−.4 10^{−99}

We have used maple’s implementation of the lattice algorithm^{4} The maple function minpoly(x,n) finds a polynomial of degree n with small integer coefficients satisfied by the real floating point number x. The output depends on the number of digits computed for the approximation x. In our session the function alpha(p,r) corresponds to α_{p}(r). Observe that we were able to verify (3.10) and (3.20) for p = 7. We found a nice quartic polynomial which appears to be satisfied by α_{2}(3) and that it appears that

α_{2}(3) = \frac{3\sqrt{6} − 2\sqrt{3}}{12}.  (4.3)

Once we have such a conjectured value for α_{p}(r_{0}) it is possible to prove the result by computing to enough digits. In this way we were able to find many initial values. These are given in Table 2.

Table 2: Initial values α_{p}(r_{0})

\begin{matrix}p & r_{0} & α_{p}(r_{0}) & p & r_{0} & α_{p}(r_{0}) \\ 2 & 3 & (3\sqrt{6} − 2\sqrt{3})/12 & 4 & 2 & (3\sqrt{2} − 2)/6 \\ 2 & 4 & (3\sqrt{2} − 2)/7 & 4 & 3 & (10\sqrt{3} − 9)/24 \\ 2 & 5 & (3\sqrt{10} − 4\sqrt{2})/12 & 5 & 2 & (6\sqrt{10} − 5\sqrt{2})/30 \\ 2 & 6 & (13\sqrt{3} − 12)/33 & 5 & 3 & (21\sqrt{15} − 25\sqrt{3})/105 \\ 2 & 6 & (13\sqrt{3} − 12)/33 & 6 & 2 & (21\sqrt{3} − 20)/39 \\ 3 & 2 & (2\sqrt{6} − \sqrt{3})/9 & 6 & 3 & (90\sqrt{3} − 17\sqrt{2})/348 \\ 3 & 4 & (4\sqrt{3} − 4)/9 & 7 & 3 & (12\sqrt{21} − 14\sqrt{7} + 7)/63 \\ 3 & 5 & (10\sqrt{15} − 14\sqrt{3})/45 & 7 & 4 & (200\sqrt{7} − 336)/525 \\ 3 & 6 & (28\sqrt{2} − 9\sqrt{3})/75 & 9 & 2 & (45\sqrt{2} − 14\sqrt{3})/81 \\ 3 & 7 & (2\sqrt{21} − 2\sqrt{7} − 1)/9 & 9 & 3 & \sqrt{49/27 + (1/3)\sqrt[3]{16} − (14/9)\sqrt[3]{4}} \end{matrix}

4.2 Modular Equations

The problem is to find a recurrence relation for the α_{n}, defined in (4.1). An examination of (3.18) reveals that we need to get m_{p,p}(p^{2}r) in terms of m_{p,p}(r). From (3.14) we see that this is equivalent to finding a relation between A_{p}(q), A_{p}(q^{p}) and A_{p}(q^{p^{2}}). Here we have written A_{p} as a function of q and not r. By using (3.15), it can be shown that A_{p}(q) is a modular form^{5} of weight two on a certain congruence subgroup.^{6}

4See the Bailey/Plouffe paper within this volume ‘Recognizing Numerical Constants’.

This implies that all three functions A_{p}(q), A_{p}(q^{p}) and A_{p}(q^{p^{2}}) are modular forms of weight two on a certain congruence subgroup and hence must satisfy an algebraic relation of the form

P (A(q), A(q^{p}), A(q^{p^{2}})) = 0,  (4.4)

where P is a certain rational homogeneous polynomial of degree k, say. The crucial observation is that the left-hand side of (4.4) is a modular form of weight 2k. It is well-known that

The dimension of the space of modular forms (above) of weight 2k ∼ c_{1} k

(for some nonzero constant c_{1}, see [11])

and

the number of monomials A_{p}^{k_{1}}(q) A_{p}^{k_{2}}(q^{p})A_{p}^{2k−k_{1}−k_{2}}(q^{p^{2}}) ∼ 2k^{2}

Hence there will always be a relation for larger enough k. Such a relation can be found and proved symbolically. This is really a linear problem. The q-series expansion of each monomial up to a certain power of q can be easily computed and stored as a column in a matrix. Then finding homogeneous relations is then equivalent to finding the nullspace of a certain matrix. Such relations can be proved by verifying them to a high enough power of q using the theory of modular forms. See [8] for more details.

We illustrate the case p = 2 with a maple^{7} session.

> read funcs:
> read findhom:
> A2:=Aseries(2);

A2 := q ⟶ 1 + 24q + 24q^{2} + ⋯ + 3744q^{99}

Γ(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ : a ≡ d ≡ 1 (mod N), c ≡ b ≡ 0 (mod N)\} .

Γ_{0}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ : c ≡ 0 (mod N)\} ,

Γ_{1}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) : a ≡ 1 (mod N)\} .

5See appendix for a definition of modular forms.

6We let Γ denote the classical modular group, i.e. the group of 2 × 2 matrices with integer coefficients with determinant +1. Let N be a positive integer. We define

A subgroup of Γ is called a congruence subgroup of level N if it contains Γ(N). The most important congruence subgroups are

7See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple2.mpl for the associated Maple form interface.

> findhom([A2(q),A2(q^{∧}2),A2(q^{∧}4)],2,100);

# of terms, 22

-----RELATIONS-----of order---,2

{X_{1}^{2} − 2X_{1}X_{2} − 7X_{2}^{2} − 8X_{2}X_{3} + 16X_{3}^{2}}

We now check the relations to O(q^{∧}t)

---RELATION----, 1, ---checks to order---

O(q^{100})

The function Aseries(2) gives the q-series expansion of A_{2}(q). Using our function findhom we found that x = A_{2}(q), y = A_{2}(q^{2}), z = A(q^{4}) seem to satisfy the equation

x^{2} − 2xy − 7y^{2} − 8yz + 16z^{2} = 0,  (4.5)

at least to O(q^{100}), which is enough for a proof. By solving this equation we see that

m_{2,2}(q^{2}) = \frac{4}{1 + \sqrt{(4 − m_{2,2}(q))(2 + m_{2,2}(q))}}.  (4.6)

4.3 Iteration Construction

Once we have m_{p,p}(p^{2}r) in terms of m_{p,p}(r) constructing a p-th order iteration is a simple matter. We illustrate the construction for p = 2. We take r_{0} = 1/2 as our initial value. We could have used any value from Table 2. We define two sequences {α_{n}}_{n=0}^{∞} and {m_{n}}_{n=0}^{∞}:

α_{n} := α_{2}(2^{2n−1}),  (4.7)
m_{n} := m_{2,2}(2^{2n−1}).  (4.8)

Here we consider m_{2,2} as a function of r. From (3.19) and (3.20) we have

α_{0} := \frac{1}{3},  (4.9)
m_{0} := 2.  (4.10)

From (3.18) and (4.6) we have for n ≥ 1

m_{n} = \frac{4}{1 + \sqrt{(4 − m_{n−1})(2 + m_{n−1})}},  (4.11)

α_{n} = m_{n−1} α_{n−1} + \frac{2^{n−1}}{3} (1 − m_{n−1}).  (4.12)

Equations (4.9)–(4.12) uniquely define the two sequences {α_{n}}_{n=0}^{∞} and {m_{n}}_{n=0}^{∞}. The convergence is quadratic. We illustrate with a maple ^{8} session.

8See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple3.mpl for the associated Maple form interface.

> read iteration1:
> it1(5,100);

\begin{matrix}1, & 3.00 … 0, & .141593 \\ 2, & 3.1407 …, & .838172 10^{−3} \\ 3, & 3.141592646 …, & .676494 10^{−8} \\ 4, & 3.14159265358979323828 …, & .175696 10^{−18} \\ 5, & 3.1415926535897932384 … 884197115 …, & .536215 10^{−40} \end{matrix}

The function it1(5,100) computes the first 5 iterations to 100 digits and an approximation for the difference between 1/α_{n} and π. We also note that this iteration is really a known quadratic iteration [7, Iteration 3.6, p. 700].

5 The Quadratic and Quartic Iterations

In the previous section we showed how to construct a quadratic iteration from the case p = 2. In this section we show how the cases p = 2 and p = 4 relate to the Borwein’s quadratic and quartic iterations. For an overview of the method see [9]. For more complete details see [5], [7]. In this and the next section we summarise the results needed and omit the proofs. Before stating these results, we first give a brief overview.

In the previous section we saw how a quadratic iteration can be constructed using the modular equation (4.6) which relates m_{2,2}(q^{2}) and m_{2,2}(q). It is useful it introduce auxiliary functions. These functions are modular forms and are usually denoted by a(q), b(q) and c(q). In each case, these auxiliary functions have some remarkable properties:

(1) a(q) is some k-th root of A_{p}(q)

(2) There exist linear relationships between the functions a(q), b(q), c(q), a(q^{p}), b(q^{p}) and c(q^{p}).

(3) There is a simple polynomial equation relating a(q), b(q) and c(q). In fact, for the quadratic and quartic iterations of this section and the cubic iteration of the next section this relation is precisely

a^{p} = b^{p} + c^{p}.  (5.1)

Instead of defining a sequence {m_{n}}_{n=0}^{∞} which comes from the function m_{p,p} we define two auxiliary sequences {s_{n}}_{n=1}^{∞} and {s_{n}^{∗}}_{n=1}^{∞} which will come from the functions

s(q) := \frac{c(q)}{a(q)},  s^{∗}(q) := \frac{b(q)}{a(q)}.  (5.2)

In this way, (5.1) becomes

s^{p} + (s^{∗})^{p} = 1.  (5.3)

It will turn out that the multiplier m_{p,p} can be written simply in terms of s(q^{p}).

Approximations to π via the Dedekind eta function 97

We now consider the first case p = 2. By [5, p.698] we have

A_{2}(q) = 2P (q^{2}) − P (q) = θ_{3}^{4}(q) + θ_{2}^{4}(q).

Following [5] we define

a(q) := θ_{3}^{4}(q) + θ_{2}^{4}(q),  (5.4)
b(q) := θ_{4}^{4}(q),  (5.5)
c(q) := 2θ_{2}^{2}(q)θ_{3}^{2}(q).  (5.6)

Then we have the following results

a^{2} = b^{2} + c^{2},  (5.7)
m_{2,2}(r) = \frac{a(q)}{a(q^{2})},  (5.8)
a(q^{2}) = \frac{a(q) + 3b(q)}{4},  (5.9)
a(q) = a(q^{2}) + 3c(q^{2}).  (5.10)

Now we let

s(q) := \frac{c(q)}{a(q)}, s^{∗}(q) := \frac{b(q)}{a(q)}. (5.11)

Then from (5.7), (5.9)–(5.11) we have

1 + 3s(q^{2}) = 1 + 3\frac{c(q^{2})}{a(q^{2})} = \frac{a(q)}{a(q^{2})},  (5.12)
1 + 3s^{∗}(q) = 1 + 3\frac{b(q)}{a(q)} = 4\frac{a(q^{2})}{a(q)},  (5.13)

and

s^{2} + (s^{∗})^{2} = 1.  (5.14)

Now, by (5.8),(5.12) and (5.13) we have

m_{2,2}(r) = [1 + 3s(q^{2})] ,  (5.15)
[1 + 3s(q^{2})] [1 + 3s^{∗}(q)] = 4,  (5.16)

and, by (3.18), and (5.15) we have

α_{2}(4r) = α_{2}(r)m_{2,2}(r) + \frac{\sqrt{2r}}{3} (1 − m_{2,2}(r))  (5.17)

= α_{2}(r) [1 + 3S(4r)] − \frac{\sqrt{2r}}{3} S(4r);

where

S(r) = s(q), q = exp(−2π\sqrt{r}/\sqrt{2}). (5.18)

Now, from (3.19), (3.20) we know that

α_{2}(2) = α_{2}(1/2) = \frac{1}{3},  (5.19)
m_{2,2}(1/2) = p = 2,  (5.20)

and so

2 = 1 + 3S(2),
S(2) = \frac{1}{3}.  (5.21)

By letting,

α_{n} := α (4^{n} · 2) ,  (5.22)
s_{n} := S (4^{n} · 2) = s (exp(−2π2^{n})) ,  (5.23)
s_{n}^{∗} := S^{∗} (4^{n} · 2) ,  (5.24)

we have the following

Theorem 5.1 (Borwein and Borwein [7, Iteration 3.6, p. 700]) Define sequences {α_{n}}, {s_{n}}, and {s_{n}^{∗}} by

α_{0} := α(2) = \frac{1}{3};  (5.25)
s_{0} : = S(2) = \frac{1}{3},  (5.26)
(s_{n})^{2} + (s_{n}^{∗})^{2} = 1;  (5.27)
(1 + 3s_{n})(1 + 3s_{n−1}^{∗}) = 4;  (5.28)
α_{n} = (1 + 3s_{n})α_{n−1} − 2^{n}s_{n}.  (5.29)

Then α_{n} converges quadratically to \frac{1}{π}.

We now consider the case p = 4 and show how it is related to Borwein and Borwein quartic iteration [9, Algorithm1] and how it coincides with another Borwein and Borwein quartic iteration [7, Iteration 3.4, p. 700]. From (3.15), [5, Chapter 9], [7, Theorem 2.2] we have

A_{4}(q) = 1/3(4P (q^{4}) − P (q)) = [θ_{3}(q)]^{4} .  (5.30)

Following [5] we define

a(q) := θ_{3}(q),  (5.31)
b(q) := θ_{4}(q),  (5.32)
c(q) := θ_{2}(q).  (5.33)

Then we have the following results

a^{4} = b^{4} + c^{4},  (5.34)
m_{4,4}(r) = [\frac{a(q)}{a(q^{4})}]^{4} ,  (5.35)
a(q^{4}) = \frac{a(q) + b(q)}{2},  (5.36)
a(q) = a(q^{4}) + c(q^{4}).  (5.37)

Approximations to π via the Dedekind eta function 99 As usual we let

s(q) := \frac{c(q)}{a(q)}, s^{∗}(q) := \frac{b(q)}{a(q)}. (5.38)

Then, by (5.34), (5.36)–(5.37) we have

1 + s(q^{4}) = 1 + \frac{c(q^{4})}{a(q^{4})} = \frac{a(q)}{a(q^{4})},  (5.39)
1 + s^{∗}(q) = 1 + \frac{b(q)}{a(q)} = 2\frac{a(q^{4})}{a(q)},  (5.40)

and

s^{4} + (s^{∗})^{4} = 1.  (5.41)

Now, from (5.35), (5.39)–(5.41) we have

m_{4,4}(r) = [1 + s(q^{4})]^{4} ,  (5.42)
[1 + s(q^{4})] [1 + s^{∗}(q)] = 2,  (5.43)
s^{4}(q) + (s^{∗})^{4}(q) = 1,  (5.44)

and, by (3.18), (5.42) we have

α_{4}(16r) = α_{4}(r)m_{4,4}(r) + \frac{2\sqrt{r}}{3} (1 − m_{4,4}(r))  (5.45)

where

S(r) = s(q), q = exp(−2π\sqrt{r}/\sqrt{4}) = exp(−π\sqrt{r}).) (5.46)

From (3.19), (3.20) we know that

α_{4}(4) = α_{4}(1/4) = \frac{1}{3},  (5.47)
m_{4,4}(1/3) = p = 4,  (5.48)

and so

4 = [1 + S(4)]^{4} ,
S(4) = \sqrt{2} − 1.  (5.49)

By letting,

α_{n} := α (16^{n} · 4) ,  (5.50)
s_{n} := S (16^{n} · 4) = s (exp(−4π4^{n})) ,  (5.51)
s_{n}^{∗} := S^{∗} (16^{n} · 4) ,  (5.52)

we have the following Theorem 5.2 (Borwein and Borwein [7, Iteration 3.4, p. 700]) Define sequences {α_{n}}, {s_{n}}, and {s_{n}^{∗}} by

α_{0} := α(1/4) = \frac{1}{3};  (5.53)
s_{0} : = S(2) = \sqrt{2} − 1,  (5.54)
(s_{n})^{4} + (s_{n}^{∗})^{4} = 1;  (5.55)
(1 + s_{n})(1 + s_{n−1}^{∗}) = 2;  (5.56)
α_{n} = (1 + s_{n})^{4}α_{n−1} + \frac{4^{n}}{3} (1 − (1 + s_{n})^{4}).  (5.57)

Then α_{n} converges quartically to \frac{1}{π}.

6 The Cubic

We summarise the results of [7, 8] and how they correspond to the case p = 3. Define

a(q) := ∑_{m,n=−∞}^{∞} q^{m^{2}+mn+n^{2}},  (6.1)

b(q) := ∑_{m,n=−∞}^{∞} ω^{(m−n)}q^{m^{2}+mn+n^{2}},  (ω = exp(2πi/3)  (6.2)

c(q) := ∑_{m,n=−∞}^{∞} q^{(m+ \frac{1}{3} )^{2}+(m+ \frac{1}{3} )(n+ \frac{1}{3} )+(n+ \frac{1}{3} )^{2}}.  (6.3)

Then

A_{3}(q) = [a(q)]^{2} ,  (6.4)
b(q) = \frac{η^{3}(τ)}{η(3τ)} = B_{3},  (6.5)
c(q) = 3\frac{η^{3}(3τ)}{η(τ)} = C_{3},  (6.6)
m_{3,3}(r) = [\frac{a(q)}{a(q^{3})}]^{2} .  (6.7)

We also mention the identity

∑_{m,n=−∞}^{∞} q^{m^{2}+mn+n^{2}} = 1 + 6 ∑_{n=1}^{∞} (\frac{n}{3}) \frac{q^{n}}{1 − q^{n}} .  (6.8)

This equation was known to Ramanujan [4, p. 199]. See [3] for a proof using theta-functions.

Approximations to π via the Dedekind eta function 101

We have

a(q^{3}) = \frac{a(q) + 2b(q)}{3},  (6.9)
a(q) = a(q^{3}) + 2c(q^{3}),  (6.10)
a^{3} = b^{3} + c^{3}.  (6.11)

As usual we let

s(q) := \frac{c(q)}{a(q)}, s^{∗}(q) := \frac{b(q)}{a(q)}. (6.12)

Then, from (6.9)–(6.12) we have

1 + 2s(q^{3}) = 1 + 2\frac{c(q^{3})}{a(q^{3})} = \frac{a(q)}{a(q^{3})},  (6.13)
1 + 2s^{∗}(q) = 1 + 2\frac{b(q)}{a(q)} = 3\frac{a(q^{3})}{a(q)},  (6.14)

and

s^{3} + (s^{∗})^{3} = 1.  (6.15)

Now, from (6.7), (6.13), (6.14) we have

m_{3,3}(r) = [1 + 2s(q^{3})]^{2} ,  (6.16)
[1 + 2s(q^{3})] [1 + 2s^{∗}(q)] = 3,

and, by (3.18), (6.16) we have

α_{3}(9r) = α_{3}(r) [1 + 2S(9r)]^{2} − \frac{4\sqrt{3r}}{3} S(9r)(1 + S(9r));  (6.17)

where

S(r) = s(q), q = exp(−2π\sqrt{r}/\sqrt{3}). (6.18)

From (3.19), (3.20) we know that

α_{3}(3) = α_{3}(1/3) = \frac{1}{3},  (6.19)
m_{3,3}(1/3) = p = 3,  (6.20)

and so

S(3) = \frac{\sqrt{3} − 1}{2}.  (6.21)

By letting,

α_{n} := α (9^{n} · 3) ,  (6.22)
s_{n} := S (9^{n} · 3) = s (exp(−2π3^{n})) ,  (6.23)
s_{n}^{∗} := S^{∗} (9^{n} · 3) ,  (6.24)

we have the following Theorem 6.1 (Borwein and Borwein [7, Iteration 3.2, p. 699]) Define sequences {α_{n}}, {s_{n}}, and {s_{n}^{∗}} by

α_{0} := α(1/3) = \frac{1}{3};  (6.25)
s_{0} : = S(3) = \frac{\sqrt{3} − 1}{2};  (6.26)
(s_{n})^{3} + (s_{n}^{∗})^{3} = 1;  (6.27)
(1 + 2s_{n})(1 + 2s_{n−1}^{∗}) = 3;  (6.28)
α_{n} = (1 + 2s_{n})^{2}α_{n−1} − 4 · 3^{n−1}(1 + s_{n})s_{n}.  (6.29)

Then α_{n} converges cubically to \frac{1}{π}.

7 The Septic

In this section we examine the case p = 7 and obtain an iteration^{9} that converges to 1/π to 7-th order. The proofs of some of the results of this section have been omitted. They will appear in a later version of this organic paper. Our idea is to mimic the cubic iteration of Section 6. The quadratic form in the cubic case is n^{2} + nm + m^{2}, which has discriminant^{10} −3 and class number^{11} h = h(−3) = 1. The case p = 7 is the next case to consider since the next odd, negative discriminant with class number 1 is −7. The corresponding quadratic form is 2n^{2} + 3mn + 2n^{2}. See [10] for a treatment of the classical theory of binary quadratic forms. We have

A_{7}(q) = [a_{7}(q)]^{2}  (7.1)

where

a_{7}(q) = 1 + 2 ∑_{n=1}^{∞} (\frac{n}{7}) \frac{q^{n}}{1 − q^{n}}  (7.2)

= ∑_{m,n=−∞}^{∞} q^{2n^{2}+3mn+2m^{2}}.  (7.3)

Also,

[a_{7}(q)]^{3} = \frac{η^{7}(τ)}{η(7τ)} + 49\frac{η^{7}(7τ)}{η(τ)} + 13η^{3}(τ)η^{3}(7τ)  (7.4)
= B_{7} + 49C_{7} + 13\sqrt{B_{7}C_{7}}
= [\sqrt{B_{7}} + 7\sqrt{C_{7}}]^{2} − \sqrt{B_{7}C_{7}}.

9See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple5.mpl for the associated Maple form interface.

10The binary quadratic form

f(x, y) = ax^{2} + bxy + cy^{2} has discriminant Δ = b^{2} − 4ac. For example, the form

2x^{2} + 3xy + 2y^{2} has discriminant Δ = −7.

11See appendix for more information on class number.

Equation (7.1) with (7.2), and (7.4) are contained in Entry 5(i) of Chapter 21 of Ramanujan’s second notebook [14], [2, p. 467]. Equation (7.3) is analogous to (6.8) but does not appear in Ramanujan’s notebooks. We expect a result like (7.3) to hold since the class number h(−7) = 1. A result of Legendre’s [12, Equation (1), p. 97] gives an equivalent formulation in terms of L-series ^{12}:

2 ∑_{n=1}^{∞} \frac{1}{n^{s}} ∑_{m=1}^{∞} (\frac{m}{7}) \frac{1}{m^{s}} = ∑_{(m,n)≠(0,0)} \frac{1}{(2n^{2} + 3mn + 2m^{2})^{s}} .  (7.5)

We define the following functions (analogous to the cubic case):

a := a(q) = ∑_{m,n=−∞}^{∞} q^{2n^{2}+3mn+2m^{2}},  (7.6)

and for j = 1, 2, 3

b_{j} := ∑_{m,n=−∞}^{∞} ω^{j(m−n)}q^{2n^{2}+3mn+2m^{2}},  (ω = exp(2πi/7))  (7.7)

c_{j} := ∑_{m,n=−∞}^{∞} q^{2(n+ \frac{j}{7} )^{2}+3(m+ \frac{j}{7} )(n+ \frac{j}{7} )+2(m+ \frac{j}{7} )^{2}}.  (7.8)

In a later version of this paper we will examine these functions under the action of the congruence subgroup, Γ_{0}(7).

The proof of the following is analogous to the cubic case [8, Lemma 2.1, p. 36]:

a(q^{7}) = \frac{1}{7}(a(q) + 2(b_{1}(q) + b_{2}(q) + b_{3}(q)),  (7.9)
c_{1}(q^{7}) = \frac{1}{7}(a + (ω^{2} + ω^{5})b_{1} + (ω^{3} + ω^{4})b_{2} + (ω + ω^{6})b_{3})  (7.10)
c_{2}(q^{7}) = \frac{1}{7}(a + (ω^{3} + ω^{4})b_{1} + (ω + ω^{6})b_{2} + (ω^{2} + ω^{5})b_{3})  (7.11)
c_{3}(q^{7}) = \frac{1}{7}(a + (ω + ω^{6})b_{1} + (ω^{2} + ω^{5})b_{2} + (ω^{3} + ω^{4})b_{3}).  (7.12)

From (7.9)–(7.12) we easily find that

c_{1}(q^{7}) + c_{2}(q^{7}) + c_{3}(q^{7}) = \frac{1}{7}(3a − b_{1} − b_{2} − b_{3})  (7.13)
= \frac{1}{2}(a(q) − a(q^{7})).  (7.14)

For j = 1, 2, 3 we define

s_{j} := \frac{c_{j}}{a} ,  (7.15)
s_{j}^{∗} := \frac{b_{j}}{a} ,  (7.16)

L(χ, s) := ∑_{n=1}^{∞} \frac{χ(n)}{n^{s}} = ∏_{p} \frac{1}{1 − χ(p)p^{−s}} ,

12For a given character (group homomorphism) χ : (Z/NZ)^{∗} → (C)^{∗}, there corresponds the Dirichlet L-series for ℜs > 1.

and

σ := s_{1} + s_{2} + s_{3},  (7.17)
σ^{∗} := s_{1}^{∗} + s_{2}^{∗} + s_{3}^{∗}.  (7.18)

From (3.14), (7.1) we have

m_{7,7}(r) = (\frac{a_{7}(q)}{a_{7}(q^{7})})^{2} .  (7.19)

Throughout this section q and r are related by

q = exp(−2π\sqrt{r}/\sqrt{7}).  (7.20)

If we write the s_{j} as functions of r then by using the multidimensional analogue [1] of (3.3) it can be shown that

s_{j} (\frac{1}{r}) = s_{j}^{∗}(r),  (7.21)

for j = 1, 2, 3.

By (7.14), (7.19) we have

m_{7,7}(r) = [1 + 2σ(q^{7})]^{2} ,  (7.22)
[1 + 2σ(q^{7})] [1 + 2σ^{∗}(q)] = 7.  (7.23)

As noted in Section 4.3 the main problem of constructing a 7-th order iteration is to find a relationship between m_{7,7}(49r) and m_{7,7}(r). In view of (7.22) and (7.23), we would like to find a relationship between σ(q) and σ^{∗}(q). Instead we get each of the s_{j}^{∗} in terms of s_{1}, s_{2}, s_{3}. This is achieved by solving a certain cubic and taking 7-th roots.

We now give some details of this construction. Let

α = (s_{1}^{∗})^{3} s_{2}^{∗},  β = (s_{2}^{∗})^{3} s_{3}^{∗},  γ = (s_{3}^{∗})^{3} s_{1}^{∗}.  (7.24)

Now we define the following functions:

g_{1} = s_{1}s_{2}s_{3},  (7.25)
g_{2} = s_{1}^{3}s_{2} + s_{2}^{3}s_{3} + s_{3}^{3}s_{1},  (7.26)
g_{3} = 1 − \frac{10}{7}g_{1} + \frac{1}{7}g_{2}  (7.27)
g_{4} = 3 − \frac{51}{7}g_{1} + \frac{10}{7}g_{2}.  (7.28)

Then by using the theory of modular functions it can be shown that

α + β + γ = g_{4},  (7.29)
(αβ + βγ + γα) = g_{3}(2g_{4} − 3g_{3}),  (7.30)
αβγ =(s_{1}^{∗}s_{2}^{∗}s_{3}^{∗})^{4} = g_{3}^{4}.  (7.31)

Also,

δ := s_{1}^{∗}s_{2}^{∗}s_{3}^{∗} = g_{3}.  (7.32)

Our functions g_{1} and g_{2} are related. We find that

g_{2} = \frac{7g_{1} + 6 + \sqrt{49 − (g_{1} − 1)(27g_{1} − 13)}}{2}.  (7.33)

Hence α, β, γ are the roots of the following cubic equation

x^{3} − g_{4}x^{2} + x g_{3}(2g_{4} − 3g_{3}) − g_{3}^{4} = 0.  (7.34)

Now observe that

\frac{α^{3}γ}{δ^{3}} = \frac{(s^{∗}_{1}^{3}s^{∗}_{2})^{3}(s^{∗}_{3}^{3}s^{∗}_{1})}{(s^{∗}_{1}s^{∗}_{2}s^{∗}_{3})^{3}} = s_{1}^{∗}^{7}.  (7.35)

Similarly we find that

s_{1}^{∗} = (\frac{α^{3}γ}{δ^{3}} )^{\frac{1}{7}} , s_{2}^{∗} = (\frac{β^{3}α}{δ^{3}})^{\frac{1}{7}} , s_{3}^{∗} = (\frac{γ^{3}β}{δ^{3}} )^{\frac{1}{7}} .  (7.36)

Thus after solving a certain cubic, and taking 7-th roots we may obtain each of the s_{j}^{∗} in terms of s_{1}, s_{2}, and s_{3}.

From (3.18) we have

α_{7}(49r) = α_{7}(r) m_{7,7}(r) + \frac{\sqrt{7r}}{3} (1 − m_{7,7}(r)).  (7.37)

Finally, to construct a septic iteration, we need some initial values. Two obvious candidates are r_{0} = 1, 1/7. From (3.10) we have

α_{7}(1) = \frac{4}{3\sqrt{7}}.  (7.38)

We need to find the values of the s_{j} for r = 1. Fortunately, by (7.21), these coincide with the values of the s_{j}^{∗}. So when r = 1, g_{1} = g_{3} and g_{2} = g_{4}. Thus we can in this case solve equations (7.27) and (7.33) to obtain

g_{1}(1) = \frac{13}{27},  g_{2}(1) = \frac{32}{27}.  (7.39)

Thus via (7.34) we find that α(1), β(1), γ(1) are roots of the cubic

27^{4} x^{3} − 27^{3} · 32 x^{2} + 325 · 27^{2} x − 13^{4} = 0.  (7.40)

Then after solving this cubic we can find the initial values s_{j}(1) = s_{j}^{∗}(1) using (7.36). To obtain the septic iteration we need only consider r = 49^{n} in our identities and equations to obtain the following theorem.

Theorem 7.1 Define sequences {α_{n}}, {s_{n}}, {s_{n}^{∗}}, and {m_{n}} by

α_{0} := \frac{4}{3\sqrt{7}};  (7.41)

s_{0} := (\frac{27}{13})^{3/7} ( \begin{matrix}y_{1} \\ y_{2} \\ y_{3} \end{matrix} ) ,  (7.42)

where

y_{1} = (x_{1}^{3}x_{3})^{\frac{1}{7}} ,  y_{2} = (x_{2}^{3}x_{1})^{\frac{1}{7}} ,  y_{3} = (x_{3}^{3}x_{2})^{\frac{1}{7}} ,  (7.43)

and x_{2} < x_{1} < x_{3} are the roots of the cubic equation

27^{4} x^{3} − 27^{3} · 32 x^{2} + 325 · 27^{2} x − 13^{4} = 0;  (7.44)

bfs_{n}^{∗} = ( \begin{matrix}(α^{3}γ/g_{3}^{3})^{1/7} \\ (β^{3}α/g_{3}^{3})^{1/7} \\ (γ^{3}β/g_{3}^{3})^{1/7} \end{matrix} ) ,  (7.45)

where β < α < γ are the roots of the cubic

x^{3} − g_{4}x^{2} + x g_{3}(2g_{4} − 3g_{3}) − g_{3}^{4} = 0;  (7.46)

where

g_{3} = 1 − \frac{10}{7}g_{1} + \frac{1}{7}g_{2},  (7.47)
g_{4} = 3 − \frac{51}{7}g_{1} + \frac{10}{7}g_{2},  (7.48)

and where

g_{1} = s_{1,n} s_{2,n} s_{3,n},  (7.49)
g_{2} = s_{1,n}^{3}s_{2,n} + s_{2,n}^{3}s_{3,n} + s_{3,n}^{3}s_{1,n},;  (7.50)
m_{n} = \frac{49}{(1 + 2s_{n}^{∗} · 1)^{2}} , (where 1 = (1, 1, 1)^{t});  (7.51)
s_{n} = \frac{1}{7} \sqrt{m_{n−1}} [M s_{n−1}^{∗} + 1]  (7.52)

where

M = (2 cos (\frac{4πij}{7} ))_{1≤i,j≤3} ;  (7.53)

and

α_{n} = m_{n−1} α_{n−1} + \sqrt{7} \frac{7^{n−1}}{3} ( 1 − m_{n−1}) .  (7.54)

Then α_{n} converges septically to \frac{1}{π}.

8 Nonic Iterations

In this section we explore nonic (ninth order) iterations. Of course, we may construct a ninth order iteration by iterating the cubic. We have found a nonic iteration which does not seem to come from the cubic. Our nonic iteration ^{13} is based on a symbolic discovery that A_{9}(q) is a nice eta-product^{14}:

A_{9}(q) = \frac{η^{10}(3τ)}{η^{3}(τ)η^{3}(9τ)},  (q = exp(2πiτ).  (8.1)

13See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple6.mpl for the associated Maple form interface.

14See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=img302.mpl for the associated Maple form interface.

This identity can be proved easily from results in Ramanujan’s notebook. From (3.15), (3.16) we have

A_{9}(q) = 1 + 3 ∑_{n=1}^{∞} \frac{nq^{n}}{1 − q^{n}} − 27 ∑_{n=1}^{∞} \frac{nq^{9n}}{1 − q^{9n}} ,  (8.2)

= \frac{η^{6}(3τ)}{η^{2}(τ)η^{2}(9τ)} \{η^{6}(τ) + 9η^{3}(τ)η^{3}(9τ) + 27η^{6}(9τ)\}^{1/3} ,

(by [2, Entry 7(i), Chapter 21, p. 475]),

= \frac{η^{10}(3τ)}{η^{3}(τ)η^{3}(9τ)},

by [2, Entry 1(iv), Chapter 20, p. 345]. In the last step we have used the classical cubic modular equation for the eta-function:

(1 + 9\frac{η^{3}(9τ)}{η^{3}(τ)})^{3} = 1 + 27 (\frac{η(3τ)}{η(τ)})^{12} .  (8.3)

See [8, Corollary 2.5]. Using (6.5), (6.6) A_{9}(q) may be written in terms of either b(q) or c(q) (defined in (6.2), (6.3)), which were involved in the cubic iteration. We have

A_{9}(q) = \frac{1}{9} \frac{c^{3}(q)}{c(q^{3})} = \frac{b^{3}(q^{3})}{b(q)}.  (8.4)

This time we let

s(q) = \frac{c(q^{3})}{a(q^{3})},  s^{∗}(q) = \frac{b(q^{3})}{a(q^{3})},  (8.5)

where a(q), b(q), c(q) are defined in (6.1)–(6.3). We define the following functions:

T (q) := 1 + 2s^{∗}(q),  (8.6)
U(q) := [9s^{∗}(q) (1 + s^{∗}(q) + (s^{∗}(q))^{2})]^{1/3} .  (8.7)

Using the results of Section 6 it can be shown that

s(q^{9}) = \frac{(1 − s^{∗}(q))^{3}}{(T (q) + 2U(q))(T^{2}(q) + T (q)U(q) + U^{2}(q))},  (8.8)
m_{9,9}(r) = \frac{27(1 + s(q) + s^{2}(q))}{T^{2}(q) + T (q)U(q) + U^{2}(q)},  (8.9)

where q = exp(−2π\sqrt{r}/3). We can also utilize the initial values for the cubic and obtain a nonic iteration.

Theorem 8.1 Define sequences {α_{n}}, {s_{n}}, and {s_{n}^{∗}} by

α_{0} := \frac{1}{3};  (8.10)
s_{1}^{∗} : = \frac{\sqrt{3} − 1}{2};  (8.11)
s_{1} : = (1 − (s_{1}^{∗})^{3})^{1/3};  (8.12)
α_{n} = m α_{n−1} + 3 · 9^{n−2}(1 − m),  (8.13)

108 J.M. Borwein, F.G. Garvan where

m := 27\frac{(1 + s_{n} + s_{n}^{2})}{t^{2} + tu + u^{2}},  (8.14)
t := 1 + 2s_{n}^{∗},  (8.15)
u := [9s_{n}^{∗}(1 + s_{n}^{∗} + (s_{n}^{∗})^{2})]^{1/3} ,  (8.16)
s_{n+1} = \frac{(1 − s_{n}^{∗})^{3}}{(t + 2u)(t^{2} + tu + u^{2})},  (8.17)

and

s_{n+1}^{∗} = (1 − s_{n+1}^{3})^{1/3}.  (8.18)

Then α_{n} converges nonically to \frac{1}{π}.

9 A Sixteenth Order Iteration

By combining two steps of the quartic iteration and simplifying one can obtain a 16-th order iteration^{15}. Theorem 9.1 Define sequences {α_{n}}, {s_{n}}, and {s_{n}^{∗}} by

α_{0} := \frac{1}{3};  (9.1)
s_{1} : = \sqrt{2} − 1;  (9.2)
s_{1}^{∗} :=(1 − (s_{1}^{∗})^{4})^{1/4};  (9.3)
α_{n} = 16m_{1} α_{n−1} + \frac{4^{2n−1}}{3} (1 − 12m_{2} − 4m_{1}),  (9.4)

where

m_{1} := (\frac{1 + s_{n}}{1 + s_{n}^{∗}} )^{4} ,  (9.5)
m_{2} := \frac{1}{(1 + s_{n}^{∗})^{4}} ;  (9.6)
s_{n+1} = \frac{(1 − s_{n}^{∗})^{4}}{(t + u)^{2}(t^{2} + u^{2})},  (9.7)

where

t := 1 + s_{n}^{∗},  (9.8)
u := (8s_{n}^{∗}(1 + (s_{n}^{∗})^{2}))^{1/4} ;  (9.9)

and

s_{n+1}^{∗} = (1 − s_{n+1}^{4})^{1/4} .  (9.10)

Then α_{n} converges to \frac{1}{π} with order sixteen.

15See http://www.cecm.sfu.ca/cgi-bin/organics/nmpform?mfname=localmaple7.mpl for the associated Maple form interface.

References

[1] R. Bellman, A brief Introduction to Theta Functions (Holt, Rinehart and Winston, New York, 1961).

[2] B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer, New York, 1991.

[3] B.C. Berndt, On a certain theta-function in a letter of Ramanujan from Fitzroy House, Ganita, 43 (1992), 33–43.1992), 33–43.

[4] B.C. Berndt and R.A. Rankin, Ramanujan — Letters and Commentary, Amer. Math. Soc. History of Mathematics Series, Vol. 9, 1995.

[5] J.M. Borwein and P.B. Borwein, Pi and the AGM — A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.

[6] J.M. Borwein and P.B. Borwein, Approximating π with Ramanujan’s modular equations, Rocky Mountain J. Math., 19 (1989), 93–102.

[7] J.M. Borwein and P.B. Borwein, A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), 691–701.

[8] J.M. Borwein, P.B. Borwein and F.G. Garvan, Some cubic modular identities of Ramanujan, Trans. Amer. Math. Soc., 343 (1994), 35–47.

[9] J.M. Borwein, P.B. Borwein and D.H. Bailey, Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi, Amer. Math. Monthly, 96 (1989) 201–219.

[10] D.A. Buell, Binary Quadratic Forms — Classical Theory and Modern Computations, Springer, New York, 1989.

[11] H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, in Modular Functions in One Variable VI, International Summer School of Modular Functions, Bonn 1976, Lecture Notes in Math., vol. 627, Springer, N.Y., 1976, 69–78.

[12] L.E. Dickson, The Theory of Numbers, Volume III, Quadratic and Higher Forms, Chelsea, New York, 1952.

[13] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer, New York, 1984.

[14] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

10 Appendix: Modular Forms

Let H denote the complex upper half plane. SL_{2}(Z) acts transitively on H by linear fractional transformations

γτ = \frac{aτ + b}{cτ + d}, where γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z).

Now we consider functions f : H ⟶ C. The notation f | [γ]_{k} is used to denote the function whose values at τ is (cτ + d)^{−k}f(γτ). The value of f | [γ]_{k} at τ is denoted by f(r) | [γ]_{k} so that

f(τ) | [γ]_{k} := (cτ + d)^{−k}f(γτ), for γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1).

We define modular forms and cusp forms for a congruence subgroup Γ′ ⊂ Γ(1) of level N. Let q_{N} denote e^{2πiτ/N} . Let k ∈ Z. A function f : H ⟶ C is said to be a modular form of weight k for Γ′ if it satisfies the following conditions:

(i) f is holomorphic on H,

(ii) f | [γ]_{k} = f for all γ ∈ Γ′,

(iii) f is holomorphic at the cusps γ(∞) for γ ∈ Γ(1); i.e. f(τ) | [γ]_{k}has the form∑_{n≥0} a_{n}q_{N}^{n}.

We let M_{k}(Γ′)denote the set of such modular forms. It turns out that M_{k}(Γ′) is a finite dimensional C-vector space. If a_{0} = 0 for all γ ∈ Γ(1) we say f is a cusp form. We let S_{k}(Γ′) denote the subspace of cusp forms.

Now we define what it means for a function to be a modular form of weight k and a character χ for the group Γ_{0}(p) when p is prime. Let χ be a Dirichlet character modulo p

χ : (Z /pZ)^{×} ⟶ C^{×}, with χ(−1) = (−1)^{k}.

A function f : H ⟶ C is said to be a modular form of weight k and character χ forΓ_{0}(p) if it satisfies the conditions (i) – (iii) above except (ii) is replaced by

f | [γ]_{k} = χ(d) f for all γ ∈ Γ_{0}(p).

In this case, condition (iii) amounts to the following two conditions:

\begin{aligned}f(τ) & = ∑_{n=0}^{∞} a_{n}q^{n},  q = e^{2πiτ} \\ τ^{−k}f(−1/pτ) & = ∑_{n=0}^{∞} b_{n}q^{n},  q = e^{2πiτ} . \end{aligned}

We call the Fourier series in (1) the q-expansion of f at ∞, and we call (2) the q-expansion at 0. We let M_{k}(p, χ) denote the spaace of such modular forms. If a_{0} = b_{0} = 0 above then f is a cusp form. We let S_{k}(p, χ) denote the subspace of cusp forms.

11 Appendix: Class Number

We let Γ denote the classical modular group, i.e. the group of 2 × 2 matrices with integer coefficients with determinant +1. The elements of Γ act on a given binary quadratic form in a natural way. Consider the form

f(x, y) = ax^{2} + bxy + cy^{2}.

The matrix

A = ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

acts on f as follows. Under the transformation

x = αx′ + βy′  (11.1)
y = γx′ + δy′  (11.2)

the form f is transformed into the form

f′(x′, y′) = a′x′^{2} + b′x′y′ + c′y′^{2}

where

a′ = aα^{2} + bαγ + cγ^{2},  (11.3)
b′ = bαδ + 2cγδ + 2aαβ + bβγ,  (11.4)
c′ = cδ^{2} + aβ^{2} + bβδ,  (11.5)

and f′ has discriminant

Δ′ = b′^{2} − 4a′c′ = (det A)^{2} Δ = Δ.

This defines an equivalence relation f ∼ f′. Also note that the discriminant is invariant. The number of equivalence classes of forms of a given discriminant Δ is called the class number h = h(Δ). Below is table of class numbers for forms with negative discriminant.

Δ h Representatives of the equivalence classes
-3 1 x^{2} + xy + y^{2}
-4 1 x^{2} + y^{2}
-7 1 x^{2} + xy + 2y^{2}
-8 1 x^{2} + 2y^{2}
-11 1 x^{2} + xy + 3y^{2}
-12 1 x^{2} + 3y^{2}
-15 2 x^{2} + xy + 4y^{2}, 2x^{2} + xy + 2y^{2}
-16 1 x^{2} + 4y^{2}
-19 1 x^{2} + xy + 5y^{2}
-20 2 x^{2} + 5y^{2}, 2x^{2} + 2xy + 3y^{2}