BIBLIOGRAPHICAL NOTES

The bibliographical entries in the main body of the text contained only references to books that filled in gaps in our presentation or described prerequisites. The main purpose of these notes is to give the reader an idea of the many relevant books and scientific manuscripts in the literature on one variable theta functions as related to the topics of the book. It is incomplete and will be maintained and updated periodically on this web site.

We (our respective e-mail addresses are

                 farkas@sunset.huji.ac.il

and

                  irwin@math.sunysb.edu)

would appreciate receiving corrections and additions from readers, especially from experts. The authors of the current volume are not experts in the field of combinatorial number theory and have attempted in their work to open this area to a wider audience. We have emphasized the function theoretic aspects and may have missed many points known to number theorists and combinatorialists. Edited communications to us will be posted on the web and acknowledged appropriately.

This is an expanded version of pages 513 – 525 of our book.

1. Books – background material

1.1. Analytic and other prerequisites. The basic facts on complex analysis come from [1]. We have assumed the reader is familiar with the material of a first year graduate course on the subject.

We refer the reader to [1, Ch. 7] for elementary facts concerning elliptic functions and curves. Many more complete treatments of function theory on tori are available in the literature including, for example, [67].

The necessary preliminaries about Riemann surfaces were taken from [39]. There are of course many other excellent books on this subject; among them we list only three: [108] (an English translation of an earlier classic), [83] and [49].

The prerequisites on automorphic forms and Fuchsian groups may be found in portions of [20], [81], [58] and [72]. The computations of the topological invariants associated to a subgroup of the modular group are based on [104].

In Chapter 3, we discussed automorphic forms. Although not extremely relevant for our work, we refer to two papers on the subject (in part because they deal with averaging processes, a topic central to this book): the first by one of the coauthors of this volume [73]; the second by McMullen [82]. The reader interested in number theory should consult [65].

1.2. Other approaches. There is some overlap between our treatment of theta functions and resulting identities (especially involving the partition function) and that of the number theoretic work [68].

[9] is a thorough treatment of the theory of partitions, particularly the interplay between combinatorial and analytic methods.

The book [54] on hypergeometric functions has many points of contact with Ramanujan identities and partition theory.

The encyclopedic book [11] contains chapters on q-series, partitions and the Rogers-Ramanujan identities.

Each of the books mentioned in this subsection contains an extensive bibliography.

1.3. Recent developments. Although it is not our purpose to record recent progress in Teichmüller theory or other fields encountered in our presentation but not directly related to the main topics under study, we mention that the problem of describing quasi-Fuchsian space for once punctured tori as a subset of ℂ^{2} is treated in [103] in the smooth, not complex analytic, setting.

2. Papers by other authors

2.1. Curves represented by subgroups of Γ. In §8.7 of Chapter 3, we described the curve \overline{ℍ^{2}/Γ(9)}. A refinement of the description can be found in [71].

2.2. The ȷ-function. Divisibility properties of the Fourier coefficients of the ȷ-function for the primes 5, 7 and 11 (the level 1 and 2 only for the prime 11) are found in [77]. The complete picture for the prime 11 is established in [14]. Multiplicative properties (see §2.4) of these Fourier coefficients are studied in [75].

2.3. The partition functions P_{N}. For N = 1, Ramanujan [94], [95] conjectured partition congruences for the primes 5, 7 and 11 and established the level 1 and 2 results. Watson [107] proved the conjecture for all levels n for the prime 5 and a suitably modified level n conjecture for the prime 7. More work is required for the prime 11 since the surface involved is no longer of genus 0. The level 1 congruence was proven by Winquist [109]; Atkin [14] established the general conjecture (see also the earlier manuscript [76]). The level 1 congruence for the prime 5 has probably received more attention than any other case; among the shortest proofs are [32] and [62]. Partition congruences (as well as properties of the Fourier coefficients of the ȷ-function) are studied in [17].

Among the papers studying the function P_{N} for arbitrary N are [16],

[56] and [63]; the computer connection is explored in [15].

For negative N, one obtains recursion identities rather than only congruences for the partition coefficients P_{N}(n). This case has been thoroughly investigated by M. Newman, whose papers on this and related topics include [85], [86], [87], [89], [88], [90], [91] and [92].

The arithmetic of partition (the properties of the function P = P_{1}) and recent progress ([59], [2] and [93]) are discussed in [3].

2.4. The Hecke connection. A series ∑_{n=0}^{∞} a_{n}x^{n} that converges on the unit disc is multiplicative if a_{nm} = a_{n}a_{m} whenever (n, m) = 1. The multiplicativity of Ramanujan’s τ-function (η^{24}), a theorem first proved by Mordell [84], was established in Chapter 4. In [33], the authors determine all the multiplicative products of η-functions. Two further papers should be mentioned here which connect the η-function to the representation theory of Lie algebras. One is the 1972 paper of Macdonald [78]; the second is the 1978 paper of V. Kac [66]. See also [79]. Hecke operators are discussed in portions of [67, Chs. VIII, IX and X].

One of the motivations of the authors in writing this volume was to show that η is in fact just one example of a theta function with rational characteristics and that introduction of these more general functions can be extremely useful in the theory of uniformizations of the Riemann surfaces which arise from congruence subgroups of the modular group and in the study of theta constant identities. While this was presumably known to Klein, it is only with the development of the theory of theta characteristics that it becomes an efficient procedure.

2.5. The η-function. Papers dealing principally or exclusively with the eta-function and having at least tangential connection to our work include [57], [33], [80] and [79].

2.6. Theta identities. This rich subject is studied in many papers. Among the papers dealing with short and elementary proofs of the Jacobi triple product formula and the quintuple product identity are [5] and [31]. Proofs of the latter also appear in [106], [102] and [55].

[53] is a follow up of some of the work in our book.

2.7. Combinatorial proofs and interpretations of some identities. Among the many papers on the subject are [6], [51] and [52]. An invariant of partitions (the crank) introduced by F. Dyson is investigated in [13] and [50].

3. Papers by the authors

Our interest in this subject may be traced to an early paper by one of the authors [35]. His interest in the subject reemerged as a consequence of the doctoral studies of one of his students resulting in [36] and [37].

The collaboration that led to this book started with [40] and [41], where the use of theta constants with rational characteristics was applied to the study of uniformization of surfaces represented by subgroups of the modular group. In [38] modified theta constants were introduced as a tool in the study of uniformization of surfaces represented by the prime level principal congruence subgroups; related topics are investigated in part of [47]. Our papers on partitions and the Ramanujan τ-function include [42], [74], [45], [43] and [46]. Our work that relates to Hecke operators includes [48]. The quintuple product identity is reproven in [44].

4. Original sources, conference proceedings

The primary sources for the mathematics of S. Ramanujan, the main subject of much of this book, are [61] and [97]; the Hardy lectures [60] are extremely useful in any attempt to evaluate Ramanujan’s genius. So are his notebooks [96], [98] and [21]. The last of these contains extensive scholarly and bibliographic information. Among the many conference proceedings devoted to Ramanujan’s legacy is [12]; [64] is among the latest. At least two journals, \underline{The Ramanujan Journal} and \underline{The Hardy-Ramanujan Journal} are named in his honor. Among the many books and papers that include the name Ramanujan in their title are [22], [23], [24], [25], [26], [28], [7], [8], [10], [29] and [30].

Among the many references Ramanujan’s unpublished manuscripts is [27].

5. Related questions

Among the many issues related but not treated in the book are the higher level congruences for ȷ and P_{N} and the Rogers-Ramanujan identities. These identities and their generalizations appear in many books and in the following papers: [99], [100], [106], [18], [19], [105] and [34], among others.

The Rogers-Ramanujan identities quoted in Chapter 7 read

\frac{1}{∏_{n=0}^{∞}(1 − x^{5n+1})(1 − x^{5n+4})} = 1 + ∑_{n=1}^{∞} \frac{x^{n^{2}}}{∏_{l=1}^{n}(1 − x^{l})}

and

\frac{1}{∏_{n=0}^{∞}(1 − x^{5n+2})(1 − x^{5n+3})} = 1 + ∑_{n=1}^{∞} \frac{x^{n^{2}+n}}{∏_{l=1}^{n}(1 − x^{l})}.

These identities are related via the Jacobi triple product formula to the theta constants which appeared in Chapter 3, namely

θ [ 1^{\frac{1}{5}} ] (0, 5τ) and θ [ 1^{\frac{3}{5}} ] (0, 5τ).

The Jacobi triple product yields

∏_{n=0}^{∞}(1 − x^{5n+1})(1 − x^{5n+4})(1 − x^{5n+5}) = ∑_{n=−∞}^{∞}(−1)^{n}x^{\frac{5n^{2}+3n}{2}}

and

∏_{n=0}^{∞}(1 − x^{5n+2})(1 − x^{5n+3})(1 − x^{5n+5}) = ∑_{n=−∞}^{∞}(−1)^{n}x^{\frac{5n^{2}+n}{2}} ,

while straightforward calculation from the definitions shows that in terms of the local coordinate x = exp(2πıτ),

θ [ 1^{\frac{1}{5}} ] (0, 5τ) = exp (\frac{πı}{10}) x^{\frac{1}{40}} ∑_{n∈ℤ}(−1)^{n}x^{\frac{n(5n+1)}{2}}

and

θ [ 1^{\frac{3}{5}} ] (0, 5τ) = exp (\frac{3πı}{10}) x^{\frac{9}{40}} ∑_{n∈ℤ}(−1)^{n}x^{\frac{n(5n+3)}{2}} .

Since

η(5τ) = x^{\frac{5}{24}} ∏_{n=1}^{∞}(1 − x^{5n}),

the left hand side (the infinite products) of the Rogers-Ramanujan identities can be interpreted as multiplicative automorphic forms for Γ(5). Part of the fascination with these identities is that the right hand side (the infinite sum) does not seem to have such an interpretation.

The fact that the Rogers-Ramanujan identities are connected to the above theta constants was known to Schur. In [101], this connection is made and Schur derives the identities in two different ways. One way uses the theta constants but he also gives a second proof using combinatorial methods. The second proof uses an identity between the binomial coefficients extended to the q-binomial coefficients and some cosine identities.

There is also a generalization of the Rogers-Ramanujan identities known as the Gollnitz-Gordon identities where the modulus 8 replaces 5. While this topic is also not treated in the text, it seems interesting to point out that some of the techniques used to study these identities seem to be related to material in this text. It is pointed out in a paper by K. Alladi [4] that proofs of these identities can be based on splitting the respective functions (defined as infinite sums) into their even and odd parts and that these parts have interesting product representations. The variable x (in most of the literature q) is related to the parameter τ by x = exp(2πıτ). Our proof of the quintuple product identity also used a decomposition of a function into even and odd parts in the variable z = exp(2πıζ). These ideas arise from certain continued fraction identities which were also considered by us in the last chapter of the book. The reader is encouraged to look at Alladi’s paper and the references given there to the work of Andrews, Gordon and others.

Theta functions with characteristics enrich the set of tools available to investigate diverse questions. We believe that the material in the text opens areas for further investigations (in particular for doctoral work) and can be used to derive many identities not considered in the book. We illustrate both points by providing two different proofs of the Köhler-Macdonald (the name comes from [80]) identity quoted in Chapter 5. It reads

x ∏_{n=1}^{∞}(1 − x^{6n})^{5}(1 − x^{3n})^{−2} = ∑_{n=1}^{∞}(−1)^{n−1} (\frac{n}{3}) nx^{n^{2}}.  (1)

The right hand side is easily seen to be the same as

x ∑_{m=0}^{∞}(−1)^{m}(3m + 1)x^{9m^{2}+6m} + x ∑_{m=0}^{∞}(−1)^{m}(3m + 2)x^{9m^{2}+12m+3}.

It now makes sense to replace x by x^{\frac{1}{3}} and rewrite the identity as

∏_{n=1}^{∞}(1 − x^{2n})^{5}(1 − x^{n})^{−2} = ∑_{m=−∞}^{∞}(−1)^{m}(3m + 1)x^{m(3m+2)}.

To translate this identity to the language of theta functions, we use the change of variable x = exp(2πıτ). The left hand side of the Köhler-Macdonald identity is thus

exp (−\frac{2πıτ}{3} ) η(2τ)^{5}η(τ)^{−2}.

We have shown that

θ′ [ 1^{\frac{2}{3}} ] (0, τ) = \frac{2πı}{3} exp \{πı\frac{1}{3}\} x^{\frac{1}{18}} ∑_{n∈ℤ}(3n + 1)(−1)^{n}x^{\frac{n}{6} (3n+2)}.

Thus the right hand side of the Köhler-Macdonald identity is

\frac{3}{2πı} exp \{−πı\frac{1}{3}\} exp \{−2πıτ \frac{1}{3}\} θ′ [ 1^{\frac{2}{3}} ] (0, 6τ).

We conclude that the function theoretic version of Köhler-Macdonald is

2πıη(2τ)^{5}η(τ)^{−2} = 3 exp \{−πı\frac{1}{3}\} θ′ [ 1^{\frac{2}{3}} ] (0, 6τ).  (2)

Our first proof of the above identity follows from the following product formula established by Y. Godin, which will appear as part of his doctoral thesis at the Hebrew University of Jerusalem. Godin proves

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ, τ)θ [ \begin{matrix}δ \\ δ′ \end{matrix} ] (ζ, 2τ)

= ∑_{μ=0}^{2} θ [ 2ϵ′ + δ′^{\frac{2μ+ϵ+δ}{3}} ] (3ζ, 6τ)θ [ ϵ′ − δ′^{\frac{2μ+ϵ−2δ}{3}} ] (0, 3τ).

It thus follows that

θ [ \begin{matrix}1 \\ 0 \end{matrix} ] (ζ, τ)θ [ \begin{matrix}1 \\ 1 \end{matrix} ] (ζ, τ)

= θ [ 1^{\frac{1}{3}} ] (0, 3τ) (θ [ 1^{\frac{2}{3}} ] (3ζ, 6τ) − θ [ 1^{\frac{2}{3}} ] (−3ζ, 6τ)) .

Differentiation (with respect to ζ) and then evaluation at ζ = 0 yields

θ′ [ 1^{\frac{2}{3}} ] (0, 6τ) = \frac{θ [ \begin{matrix}1 \\ 0 \end{matrix} ] (0, τ)θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 2τ)}{6θ [ 1^{\frac{1}{3}} ] (0, 3τ)}.

An application of the Jacobi triple product formula completes the proof of (2).

It is appropriate to illustrate the combinatorial content of the Köhler-Macdonald identity. The identity may be rewritten as

∑_{n=−∞}^{∞}(−1)^{n}(3n + 1)x^{3n^{2}+2n} = ∏_{n=1}^{∞}(1 − x^{4n})^{3}(1 − x^{4n−2})(1 + x^{2n−1})^{2}.

In this form the combinatorial content is as follows: Let S be the set containing 3 copies of the positive integers congruent to 0 modulo 4, 1 copy of the positive integers congruent to 2 modulo 4, and 2 copies of the odd positive integers. Call such a partition even if it has an even number of even parts and call it odd otherwise. Denote by E(N) and O(N) the number of even and odd partitions of N ∈ ℤ^{+} using parts from the set S. The Köhler-Macdonald identity says that for all N ∈ ℤ^{+},

E(N)−O(N) = \{ \begin{matrix}0 & for N ≠ 3m^{2} + 2m with m ∈ ℤ \\ (−1)^{m}(3m + 1) & for N = 3m^{2} + 2m with m ∈ ℤ \end{matrix} .

The second proof of the identity, more in the spirit of the book, is based on the connection of θ-constants to automorphic forms. It starts by “rewriting” (2) as

(2π)^{24}η(2τ)^{120}η(τ)^{−48} = 3^{24} (θ′ [ 1^{\frac{2}{3}} ] (0, 6τ))^{24} .  (3)

The equivalence class of the characteristic [ 1^{\frac{2}{3}} ] is fixed by the group

Γ^{o}(6). Using this observation one shows that

f(τ) = (θ′ [ 1^{\frac{2}{3}} ] (0, 6τ))^{24}

is a multiplicative 6-form for Γ_{o}(6) (perhaps with a nontrivial multiplier system); that is for all τ ∈ ℍ^{2} and all γ ∈ Γ_{o}(6),

(θ′ [ 1^{\frac{2}{3}} ] (0, 6γ(τ)))^{24} (γ′(τ))^{6} = c_{γ} (θ′ [ 1^{\frac{2}{3}} ] (0, 6τ))^{24} ,

where c_{γ} is a constant of absolute value 1. The four punctures on ℍ^{2}/Γ_{o}(6) are P_{∞}, P_{\frac{1}{2}} , P_{0} and P_{\frac{1}{3}} . The divisor (f) is computed to be

P_{∞}^{8}P_{0}^{9}P 18_{\frac{1}{2}}P_{\frac{1}{3}} . The function \frac{η(2τ)^{120}}{η(τ)^{48}} is a multiplicative 6-form for Γ_{o}(2), hence certainly for Γ_{o}(6). It has the same divisor as the automorphic form f. The formulae for the divisors show that these are ordinary cusp forms. Hence, except for the constants, (3) follows. If we extract a 24-th root and evaluate the one (so far undetermined) constant, we obtain (2).

The above two proofs are based on recognizing that both sides of (1) can be reinterpreted in terms of θ-constants. A proof of Köhler-Macdonald based on theta functions (in particular on [69]) can be found in [70]. It is different in flavor from the second of our proofs because it relies more directly and more exclusively on the properties of the η-function. Our second proof has an unexpected dividend. While it

is more or less obvious that θ′ [ 1^{\frac{2}{3}} ] (0, 6τ) is a multiplicative form for

Γ_{o}(6), we have shown that it is a form for the bigger group Γ_{o}(2). As a special case of a more general formula Godin also proves

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (0, τ)θ [ \begin{matrix}δ \\ δ′ \end{matrix} ] (0, 4τ)

= ∑_{n=0}^{4} θ [ 4ϵ′ + δ′^{\frac{2μ+ϵ+δ}{5}} ] (0, 20τ)θ [ −ϵ′ + δ′^{\frac{−2μ−ϵ+4δ}{5}} ] (0, 5τ).

Choosing [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] = [ \begin{matrix}1 \\ 0 \end{matrix} ] and [ \begin{matrix}δ \\ δ′ \end{matrix} ] = [ \begin{matrix}0 \\ 1 \end{matrix} ] we obtain after some

algebraic manipulation

∏_{n=1}^{∞}(1 + x^{2n−1})^{2}

= \frac{1}{∏_{n=0}^{∞}(1 − x^{5n+1})(1 − x^{5n+4})(1 − x^{4(5n+1)})(1 − x^{4(5n+4)})}

+\frac{x}{∏_{n=0}^{∞}(1 − x^{5n+2})(1 − x^{5n+3})(1 − x^{4(5n+2)})(1 − x^{4(5n+3)})},

a formula, related to the Rogers-Ramanujan identities, which also has a clear combinatorial interpretation which we now give.

Let S be the set consisting of two copies of the odd positive integers. Denote the number of partitions of N from S by S(N). Let T be the set consisting of the positive integers congruent to ±1 modulo 5 with an extra copy of those which are also congruent to 0 modulo 4. Denote the number of partitions of N from T by T (N). Finally, let U be the set of the positive integers congruent to ±2 modulo 5 with an extra copy of those which are also congruent to 0 modulo 4. Denote the number of partitions of N from U by U(N). The above identity says that for all N ∈ ℤ, N ≥ 2,

S(N) = T (N) + U(N − 1).

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