A FUNCTION THEORETIC APPROACH TO THE RAMANUJAN PARTITION IDENTITIES WITH APPLICATIONS TO COMBINATORIAL

NUMBER THEORY

HERSHEL M. FARKAS AND IRWIN KRA

1. Introduction

This article, an expanded version of talks delivered at the First Iberoamerican Congress on Geometry, has two parts. The first part (§§2–10) outlines the development of the theory of what we call Ramanujan partition identities from a function theoretic point of view. The second part (§11) deals with applications to combinatorial number theory. The mathematics involved is an interesting combination of core areas of the subject displaying a deep interplay between complex analysis (function theory), number theory, algebra and combinatorics. It has nontrivial connections to representation theory and mathematical physics. This development is based on beautiful AND simple concepts. It yields elegant formulae spanning the gambit from elementary to very complicated. It confronts the researcher with very basic questions; for example, when is a multivalued function on a Riemann surface single valued. Much is known about the subject. There is a rich literature and work in the field involves communing with giants of the past. However, it also reinforces ones sense of modesty since almost every discovery a new researcher in the area makes can be found some place in the works of the previous generations. The development of the subject may also be based on combinatorics (see, for example, [1]) or hypergeometric functions ([12], [15]).

We study the Ramanujan partition functions

P_{N} : ℤ^{+} ∪ {0} → ℤ, N ∈ ℤ,

defined by convergent products and sums on the unit disc Δ

P^{N}(x) = ∏_{n=1}^{∞} \frac{1}{(1 − x^{n})^{N}} = ∑_{n=0}^{∞} P_{N}(n)x^{n}, x ∈ Δ.

Write P for P_{1}. Ramanujan [21] proved that for all m ∈ ℤ^{+} ∪ {0},

P (5m + 4) ≡ 0 mod 5, P (7m + 5) ≡ 0 mod 7 and P (11m + 6) ≡ 0 mod 11,

and more or less conjectured (see [22] and [17, Ch. 7]) and partially proved that (here n ∈ ℤ^{+}):

(a) If 24m ≡ 1 mod 5^{n}, then P (m) ≡ 0 mod 5^{n}.

(b) If 24m ≡ 1 mod 7^{n}, then P (m) ≡ 0 mod 7^{1+[\frac{n}{2}]}.

(c) If 24m ≡ 1 mod 11^{n}, then P (m) ≡ 0 mod 11^{n}.

Research by IK supported in part by NSF Grant DMS 9500557. Research by HMF supported in part by the Gabriella and Paul Rosenbaum Foundation and the Edmund Landau Center for Research in Mathematical Analysis sponsored by the Minerva Foundation Germany. Research of HMF and IK supported in part by a US-Israel BSF Grant 95-348.

We refer to these as the Ramanujan level n congruences for the primes 5, 7 and 11, respectively.

Several partition functions P_{N} have been investigated more than the general case; in part, because of various applications and/or interpretations. Among these are the cases N = 1 (studied by Ramanujan), N = −1 (Euler), N = −3 (Jacobi) and N = −24 (Lehmer). In particular, only for N = −1 and −3 do we have (see §2.5 below) a closed formula for the partition coefficients P_{N}(n), n ∈ ℤ^{+}∪{0}. Other significant contributions to the theory were made by G.H. Hardy, G.N. Watson, L.J. Mordel, J. Lehner, A.O.L. Atkin and M. Newman, among others.

There are also congruences for some other primes; for example, 2, 3 and 13.

(d) If 8m ≡ 1 mod 3^{n}, then P_{3}(m) ≡ 0 mod 3^{[^{\frac{3n}{2}}]+1}.

(e) If 3m ≡ 1 mod 2^{n}, then P_{8}(m) ≡ 0 mod 2^{3[[\frac{n+1}{2}]]}. The results for 13 are different in flavor from the previous ones. A sample formula is (f)

P (13m + 6) ≡ 11P_{−11}(m) mod 13.

Several immediate questions should be addressed. What primes appear in the theory? Why? What is the significance of the integer 24 (and its factors 3 and 8)? There are many conjectures (see, for example, [5]) and some of these have been established (for example, [18, Ch. 6] contains Atkin’s proofs of the conjectures of [5]). However, it is our opinion that much more remains to be done. In particular, this beautiful chapter in mathematics seems to have been diverted from a main stream in complex analysis. We hope that our contributions will in part revive the interests of those working on Riemann surfaces in applications to number theoretic questions.

We start with a combinatorial interpretation of the partition functions. It is, of course, obvious that for all N ∈ ℤ, P_{N}(0) = 1 and P_{N}(1) = N. It is easily seen that in general for n ∈ ℤ^{+}, P_{N}(n) is a polynomial in N of degree n, zero constant term (thus a root at N = 0) and all the rest of the coefficients are positive rationals (hence no roots in ℝ^{+}) . These polynomials may, however, have roots in −ℤ^{+}, the negative integers. It is a nontrivial matter of some interest to compute these polynomials and their real roots^{1}. Assume that n ∈ ℤ^{+}. For N ∈ ℤ^{+}, P_{N}(n) counts the total number of partitions of n using positive integers of N colors. For −N ∈ ℤ^{+}, P_{N}(n) counts the difference between the total number of \underline{even} and \underline{odd} partitions of n WITHOUT REPETITIONS using positive integers of −N colors. A partition of n ∈ ℤ^{+} is a sum of positive integers totaling n. Thus the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The parity of the partition is the parity of the number of summands. One easily checks from the definitions or the last enumeration that

P (4) = 5, P_{−1}(4) = 0, P_{−3}(4) = 0, P_{2}(4) = 20, P_{−2}(4) = 1.

A sample formula, hinting at the underlying function theory, that already appears in [21] is

∑_{m=1}^{∞} P (5m − 1)x^{m} = 5 x \frac{∏_{n=1}^{∞}(1 − x^{5n})^{5}}{∏_{n=1}^{∞}(1 − x^{n})^{6}}

1For 0 ≤ n ≤ 10, Newman [19] computed the polynomials P_{·}(n) and their roots in ℤ. Further results will be found in [?].

which, for reasons that should become apparent in due course, is more conveniently rewritten as

∏_{n=1}^{∞}(1 − x^{5n}) ∑_{m=1}^{∞} P (5m − 1)x^{m} = 5 x (∏_{n=1}^{∞} \frac{1 − x^{5n}}{1 − x^{n}} )^{6} .

To unify the various identities, we have been discussing we introduce

Definition 1. Let

e : ℤ^{+} ∪ {0} → ℤ^{+} ∪ {0}

be a function and let F be a meromorphic function on a Riemann surface X. Let x ∈ X and let z be a local coordinate on X vanishing at x. We say the F satisfies a level n congruence for the prime k (with respect to the local coordinate z) of type e if the Laurent series expansion of F in terms of z is

z^{I_{o}} ∑_{i=0}^{∞} a_{i}z^{i}, with I_{o} and a_{i} ∈ ℤ, a_{i} ≡ 0 mod k^{e(n)}.

It turns out that all the congruences discussed so far arise in this way.

The aim of our research in this area is the study of function theory on Riemann surfaces represented by subgroups of the modular group SL(2, ℤ) acting on the upper half plane

ℍ^{2} = {z ∈ ℂ; ℑz > 0}.

The function theoretic approach to some problems in combinatorial number theory is a pleasant byproduct.

We are reporting on a body of work that originates with fundamental ideas and results of Srinivasa Ramanujan. The proceedings of a centenary conference in his honor [2] show the tremendous influence his work has had on 20th century mathematics; especially relevant for us are the articles [7], [14] and [6].

2. The needed tools

We summarize the ingredients in our development of this theory.

2.1. Compact Riemann surfaces of genus p = 0 and 1. We need to describe some properties of the field K(X) of meromorphic functions on a closed Riemann surface X of genus p. Any two elements of K(X) are, of course, algebraically dependent. It is a nontrivial task, in general, to determine the equations satisfied by the two functions. It is fortunate that for most applications, the cases p = 0 and 1 suffice. For p = 0, K(X) ≅ ℂ(z), the field of complex valued rational functions. In particular, if we fix x ∈ X, and let F_{N} ∈ K(X) have pole at x of order N > 0 and no other poles, then there exist constants c_{i} ∈ ℂ such that

F_{N} = ∑_{i=0}^{N} c_{i}F_{1}^{i}.  (1)

For p = 1, K(X) ≅ ℂ(x)[y]/M, where M is the principal maximal ideal generated by

y^{2} − 4(x − e_{1})(x − e_{2})(x − e_{3}),

with e_{1}, e_{2}, e_{3} three distinct complex numbers. With x and F_{N} as above^{2}, now (N ≥ 2)

F_{N} = (\begin{matrix}[ & \frac{N−3}{2} & ] \\ & ∑ \\ & i=0 \end{matrix} c_{i}F_{2}^{i}) F_{3} + \begin{matrix}[ & \frac{N}{2} & ] \\ ∑ \\ j=0 \end{matrix} d_{j}F_{2}^{j}, c_{i} ∈ ℂ, d_{i} ∈ ℂ.  (2)

We may (but are not required to) use x = ℘ and y = ℘′, the Weierstrass ℘-function and its derivative (in this case we may take e_{1} = ℘ (\frac{1}{2}), e_{2} = ℘ (\frac{τ}{2}) and e_{3} = ℘ (\frac{1+τ}{2} ), where X = ℂ/G and G is the group of translations by n + mτ with n, m ∈ ℤ and τ ∈ ℍ^{2}.

The undetermined constants (c_{i} and d_{i}) in equations (1) and (2) are usually computed using simple linear algebra.

2.2. Theta functions. The (one variable) theta function with characteristic [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] ∈ ℝ^{2},

argument z ∈ ℂ and parameter τ ∈ ℍ^{2} is defined by the infinite series

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (z, τ) = ∑_{n=−∞}^{∞} exp 2πı (\frac{1}{2} (n + \frac{ϵ}{2})^{2} τ + (n + \frac{ϵ}{2}) (z + \frac{ϵ′}{2} ))

that, for fixed characteristic, converges absolutely and uniformly on compact subsets of ℂ × ℍ^{2}. Two groups act on θ-functions. For (m, n) ∈ ℤ^{2},

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (z + m + nτ, τ) = exp 2πı (\frac{mϵ − nϵ′}{2} − nz − n^{2} \frac{τ}{2}) θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (z, τ).

More interesting, for our applications, is the transformation formula obtained by studying

the action of γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2, ℤ) on our functions:

\frac{exp πı \{\frac{−cz^{2}}{cτ+d}\} θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (\frac{z}{cτ+d}, \frac{aτ+b}{cτ+d})}{θ [ \begin{matrix}aϵ + cϵ′ − ac \\ bϵ + dϵ′ + bd \end{matrix} ] (z, τ)} = κ ([ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] , γ) (cτ + d)^{\frac{1}{2}} ,

where κ ([ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] , γ) is a constant of absolute value one that depends only on the motion

γ and the characteristic [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ]. It is a root of unity if the characteristic is rational. Its

dependence on the motion and characteristic is rather complicated.

We are interested mostly in the case z = 0, and we hence abbreviate the θ-constant

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (0, ·) by θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ]; the θ-constant is obviously a holomorphic function on ℍ^{2}. The

transformation formula clearly simplifies in this case, and implies that for certain characteristics, a sufficiently high power of the θ-constant is a cusp form for an appropriate subgroup of the modular group.

2Throughout this paper, [x] stands for the integral part of x ∈ ℝ (the greatest integer ≤ x), and [[x]] for the smallest integer ≥ x. Thus [[x]] = −[−x].

2.3. Fuchsian groups. We use certain families of subgroups of the modular group Γ = PSL(2, ℤ). The theory of θ-constants with rational characteristics is most suitable for the study of the function theory for the level N principal congruence subgroup, Γ(N), of the modular group (here N ∈ ℤ^{+}) defined by

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

This point of view was pursued in [9] and [10]. In the present context, we rely mostly on

the odd integral characteristic [ \begin{matrix}1 \\ 1 \end{matrix} ] and a different family of groups. Let p and q ∈ ℤ^{+}. We

define

Γ(p, q) = \{[ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ Γ; p|b and q|c\} .

The formulae

Γ(1, q) = Γ_{o}(q) and Γ(p, 1) = Γ^{o}(p)

relate this terminology to a more standard one. It follows readily that

Γ(p, q) = Γ^{o}(p) ⋂ Γ_{o}(q) ≅ Γ_{o}(pq).

Since we are dealing with finite index subgroups G of the modular group Γ, that contains only 2 and 3 torsion, it makes sense to define

μ = μ(G) = [Γ : G], the index of G in Γ, ν_{i} = ν_{i}(G), the number of G-inequivalent elliptic fixed points (in ℍ^{2}) of order i = 2, 3, and ν_{∞} = ν_{∞}(G), the number of G-inequivalent parabolic fixed points (in ℚ ∪ {∞}).

While ν_{2} and ν_{3} are only nonnegative, ν_{∞} must be positive (because of the finite index assumption). We call the quadruple {μ; ν_{2}, ν_{3}, ν_{∞}}, the branch schema for G. This quadruple is a topological invariant for the orbifold ℍ^{2}/G; its genus p = p(G) is computed, using Riemann-Hurwitz, as

p = 1 + \frac{μ}{12} − \frac{ν_{2}}{4} − \frac{ν_{3}}{3} − \frac{ν_{∞}}{2} .

The canonical projection

P = P (G) : ℍ^{2} → ℍ^{2}/G

extends to the set of fixed points, ℚ_{𝔾} ⊂ ℚ ∪ {∞}, of parabolic elements of G; we will use the same symbol P to denote the extension, and we will often write P_{x} for P (x) with x ∈ ℍ^{2} ∪ ℚ_{G}. The compactification of ℍ^{2}/G obtained by filling in the punctures will be denoted by \overline{ℍ^{2}/G}. We thus also view

P : ℍ^{2} ∪ ℚ_{𝔾} → \overline{ℍ^{⊭}/𝔾}.

For most of our groups, ℚ_{𝔾} = ℚ ∪ {∞}. The following two tables describe the topology of the most common surfaces encountered on our journey and hint (look for surfaces of genus 0) why congruences for some primes k are easier to describe.

Branch schema and genera for some groups Γ_{o}(k).

\begin{matrix}\underline{k} & \underline{μ} & \underline{ν_{2}} & \underline{ν_{3}} & \underline{ν_{∞}} & \underline{p} \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 2 & 3 & 1 & 0 & 2 & 0 \\ 3 & 4 & 0 & 1 & 2 & 0 \\ 4 & 6 & 0 & 0 & 3 & 0 \\ 5 & 6 & 2 & 0 & 2 & 0 \\ 6 & 12 & 0 & 0 & 4 & 0 \\ 7 & 8 & 0 & 2 & 2 & 0 \\ 8 & 12 & 0 & 0 & 4 & 0 \\ 9 & 12 & 0 & 0 & 4 & 0 \\ 10 & 18 & 2 & 0 & 4 & 0 \\ 11 & 12 & 0 & 0 & 2 & 1 \\ 12 & 24 & 0 & 0 & 6 & 0 \\ 13 & 14 & 2 & 2 & 2 & 0 \\ 17 & 18 & 2 & 0 & 2 & 1 \\ 19 & 20 & 0 & 2 & 2 & 1 \\ 23 & 24 & 0 & 0 & 2 & 2 \end{matrix}

Branch schema and genera of some groups Γ(k, k).

\begin{matrix}\underline{k} & \underline{μ} & \underline{ν_{2}} & \underline{ν_{3}} & \underline{ν_{∞}} & \underline{p} \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 2 & 6 & 0 & 0 & 3 & 0 \\ 3 & 12 & 0 & 0 & 4 & 0 \\ 4 & 24 & 0 & 0 & 6 & 0 \\ 5 & 30 & 2 & 0 & 6 & 0 \\ 6 & 72 & 0 & 0 & 10 & 2 \\ 7 & 56 & 0 & 2 & 8 & 1 \\ 8 & 96 & 0 & 0 & 12 & 3 \\ 9 & 108 & 0 & 0 & 12 & 4 \\ 10 & 180 & 0 & 0 & 18 & 6 \\ 11 & 132 & 0 & 0 & 12 & 6 \\ 12 & 288 & 0 & 0 & 28 & 10 \\ 13 & 182 & 2 & 2 & 14 & 8 \end{matrix}

To introduce the groups of primary concern to us, we fix a prime k. Let m ∈ ℤ^{+} ∪{0}. Then

[Γ_{o}(k) : Γ(k^{m}, k)] = k^{m}

and the motions

B^{l} = [ \begin{matrix}1 & l \\ 0 & 1 \end{matrix} ]; l = 0, 1, ..., k^{m} − 1,

form representatives for the Γ(k^{m}, k)-cosets in Γ_{o}(k). Thus the averaging operator

\frac{1}{k^{m}} ∑_{l=0}^{k^{m}−1} f ∘ B^{l}

takes a Γ(k^{m}, k)-invariant function f to a Γ_{o}(k)-invariant function.

2.4. Automorphic functions and forms. The function

τ ↦ \frac{1}{(2π)^{8}} (θ′)^{8} [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ)

is a basis for the one dimensional space of cusp forms^{3} 𝔸_{6}(ℍ^{2}, Γ) for the modular group. Its Fourier series expansion in terms of the variable x = exp(2πıτ) is

x + ∑_{n=2}^{∞} a_{n}x^{n},

and it has no zeros in ℍ^{2}. It hence make sense to extract an appropriate root of this function; a 24-th root in this case. We have obtained the classical η-function. Two formulae relate the various functions under study. The first identity

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

is an easy consequence of the fact that ℍ^{2}/Γ has genus zero (see, for example, [11]); the second

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

follows directly from our definitions. The group Γ acts naturally on the η-function. We have

η(τ + 1) = exp (\frac{πı}{12}) η(τ) and η (−\frac{1}{τ}) = exp (−\frac{πı}{4}) η(τ)\sqrt{−τ}.

The above invariance formulae (part of the multiplier system for the η-function) follow almost immediately from the transformation formula for the θ-function and the expressions for κ obtained in [9]. We need, however, complete information for the multiplier system of this function. It is given in the following proposition whose proof can be found in [17, Ch. 3], where the definitions of the Jacobi and Legendre symbols (\frac{·}{·})^{∗} and (\frac{·}{·})_{∗} (whose values are ±1) can also be found. The proof given there has a strong number theoretic flavor.

Proposition 1. The η-function satisfies the functional equations

η(γ(τ)) = v(γ)(cτ + d)^{\frac{1}{2}} η(τ), τ ∈ ℍ^{2}, γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL(2, ℤ),

where

v(γ) = \{ \begin{matrix}(\frac{d}{c})^{∗} exp \{\frac{πı}{12}[(a + d)c − bd(c^{2} − 1) − 3c]\} , c odd, \\ (\frac{c}{d})_{∗} exp \{\frac{πı}{12}[(a + d)c − bd(c^{2} − 1) + 3(d − 1 − c]\} , c even. \end{matrix}

2.5. Products and sums. The only closed expressions for the partition functions P_{N} known to us are the Jacobi identity (the case N = −3)

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

and the Euler identity (N = −1)

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

3For elements φ in this space φ(z)dz^{6} is invariantly defined.

From any (in particular, from a combinatorial) proof (see [16, §19.11] for an argument based on [13]) of this second identity, one can obtain the first one. Euler’s identity is also a consequence of the proof of the Jacobi triple product

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

We hence conclude that in terms of x = exp(2πıτ),

η(τ) = x^{\frac{1}{24}} ∑_{n=−∞}^{∞}(−1)^{n}x^{\frac{n(3n+1)}{2}} = x^{\frac{1}{24}} ∏_{n=1}^{∞}(1 − x^{n})

for which one can give a purely function theoretic proof (see [?]).

2.6. The final ingredient in our recipe is the construction of the same meromorphic function on a compact Riemann surface in two or more ways. We then express the resulting equality in terms of “good” local coordinates.

2.7. We summarize the key points for the construction of partition identities. Start with a fixed Riemann surface Y and a meromorphic function on it. Choose (if it exists) a group G acting analytically on the surface. Average the function over the group. Obtain a meromorphic function on the quotient surface Y/G. Obtain an alternate description of the meromorphic function constructed. Rewrite the resulting equation in terms of appropriate local coordinates. In many cases, interesting formulae are consequences of this recipe.

3. Constructions of meromorphic functions

It is convenient to start with

Definition 2. For the positive prime k, let

α(k) = \{ \begin{matrix}8 for k = 2 \\ 3 for k = 3 \\ 1 for k ≥ 5 \end{matrix} and β(k) = \{ \begin{matrix}1 for k ≤ 5 \\ \frac{k^{2}−1}{24} for k ≥ 5 \end{matrix} .

Note that for all primes k,

\frac{k^{2} − 1}{24} = \frac{β(k)}{α(k)}.

For a finite index subgroup G of Γ, we identify the field of meromorphic functions^{4} K(\overline{ℍ^{2}/G}) on \overline{ℍ^{2}/G} with the field K(G) of G-invariant meromorphic functions on ℍ^{2} with continuous extensions (with values in ℂ ∪ {∞}) to ℍ^{2} ∪ ℚ_{G}.

If for some prime k, \overline{ℍ^{2}/Γ_{o}(k)} has genus zero (this happens only for k = 2, 3, 5, 7 and 13),

then f_{k}(τ) = (\frac{η(kτ)}{η(τ)})^{\frac{24}{k−1}} ∈ K(\overline{ℍ^{2}/Γ_{o}(k)}) with divisor \frac{P_{∞}}{P_{0}} . It is also useful to observe that for

k = 2, 3 and 5, the surface \overline{ℍ^{2}/Γ(k, k)} has genus zero, and h_{k}(τ) = (\frac{η(kτ)}{η(\frac{τ}{k})})^{\frac{24}{k^{2}−1}} ∈ K(Γ(k, k))

with divisor \frac{P_{∞}}{P_{0}} . The interplay between these functions is an important ingredient in our study.

4To avoid notational proliferation, we write f(τ) to indicate that f is a function defined for τ ∈ ℍ^{2} ∪ ℚ ∪ {∞}. We are thus identifying, as in earlier classical notation, a function and its value at a point.

The next calculation (theorem below) is more involved. Before we state the result, it is instructive to analyze the role of α(k) and β(k) in our theory. We are mostly concerned with the family of groups Γ(k^{m}, k). The most important cases being m = 0 and 1. The surfaces represented by such groups have two distinguished punctures: P_{∞} and P_{0}. We define two linear subspaces of the field K(Γ(k^{m}, k)):

K(Γ(k^{m}, k))_{∞} = {f ∈ K(Γ(k^{m}, k)); f is regular except at P_{∞}}

and

K(Γ(k^{m}, k))_{0} = {f ∈ K(Γ(k^{m}, k)); f(∞) = 0 and f is regular except at P_{0}}.

Definition 3. We define β(Γ(k^{m}, k)) as the smallest positive integer β with P_{∞}^{β}P_{0}^{−β} a principal divisor on the compact Riemann surface \overline{ℍ^{2}/Γ(k^{m}, k)}. We let β_{P_{∞}}(Γ(k^{m}, k)) (respectively, β_{P_{0}}(Γ(k^{m}, k))) be the smallest nongap at P_{∞} (P_{0}).

At this point, it is easy to see (as a consequence of Proposition 1) that α(k) is the smallest positive integer α such that

(\frac{η(kτ)}{η (\frac{τ}{k^{m}} )})^{α} , m odd and (\frac{η(τ)}{η (\frac{τ}{k^{m}} )})^{α} , m even,

is a Γ(k^{m}, k)-invariant function. For the β-invariants, note that

β(Γ(k^{m}, k)) ≤ β_{P_{∞}}(Γ(k^{m}, k)), β(Γ(k^{m}, k)) ≤ β_{P_{0}}(Γ(k^{m}, k)) and β(Γ(k, k))|β(k).

We have established that for small primes k and conjecture that for all primes k,

β(Γ(k, k)) = β(k).

The three numbers β(Γ(k^{m}, k)), β_{P_{∞}}(Γ(k^{m}, k)) and β_{P_{0}}(Γ(k^{m}, k)) are important function theoretic invariants whose properties will be investigated in future studies.

Theorem 1. Let k be a prime and N ∈ ℤ.

(a) For odd n ∈ ℤ^{+}, ^{5}

F_{k,n,N}(τ) = \frac{1}{k^{n}} ∑_{l=0}^{k^{n}−1} (\frac{η(k(τ + l))}{η(\frac{τ+l}{k^{n}})})^{α(k)N}

defines a Γ_{o}(k)-invariant function that is holomorphic, except possibly at the punctures, with

ord_{P_{∞}}F_{k,n,N} ≥ [[\frac{α(k)N}{24} (k − \frac{1}{k^{n}} )]];

equality holds provided N > 0. Further

ord_{P_{0}}F_{k,n,N} = \frac{α(k)N}{24} (1 − k^{n+1}) for N > 0.

(b) For even n ∈ ℤ^{+} ∪ {0},

F_{k,n,N}(τ) = \frac{1}{k^{n}} ∑_{l=0}^{k^{n}−1} (\frac{η(τ + l)}{η(\frac{τ+l}{k^{n}})})^{α(k)N}

5Abbreviate F_{k,1,N} = F_{k,N}.

defines a Γ_{o}(k)-invariant function that is holomorphic, except possibly at the punctures, with

ord_{P_{∞}}F_{k,n,N} = [[\frac{α(k)N}{24} (1 − \frac{1}{k^{n}} )]] ,

and

ord_{P_{0}}F_{k,n,N} = \frac{α(k)Nk}{24} (1 − k^{n}) for N > 0.

The basic fact needed to establish the theorem is that

(\frac{η(kτ)}{η (\frac{τ}{k^{n}} )})^{α(k)} , n odd and (\frac{η(τ)}{η (\frac{τ}{k^{n}} )})^{α(k)} , n even,

is a Γ(k^{n}, k)-invariant function with poles only on points Γ_{o}(k)/Γ(k^{n}, k)-equivalent to P_{0}. When its N-th power is averaged over Γ_{o}(k)/Γ(k^{n}, k) it produces a Γ_{o}(k)-invariant function F_{k,n,N} that has only singularities at P_{0}, if N > 0.

The significance of the last theorem for partition identities becomes evident once we specialize to low values for the prime k and we obtain alternate expressions for the functions F_{k,n,N}.

Corollary 1. For k = 2, 3, 5, 7 and 13, and N ∈ ℤ^{+},

F_{k,n,N} = ∑_{i=[[\frac{α(k)N}{24} (k− \frac{1}{k^{n}})]]}^{\frac{α(k)N}{24} (k^{n+1}−1)} c_{i}f_{k}^{i}, n odd, and F_{k,n,N} = ∑_{i=[[\frac{α(k)N}{24} (1− \frac{1}{k^{n}})]]}^{\frac{α(k)N}{24} (k^{n+1}−k)} c_{i}f_{k}^{i}, n even.

Further, the extreme (that is, c[[_{\frac{α(k)N}{24}} _{(k− \frac{1}{k^{n}})}]] and c_{\frac{α(k)N}{24} (k^{n+1}−1)} (c[[_{\frac{α(k)N}{24}} _{(1− \frac{1}{k^{n}})}]] and c_{\frac{α(k)N}{24} (k^{n+1}−k)}) for odd (even) n) coefficients are nonzero. Thus for odd n (in terms of the local coordinate x = exp(2πıτ) at P_{∞} on \overline{ℍ^{2}/Γ_{o}(k)},

F_{k,n,N}(τ) = ∏_{m=1}^{∞}(1 − x^{km})^{α(k)N} ∑_{m=[[\frac{α(k)N}{24} (k− \frac{1}{k^{n}})]]}^{∞} P_{α(k)N} (k^{n}m − \frac{α(k)N(k^{n+1} − 1)}{24}) x^{m}

= ∑_{i=[[\frac{α(k)N}{24} (k− \frac{1}{k^{n}})]]}^{\frac{α(k)N}{24} (k^{n+1}−1)} c_{i} [x\frac{∏_{m=1}^{∞}(1 − x^{km})^{\frac{24}{k−1}}}{∏_{n=1}^{∞}(1 − x^{m})^{\frac{24}{k−1}}} ]^{i} ,

c[[_{\frac{α(k)N}{24}} _{(k− \frac{1}{k^{n}})}]] = P_{α(k)N} (k^{n} − \frac{α(k)N(k^{n+1} − 1)}{24}) .

For even n,

F_{k,n,N}(τ) = ∏_{m=1}^{∞}(1 − x^{m})^{α(k)N} ∑_{m=[[\frac{α(k)N}{24} (1− \frac{1}{k^{n}})]]}^{∞} P_{α(k)N} (k^{n}m − \frac{α(k)N(k^{n} − 1)}{24}) x^{m}

= ∑_{i=[[\frac{α(k)N}{24} (1− \frac{1}{k^{n}})]]}^{\frac{α(k)Nk}{24} (k^{n}−1)} c_{i} [x\frac{∏_{m=1}^{∞}(1 − x^{km})^{\frac{24}{k−1}}}{∏_{m=1}^{∞}(1 − x^{m})^{\frac{24}{k−1}}} ]^{i} ,

c[[_{\frac{α(k)N}{24}} _{(1− \frac{1}{k^{n}})}]] = P_{α(k)N} (k^{n} − \frac{α(k)N(k^{n} − 1)}{24}) .

To explain the connection between partition congruences and the last corollary, it is convenient to define three integers. Assume. as before, that we restrict the prime k to the values 2, 3, 5, 7 and 13. Let us fix n ∈ ℤ^{+} ∪ {0} and N ∈ ℤ^{+}. We start with an indexing set

I = {I_{1}, I_{1} + 1, ..., I_{2}},

where

I_{1} = \{ \begin{matrix}[[α(k)\frac{N}{24} (k − \frac{1}{k^{n}} )]] for n odd \\ [[α(k)\frac{N}{24} (1 − \frac{1}{k^{n}} )]] for n even \end{matrix} and I_{2} = \{ \begin{matrix}α(k)\frac{N}{24}(k^{n+1} − 1) for n odd \\ α(k)\frac{Nk}{24}(k^{n} − 1) for n even \end{matrix} .

Theorem 2. Under the hypothesis (and using the notation) of the last corollary, c_{i} ∈ ℤ and

c = d_{1} = d,

where

c = c(k, n, N) = gcd {c_{i}; i ∈ I},

d_{1} = d_{1}(k, n, N) = gcd \{P_{α(k)N} (k^{n}m − α(k)N \frac{k^{2[\frac{n+1}{2}]} − 1}{24}); m ∈ I\}

and

d = d(k, n, N) = gcd \{P_{α(k)N} (k^{n}m − α(k)N \frac{k^{2[\frac{n+1}{2}]} − 1}{24}); m ≥ I_{1}\} .

Remark 1. The above theorem is based on the properties of the functions F_{k,n,N} with N ∈ ℤ^{+}. A similar result holds for the functions F_{k,N} (that is for n = 1 only) with −N ∈ ℤ^{+}. It is not known whether such results can be extended to the case n > 1.

In principle, the last two theorems tell us much information about many partition congruences for the primes 2, 3, 5, 7 and 13. Any particular case (a choice of k, n and N) can, of course be resolved by a finite calculation (of d_{1}(k, n, N)). Plenty of work remains to obtain general theorems. The known methods involve a calculation of F_{k,1} = F_{k,1,1} (or more generally F_{k,N} = F_{k,1,N} for small |N|) followed by an induction argument on n (n and N) to obtain information on divisibility properties of c(k, n, N). The known general results yield only information about the divisiblity of d(k, n, N) by powers of the prime k. The induction argument has three ingredients: linear operators acting on linear subspaces of K(Γ_{o}(k)), relations obtained by viewing certain Γ_{o}(k)-invariant functions as Γ(k, k)-functions and delicate combinatorial analysis of congruence properties of the integers c_{i}. The first two of these ingredients are discussed in the next two sections.

4. Linear operators

Fix a prime k. For each nonnegative integer n, we have encountered a linear operator

U_{k,n} : K(Γ(k^{n}, k)) → K(Γ_{o}(k))

defined by

U_{k,n}(f)(τ) = \frac{1}{k^{n}} ∑_{l=0}^{k^{n}−1} f(τ + l).

We abbreviate U_{k,1} = U_{k}. Observe that for n ∈ ℤ^{+} ∪ {0},

U_{k,2n+1} ((\frac{η(k·)}{η (\frac{·}{k^{2n+1}} )})^{α(k)N} ) = F_{k,2n+1,N} and U_{k,2n} ((\frac{η}{η (\frac{·}{k^{2n}} )})^{α(k)N} ) = F_{k,2n,N}.

In particular,

U_{k} ((\frac{η(k·)}{η (\frac{·}{k})})^{α(k)N} ) = F_{k,N} and (\frac{η(k·)}{η (\frac{·}{k})})^{α(k)} = h_{k} for k = 2, 3 and 5.

More important for our work is the ℂ-linear operator

V_{k} : K(Γ_{o}(k)) → K(Γ_{o}(k)), V_{k}(f)(τ) = \frac{1}{k} ∑_{l=0}^{k−1} f (\frac{τ + l}{k}) ,

that is part of a family of operators V_{k,M}, M ∈ ℤ (V_{k} = V_{k,0}), defined by

V_{k,M}(f)(τ) = \frac{1}{k} ∑_{l=0}^{k−1} (\frac{η(k(τ + l))}{η (\frac{τ+l}{k})})^{α(k)M} f (\frac{τ + l}{k}) , τ ∈ ℍ^{2}.

If k = 2, 3, 5, 7 or 13, then f_{k}^{N} ∈ K(Γ_{o}(k))_{0} (f_{k}^{N} ∈ K(Γ_{o}(k))_{∞}) provided N > 0 (N < 0). Calculations tell us that^{6} in these cases

V_{k,M}(f_{k}^{N}) = f_{k}^{−N}F_{k,M+ \frac{k+1}{β(k)} N}.

For all primes k, all N ∈ ℤ and all n ∈ ℤ^{+},

V_{k}(F_{k,2n−1,N}) = F_{k,2n,N} and V_{k,N}(F_{k,2n,N}) = F_{k,2n+1,N}.

The last two formulae are the basis for one of the two ingredients in induction arguments on n (the level) for congruences for the partition functions P_{N}. We need a tool for induction on N. It turns out that the same tool is the second ingredient for induction on n. It is discussed in the next section.

5. Modular equations

Every Γ_{o}(k)-invariant function is of course Γ(k, k)-invariant. In particular, for k = 2, 3, 5,

7 and 13, f_{k}^{−1}(τ) = (\frac{η(τ)}{η(kτ)})^{\frac{24}{k−1}} is a Γ_{o}(k)-invariant function with divisor P_{0}P_{∞}^{−1}. Its divisor

on \overline{ℍ^{2}/Γ(k, k)} is P_{0}P_{1}...P_{k−1}P_{∞}^{−k}. For k = 2, 3 and 5, h_{k}^{−1}(τ) = (\frac{η(\frac{τ}{k})}{η(kτ)})^{\frac{24}{k^{2}−1}} is a Γ(k, k)-

invariant function with divisor P_{0}P_{∞}^{−1}. Hence for these k, (\frac{η(\frac{τ}{k})}{η(kτ)})^{\frac{24}{k^{2}−1}} is a polynomial of

degree k in (\frac{η(τ)}{η(kτ)})^{\frac{24}{k−1}} with zero constant term. We call these and similar relations for functions on Γ(k, k), modular equations for the prime k. It is of interest to also consider the

\frac{k + 1}{β(k)} = \{ \begin{matrix}k + 1 for k = 2, 3, 5 \\ \frac{24}{k−1} for k = 5, 7, 13 \end{matrix} .

6Note that

Γ^{o}(k)-invariant function (\frac{η(τ)}{η(\frac{τ}{k})})^{\frac{24}{k−1}} with divisor P_{∞}P_{0}^{−1}; its projection to \overline{ℍ^{2}/Γ(k, k)} has

divisor P_{∞}P_{1}...P_{\frac{1}{k−1}} P_{0}^{−k}. For k = 2, 3 and 5 it is a polynomial of degree k in (\frac{η(kτ)}{η(\frac{τ}{k})})^{\frac{24}{k^{2}−1}}

with zero constant term.

5.1. k = 2. A modular equation in this case is

f_{2}^{−1} = 2^{4}h_{2}^{−1} + h_{2}^{−2} or h_{2}^{2+N} = f_{2}(2^{4}h_{2}^{1+N} + h_{2}^{N});

the first of these translates, after some elementary algebraic manipulation, to the power series identity

∏_{n=1}^{∞}(1 + x^{n})^{16} = 2^{4}x ∏_{n=1}^{∞}(1 + x^{n})^{8} ∏_{n=1}^{∞}(1 + x^{2n})^{16} + ∏_{n=1}^{∞}(1 + x^{2n})^{8}.

The second identity tell us that

F_{2,2+N} = f_{2}(2^{4}F_{2,1+N} + F_{2,N})

which along with (the easily computed expressions)

F_{2,0} = 1 and F_{2,1} = 2^{3}f_{2}

gives us a recursive way for computing F_{2,N} (for all N ∈ ℤ). It is convenient to introduce the following

Definition 4. For each prime k and each n ∈ ℤ − {0}, we let π_{k}(n) be the highest power of k that divides n. (We define π_{k}(0) = +∞.)

Lemma 1. For each N ∈ ℤ^{+},

F_{2,N} = ∑_{i=[[\frac{N}{2} ]]}^{N} c_{N,i}f_{2}^{i} with c_{N,i} ∈ ℤ^{+},

and

π_{2}(c_{N,i}) \{ \begin{matrix}= 0 if N is even and i = \frac{N}{2} \\ ≥ 7 + 8 (i − \frac{N}{2} − 1) if N is even and i > \frac{N}{2} \\ = 3 if N is odd and i = \frac{N+1}{2} \\ ≥ 3 + 8 (i − \frac{N+1}{2}) if N is odd and i > \frac{N+1}{2} \end{matrix} .

In particular,

π_{2}(c_{N,i}) ≥ 3 + 8 (i − \frac{N + 1}{2}) .

5.2. k = 3. The basic equation is now

f_{3}^{−1} = 3^{3}h_{3}^{−1} + 3^{2}h_{3}^{−2} + h_{3}^{−3} or h_{3}^{3+N} = f_{3}(3^{3}h_{3}^{2+N} + 3^{2}h_{3}^{1+N} + h_{3}^{N}).

The first of these leads to the power series identity

∏_{n=1}^{∞}(1 − x^{3n})^{12} =

3^{3}x^{2} ∏_{n=1}^{∞}(1 − x^{n})^{3} ∏_{n=1}^{∞}(1 − x^{9n})^{9} + 3^{2}x ∏_{n=1}^{∞}(1 − x^{n})^{6} ∏_{n=1}^{∞}(1 − x^{9n})^{6} + ∏_{n=1}^{∞}(1 − x^{n})^{9} ∏_{n=1}^{∞}(1 − x^{9n})^{3};

the second to the recursion formula (N ∈ ℤ)

F_{3,3+N} = f_{3}(3^{3}F_{3,2+N} + 3^{2}F_{3,1+N} + F_{3,N}).

5.3. k = 5. In this case, the basic functional equation is

f_{5}^{−1} = 5^{2}h_{5}^{−1} + 5^{2}h_{5}^{−2} + 3 · 5h_{5}^{−3} + 5h_{5}^{−4} + h_{5}^{−5};  (6)

it is equivalent to the identity between infinite products

∏_{n=1}^{∞}(1 − x^{5n})^{6} = 5^{2}x^{4} ∏_{n=1}^{∞}(1 − x^{n}) ∏_{n=1}^{∞}(1 − x^{25n})^{5} + 5^{2}x^{3} ∏_{n=1}^{∞}(1 − x^{n})^{2} ∏_{n=1}^{∞}(1 − x^{25n})^{4}+

3 · 5x^{2} ∏_{n=1}^{∞}(1 − x^{n})^{3} ∏_{n=1}^{∞}(1 − x^{25n})^{3} + 5x ∏_{n=1}^{∞}(1 − x^{n})^{4} ∏_{n=1}^{∞}(1 − x^{25n})^{2} + ∏_{n=1}^{∞}(1 − x^{n})^{5} ∏_{n=1}^{∞}(1 − x^{25n}).

We rewrite the modular equations in more general form. For each integer N,

h_{5}^{5+N} = f_{5}h_{5}^{N}(5^{2}h_{5}^{4} + 5^{2}h_{5}^{3} + 3 · 5h_{5}^{2} + 5h_{5} + 1)

and hence (applying the operator U_{5})

F_{5,5+N} = f_{5}(5^{2}F_{5,4+N} + 5^{2}F_{5,3+N} + 3 · 5F_{5,2+N} + 5F_{5,1+N} + F_{5,N}).  (7)

We know that

F_{5,0} = 1, F_{5,1} = 5f_{5}, F_{5,2} = 5^{3}f_{5}^{2} + 2 · 5f_{5},
F_{5,3} = 5^{5}f_{5}^{3} + 3 · 5^{3}f_{5}^{2} + 3^{2}f_{5} and F_{5,4} = 5^{7}f_{5}^{4} + 2^{2}5^{5}f_{5}^{3} + 2 · 5^{2}11f_{5}^{2} + 2^{2}f_{5}.

We have previously seen that for all N ∈ ℤ^{+},

F_{5,N} = ∑_{i=[[\frac{N}{5} ]]}^{N} c_{N,i}f_{5}^{i}

with c_{N,i} ∈ ℤ (Theorem 2). As a consequence of (7) we have an alternate proof of this fact, and we also have a recursive way for computing these constants. We need the following stronger version of the last formula.

Lemma 2. For each N ∈ ℤ^{+} ∪ {0},

F_{5,N} = ∑_{i=[[\frac{N}{5} ]]}^{N} c_{N,i}f_{5}^{i} with c_{N,i} ∈ ℤ^{+} and 5^{[^{\frac{5i−N−1}{2}}]}|c_{N,i}.

In particular, 5|c_{N,i} for all i provided N ≡ 1 or 2 mod 5, and 5|c_{N,i} for all i > [[\frac{N}{5} ]].

The sum is to be interpreted as 1 for N = 0 and 5^{[^{\frac{5i−N−1}{2}}]} is to be replaced by 1 for 5i − N ≤ 2.

Proof. Direct computations establish the claim for 0 ≤ N ≤ 4. For an induction (on N) argument assume that the index i in the sum runs from 1 to N (with c_{N,i} = 0 for i < [[\frac{N}{5} ]]). From the basic recursion formula (7),

c_{N+5,i} = 5^{2}c_{N+4,i−1} + 5^{2}c_{N+3,i−1} + 3 · 5c_{N+2,i−1} + 5c_{N+1,i−1} + c_{N,i−1}.

Thus by induction 5^{[^{\frac{5i−(N+5)−1}{2}}]}|c_{N+5,i}.

The group PSL(2, ℤ) acts on functions by composition. Applying the operator A to (6), we obtain another modular equation

f_{1,5}^{6} = h_{5} + 5h_{5}^{2} + 3 · 5h_{5}^{3} + 5^{2}h_{5}^{4} + 5^{2}h_{5}^{5}.

5.4. k = 7. This is a more complicated case because h_{7} is not Γ(7, 7)-invariant, but h_{7}^{2} is^{7}. On \overline{ℍ^{2}/Γ(7, 7)},

(f_{7}^{−1}) = P_{0} P_{1} ... P_{6} P_{∞}^{−7} and (h_{7}^{−2}) = P_{0}^{2} P_{∞}^{−2}.

We can certainly choose constants c_{i} and d_{j} so that

H = f_{7}^{−2} − f_{7}^{−1} ∑_{i=0}^{3} c_{i}h_{7}^{−2i} − ∑_{j=1}^{7} d_{j}h_{7}^{−2j}

has a pole of order ≤ 5 at P_{∞} and its Laurent series expansion at this point in terms of the local coordinate x = exp (\frac{2πıτ}{7} ) has zero coefficients for the exponents −4 and −2. It turns out that with this choice of constants, the coefficients for the exponents −5, −3, −1 and 0 also vanish. So the function H is identically zero for this choice of coefficients. We thus have obtained the identity

f_{7}^{−2} = f_{7}^{−1} (7h_{7}^{−6} + 5 · 7h_{7}^{−4} + 7^{2}h_{7}^{−2}) +
h_{7}^{−14} + 7h_{7}^{−12} + 3 · 7h_{7}^{−10} + 7^{2}h_{7}^{−8} + 3 · 7^{2}h_{7}^{−6} + 7^{3}h_{7}^{−4} + 7^{3}h_{7}^{−2}.

Applying the operator U_{7} yields

F_{7,7+N} = f_{7} (7F_{7,4+N} + 5 · 7F_{7,5+N} + 7^{2}F_{7,6+N}) +
f_{7}^{2} (F_{7,N} + 7F_{7,1+N} + 3 · 7F_{7,2+N} + 7^{2}F_{7,3+N} + 3 · 7^{2}F_{7,4+N} + 7^{3}F_{7,5+N} + 7^{3}F_{7,6+N}) .

Remark 2. Not all the coefficients in the recursion formulae for F_{k,N} are divisible by the prime k; this is a main reason why congruence relations are not easily obtained.

5.5. k = 13. On \overline{ℍ^{2}/Γ(13, 13)}

(f_{13}^{−1}) = P_{0} P_{1} ... P_{12} P_{∞}^{−13} and (h_{13}^{−7}) = P_{0}^{7} P_{∞}^{−7}.

Any two meromorphic functions on the same Riemann surface are, of course, algebraically related. It is not clear how to use the complicated identity that connects the above two functions.

6. Constant functions

Fix a prime k. The functions F_{k,N} are never constant if N ∈ ℤ^{+}. Of course, F_{k,0} = 1. Assume that −N ∈ ℤ^{+}. The function F_{k,N} is constant for many choices of k and N. Since the only possible pole of this function is at P_{∞}, it will be constant if and only if it is regular there. The condition that the function F_{k,N} be regular at P_{∞} is that the coefficients for negative indices of an appropriate Laurent series expansion vanish. This is our next

Theorem 3. Let k be a positive prime and let N ∈ ℤ^{+}. The following conditions are equivalent:

(a) F_{k,−N} is a constant function.

(b) P_{−α(k)N}(mk + β(k)N) = 0 for m = − [\frac{β(k)N}{k} ] , ..., −2, −1.

7The surface \overline{ℍ^{2}/Γ(7, 7)} has genus one.

(c) the function F_{k,−N} is invariant under the map A_{k} : τ ↦ −\frac{1}{kτ} . Under these conditions, the constant value of F_{k,−N}(τ) is P_{−α(k)N}(β(k)N).

Proof. Only that (c) implies (a) requires some comment. If F_{k,−N} is invariant under the automorphism A_{k} of \overline{ℍ^{2}/Γ_{o}(k)}, then (since this function is holomorphic at P_{0}) it is holomorphic at P_{∞}; hence constant.

We illustrate in one case. For k = 5, the functions F_{5,−N} for N = 1, 2, 3 and 4, are constant for trivial reasons (this function has a pole of order at most [\frac{N}{5} ] at P_{∞} and is regular elsewhere on \overline{ℍ^{2}/Γ_{o}(5)}). In addition, only F_{5,−8} and F_{5,−14} are constant. This follows from two facts. The first of these are the general formulae P_{−N}(1) = −N (used for N ≡ 1 mod 5), P_{−N}(2) = \frac{N(N−3)}{2} (used for N ≡ 2 mod 5), P_{−N}(3) = −\frac{N(N−1)(N−8)}{3!} (used for N ≡ 3 mod 5), P_{−N}(4) = \frac{N(N−1)(N−3)(N−14)}{4!} (used for N ≡ 4 mod 5) and P_{−N}(0) = 1 (used for N ≡ 0 mod 5). The second fact we need is the calculation 0 = P_{−14}(4) = P_{−14}(9). We also use that P_{−8}(8) = −5^{3} and P_{−14}(14) = 5^{6}.

It is obvious that F_{k,n,0} = 1 for all k and n ∈ ℤ^{+}. If for some odd n, F_{k,m,N} = const^{8} (thus N ≤ 0), then also F_{k,n+1,N} = const; while if for some even n, F_{k,n,N} = const, then F_{k,n+1,N} = const F_{k,N}. This observation is the basis for the next theorem and provides an alternative way to establish higher level congruences for P_{−N}, N ∈ ℤ^{+} .

Theorem 4. If F_{k,−N} is constant (= P_{−α(k)N}(β(k)N)) for the prime k and N ∈ ℤ^{+}, then so is F_{k,n,−N} (= P_{−α(k)N}(β(k)N)^{[^{\frac{n+1}{2}}]}) for all n ∈ ℤ^{+}.

Proof. Let c = P_{−α(k)N}(β(k)N). Assume by induction that F_{k,n,−N} is constant for n ∈ ℤ^{+}. If n is odd, then

F_{k,n+1,−N} = V_{k}(c^{[^{\frac{n+1}{2}}]}) = c^{[^{\frac{n+2}{2}}]}.

For even n,

F_{k,n+1,−N} = V_{k}(c^{[^{\frac{n+1}{2}}]}f_{k^{2},1}^{−α(k)N}) = c^{[^{\frac{n}{2}}]}F_{k,−N} = c^{[^{\frac{n+2}{2}}]}.

As a consequence we have, for example, for k = 5, N = −4 and n ∈ ℤ^{+},

∏_{m=1}^{∞}(1 − x^{5m})^{−4} ∑_{m=0}^{∞} P_{−4} (5^{n}m + \frac{5^{2[\frac{n+1}{2}]} − 1}{6}) x^{m} = (−5)^{[^{\frac{n+1}{2}}]}.

7. Partition identities, positive N

There are quite general results about π_{k}(d(k, n, N)) for k = 2, 3, 5, 7 and 13 (see, for example, [4]; these are the cases with \overline{ℍ^{2}/Γ_{o}(k)} of genus 0); the corresponding results ([3]) for k = 11, for example, are not as general. These divisibility properties of d are partition congruences. As a sample, we illustrate for the prime 5.

Theorem 5. For all n and m ∈ ℤ^{+}, (a)

P (5^{n}m − \frac{5^{2[\frac{n+1}{2}]} − 1}{24}) ≡ 0 mod 5^{n}

8The symbol const will be used to represent a (usually nonzero) constant whose value might change with multiple uses in a single equation.

and (b)

P_{2} (5^{n}n − \frac{5^{2[\frac{n+1}{2}]} − 1}{12}) ≡ 0 mod 5^{1+[\frac{n}{2}]}.

Similar congruences exist for P_{N} for all N ∈ ℤ^{+}. The results for the primes 2, 3 and 7 are similar. Those for the prime 13, are much more restrictive. Among them are general results of the type

P_{10}(13m + 8) ≡ 0 mod 13, all m ∈ ℤ^{+} ∪ {0},

and more specialized, perhaps accidental, formulae such as

P_{−15}(13m + 1) ≡ 0 mod 5, all m ∈ ℤ^{+} ∪ {0}.

The congruence ([23], [20])

P (13m − 7) ≡ 11P_{−11}(m − 1) mod 13, m ∈ ℤ^{+},

hints at possible deeper undiscovered results.

8. Partition identities, negative N

Although Theorem 1 holds for all N ∈ ℤ, the consequences (Corollary 1 and Theorem 2) needed for applications hold for N ∈ ℤ^{+}. For negative N, these two facts hold for n = 1. We must, in this case, replace I_{2} (and all relevant upper limit for sums involving the coefficients c_{i}) by 0. In some real sense, the case of negative N is easier to handle because of the appearance of constant (and, more generally, low degree) functions. It leads to stronger results; replacing congruences by equalities.

Once again, we illustrate for k = 5. We already saw that F_{5,−N} is a constant functions for N = 1, 2, 3, 4, 8 and 14. Hence, for these values of N, we have nontrivial two term level one congruences.^{9} They are

P_{−1}(5m + 1) = −P_{−1} (\frac{m}{5}) , P_{−2}(5m + 2) = −P_{−2} (\frac{m}{5}) ,

P_{−3}(5m + 3) = 5P_{−3} (\frac{m}{5}) , P_{−4}(5m + 4) = −5P_{−4} (\frac{m}{5}) ,

P_{−8}(5m + 3) = −5^{3}P_{−8} (\frac{m − 1}{5}) and P_{−14}(5m + 4) = 5^{6}P_{−4} (\frac{m − 2}{5}) .

Each of these has a higher level analogue (for each n ∈ ℤ^{+} ∪ {0}):

P_{−1} (5^{2n}m + \frac{5^{2n} − 1}{24}) = (−1)^{n}P_{−1}(m) and P_{−1} (5^{2n+1}m + \frac{5^{2n+2} − 1}{24}) = (−1)^{n+1}P_{−1} (\frac{m}{5}) ,

P_{−2} (5^{2n}m + \frac{5^{2n} − 1}{12}) = (−1)^{n}P_{−2}(m) and P_{−2} (5^{2n+1}m + \frac{5^{2n+2} − 1}{12}) = (−1)^{n+1}P_{−2} (\frac{m}{5}) ,

P_{−3} (5^{2n}m + \frac{5^{2n} − 1}{8}) = 5^{n}P_{−3}(m) and P_{−3} (5^{2n+1}m + \frac{5^{2n+2} − 1}{8}) = 5^{n+1}P_{−3} (\frac{m}{5}) ,

P_{−4} (5^{2n}m + \frac{5^{2n} − 1}{6}) = (−5)^{n}P_{−4}(m) and P_{−4} (5^{2n+1}m + \frac{5^{2n+2} − 1}{6}) = (−5)^{n+1}P_{−4} (\frac{m}{5}) ,

9Since we have formulae for P_{N}(m) for N = −1 and −3, all m ∈ ℤ^{+} ∪ {0}. Congruence for P_{−3} and P_{−1} should follow directly from (3) and (4), respectively. In this approach, they are part of a more general phenomenum.

P_{−8} (5^{2n}m + \frac{5^{2n} − 1}{3}) = (−5)^{3n}P_{−8}(m) and P_{−8} (5^{2n+1}m + \frac{5^{2n+2} − 1}{3}) = (−5)^{3(n+1)}P_{−8} (\frac{m}{5}) ,

P_{−14} (5^{2n}m + 7\frac{5^{2n} − 1}{12}) = 5^{6n}P_{−14}(m) and P_{−14} (5^{2n+1}m + 7\frac{5^{2n+2} − 1}{12}) = 5^{6(n+1)}P_{−14} (\frac{m}{5}) .

Partition identities for P_{N} with negative N are explored in more detail in [8] and the literature quoted there. As remarked earlier, the above two term identities arise because certain functions are constant. Three term identities result from the construction of functions of degree one. For the prime 5, the resulting identity is the recursion formula for P_{−6}

P_{−6}(5m + 1) = −5^{2}P_{−6} (\frac{m − 1}{5}) − 2 · 3P_{−6}(m).

9. The primes k = 11, 17 and 19

The closed surface \overline{ℍ^{2}/Γ_{o}(k)} has genus one for k = 11, 17 and 19. A good theory for congruences for P has been developed for the first of these cases, [4]. For smaller primes, the modular equation is a key tool for deriving higher level congruences from the one for level one. For k = 11, this tool is not easily available and substitutes must be found. Level one congruences can be derived many ways (see, for example, [8].) Results for the primes 17 and 19 are even less accessible.

Theorem 6. For all n and m ∈ ℤ^{+},

P (11^{n}m − \frac{11^{2[\frac{n+1}{2}]} − 1}{24}) ≡ 0 mod 11^{n}.

There are no known comparable results for the higher partition functions P_{N}, N ≥ 2. When combined with the congruences for the primes below 11, the last theorem yields the following

Corollary 2. (a) If 3m ≡ 1 mod 2^{a}, then

P_{8}(m) ≡ 0 mod 2^{3[[\frac{a+1}{2}]]}.

(b) If 8m ≡ 1 mod 3^{b}, then

P_{3}(m) ≡ 0 mod 3^{1+[\frac{3b}{2}]}.

(c) If 24m ≡ 1 mod 5^{c}7^{d}11^{e}, then

P (m) ≡ 0 mod 5^{c}7^{1+[\frac{d}{2}]}11^{e}.

This last result should be contrasted with the identity established by Ramanujan for the Klein ȷ-function

ȷ(τ) = ∑_{n=−1}^{∞} c_{n}x^{n} = \frac{(1 + 240 ∑_{n=1}^{∞} σ_{3}(n)x^{n})^{3}}{xP^{−24}(x) − 744} , x = exp(2πıτ), τ ∈ ℍ^{2},

where for N ∈ ℤ^{+},

σ_{N}(n) = ∑_{d∈ℤ^{+}; d|n}d^{N}.

Ramanujan’s result is: if m ≡ 0 mod 2^{a}3^{b}5^{c}7^{d}11^{e}, then

c(m) ≡ 0 mod 2^{3a+2^{3}}3^{2b+3}5^{c}7^{d}11^{e}.

10. Abelian differentials of the third kind on spheres

Let G be a subgroup of Γ such that \overline{ℍ^{2}/G} is conformally equivalent to the Riemann sphere ℂ ∪ \{∞\}. Let f be a meromorphic function of degree one on \overline{ℍ^{2}/G}. If f has a zero and pole at x and y ∈ \overline{ℍ^{2}/G}, respectively, then ω = \frac{df}{f} is an abelian differential of the third kind on \overline{ℍ^{2}/G} regular except for simple poles at x and y with residues +1 and −1, respectively. Independent constructions of f and ω will lead to interesting identities connecting partition functions with the σ-function, involving the divisors of integers. See, for example, Theorem 8, below.

11. Combinatorial interpretations of some identities

The purpose of this second part of our paper is to convince the reader that some of the identities discussed in the first part have extremely interesting combinatorial and number theoretic interpretations. It is not clear whether all the results we expose require these identities, but the fact that they follow so easily seems worthy of remark and note. In fact we shall see that some of the results also follow from simpler well known classical identities. A popular title for this section could very well be ”Curious properties of integers.” We describe these properties in the form of theorems. The theorems all involve in some way partitions of the positive integers.

11.1. Let S_{0} denote the set consisting of 8 copies of the positive even integers and S_{1} denote the set consisting of 8 copies of the positive odd integers. We shall consider partitions of the integers with parts (summands) taken from the sets S_{0} and (or) S_{1}. It is of course immediate that there are no partitions of an odd integer from the set S_{0}.

Denote by L_{0}^{8}(2N) the number of partitions of the positive even integer 2N with parts taken from S_{0} and by L_{1}^{8}(N) the number of partitions of the positive integer N with parts from S_{1}.

Theorem 7. For all N ∈ ℤ^{+},

L_{1}^{8}(2N + 1) = 8L_{0}^{8}(2N).

Proof. The theorem is a consequence of the theta identity

η^{8} (\frac{τ}{2}) + η^{8} (\frac{τ + 1}{2}) = −2^{4}η^{8}(2τ)

(part of the theory for the prime k = 2) that translates to the infinite product identity (for x = exp{πıτ})

∏_{n=1}^{∞}(1 − x^{2n−1})^{8} − ∏_{n=1}^{∞}(1 + x^{2n−1})^{8} = −2^{4}x ∏_{n=1}^{∞}(1 + x^{2n})^{8}.  (8)

We find the above result quite striking but can understand that one could react to it with ”so what?” Why are statements about partitions interesting? In order to answer this question we present another example which on the one hand attempts to answer this question and on the other presents another curious property. The above identity is also an immediate consequence of the Jacobi quartic identity.

Remark 3. The modular equation for k = 2 is equivalent to (8). The theory for k = 4 leads to the identity

2^{4} ((\frac{θ [ \begin{matrix}0 \\ 0 \end{matrix} ] (0, 2τ)}{θ [ \begin{matrix}1 \\ 0 \end{matrix} ] (0, 2τ)})^{4} − 1) = (\frac{η(τ)}{η(4τ)})^{8}

that also translates to the product identity (8).

11.2. Let S be the set consisting of 4 copies of the odd positive integers and 4 copies of the even positive integers which are congruent to zero mod 4. A partition of N ∈ ℤ^{+} taken from the set S is called even (odd) provided it contains an even (odd) number of even parts (summands); d(N) = e(N) − o(N) denotes the difference between the number of even and odd partitions of N. Denote by T_{4}(N) the number of representations of N as a sum of four triangular numbers (a number of the form \frac{m(m+1)}{2} with m ∈ ℤ^{+} ∪ \{0\}. Each representation of N is a vector (n_{1}, n_{2}, n_{3}, n_{4}) with n_{j} triangular and ∑_{j=1}^{4} n_{j} = N with different vectors corresponding to distinct representations. As usual σ(N) is the sum of the positive divisors of N.

Theorem 8. For each N ∈ ℤ^{+},

d(N) = σ(2N + 1) = T_{4}(N).

Proof. The k = 2 theory leads to the theta identity

const \frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ]^{\frac{8}{3}} (0, 2τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ]^{\frac{4}{3}} (0, τ)} = \frac{d}{dτ} log \frac{θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] (0, τ)}{θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] (0, τ)}.

The identity is a consequence of the fact that both sides of the equality are meromorphic one forms on the Riemann surface which is the compactification of the upper half plane factored by the group Γ(2, 2) with the same divisors. This identity translates to the infinite product identity

x ∏_{n=1}^{∞}(1 − x^{2n})^{4}(1 + x^{2n})^{8} = ∑_{n=1}^{∞} σ(2n − 1)x^{2n−1}.

We explain how the combinatorial results follow.

Simple algebraic manipulations (using 1 − y^{2} = (1 − y)(1 + y)) followed by a change of variable (replacing x^{2} by x) and some more algebraic manipulation leads to

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

whose combinatorial content is the first equality of the theorem.

It remains to make the connection with T_{4}(N). This follows from the identity

∏_{n=1}^{∞}(1 − x^{4n})(1 + x^{2n−1}) = ∑_{n=0}^{∞} x^{\frac{n^{2}+n}{2}}  (9)

which is a consequence of the Jacobi triple product formula (5).

The above results are consequences of the k = 2 theory and as such are in a sense classical since this theory has been well understood for a long while. We do not know if the above interpretations have been made but they probably have. Before we leave this case however it seems worthwhile to point out that the above relates partition theory to one of the most basic functions of number theory, the σ-function. As a consequence of this relation we have the following characterization of primes.

Theorem 9. The odd number 2l + 1 is a prime if and only if

d(l) = T_{4}(l) = 2(l + 1).

We now turn to the case k = 3 where we expect some new results. The first of these again involves the difference between the number of even and odd partitions.

11.3. Let S be the set consisting of 4 copies of those positive integers congruent to 0 mod 9, 3 copies of the positive integers which are relatively prime to 9, and 1 copy of those positive integers which are congruent to either 3 or 6 mod 9. As before, the parity of a partition is the parity of the number of its parts. We shall let E(N) and O(N) denote the number of even and odd partitions of N.

Theorem 10. For all N ∈ ℤ^{+}, N ≥ 2,

E(N − 1) − O(N − 1) = \{ \begin{matrix}0 if N ≡ 0 mod 3 \\ σ(N) if N ≡ 1 mod 3 \\ −σ(N) if N ≡ 2 mod 3 \end{matrix} .

Proof. The theorem follows from, once again, a theta constant identity:

const \frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 3τ)θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{3})}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ]^{\frac{2}{3}} (0, τ)} = \frac{d}{dτ} log \frac{θ [ \frac{1}{3} \frac{1}{3} ] (0, τ)}{θ [ \frac{1}{3} \frac{5}{3} ] (0, τ)},

which in terms of the η-function is

const \frac{η^{3}(3τ) η^{3} (\frac{τ}{3})}{η^{2}(τ)} = \frac{d}{dτ} log \frac{θ [ \frac{1}{3} \frac{1}{3} ]^{3} (0, τ)}{θ [ \frac{1}{3} \frac{5}{3} ]^{3} (0, τ)},

whose translation in terms of infinite products and sums (with x = exp (\frac{2πıτ}{3} )) is

x ∏_{n=1}^{∞} \frac{(1 − x^{9n})^{3}(1 − x^{n})^{3}}{(1 − x^{3n})^{2}} = ∑_{n=0}^{∞} (σ(3n + 1)x^{3n+1} − σ(3n + 2)x^{3n+2}) .

We see that 9 plays a special role in the previous theorem. It seems to be the case that 9 is always special in the theory for k = 3. We give another example to demonstrate this point.

11.4. Two different constructions of a certain meromorphic differentials on the Riemann surface \overline{ℍ^{2}/Γ(3)} yield the theta identity

const \frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ]^{\frac{10}{3}} (0, τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 3τ)θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{3})} = \frac{d}{dτ} log \frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 3τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{3})};

which in terms of the η-function can be written

const \frac{η^{10}(τ)}{η^{3}(3τ) η^{3} (\frac{τ}{3})} = \frac{d}{dτ} log \frac{η^{3}(3τ)}{η^{3} (\frac{τ}{3})},

that translates in terms of the variable x = exp(\frac{2πıτ}{3} ) to the identity

∏_{n=1}^{∞} \frac{(1 − x^{3n})^{10}}{(1 − x^{n})^{3}(1 − x^{9n})^{3}} = 1 + 3 ∑_{n=1}^{∞} σ(n)(x^{n} − 9x^{9n}) = 1 + 3 ∑_{n=1}^{∞} σ̂(n)x^{n},  (10)

where σ̂(N) is the sum of the divisors of N which are not congruent to zero mod 9. A different variant of the σ-function, is obtained by setting σ̃(N) to be the sum of the divisors of N which are not congruent to zero mod 3. Averaging the meromorphic one form on ℍ^{2}/Γ(3, 3) = ℍ^{2}/Γ(3) given above over a group of translations, we obtain a one form on ℍ^{2}/Γ_{o}(3) that yields the theta identity

const \frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ]^{\frac{10}{3}} (0, τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 3τ)} ∑_{l=0}^{2} \frac{exp (\frac{2πıl}{24})}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{3})} = \frac{d}{dτ} log (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 3τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ)})^{4} .

This theta identity translates to

∏_{n=1}^{∞} \frac{(1 − x^{n})^{10}}{(1 − x^{3n})^{3}} (∑_{n=0}^{∞} P_{3}(3n)x^{n}) = 1 + 12 ∑_{n=1}^{∞} σ̃(n)x^{n}.

Replacing x by x^{3}, multiplying by ∏_{n=1}^{∞} \frac{1}{(1−x^{n})^{3}} and using (10) tells us that

(1 + 3 ∑_{n=1}^{∞} σ̂(n)x^{n}) ∏_{n=1}^{∞}(1 − x^{n})^{3} = \frac{1 + 12 ∑_{n=1}^{∞} σ̃(n)x^{3n}}{∑_{n=0}^{∞} P_{3}(3n)x^{3n}};

which as a consequence of the Jacobi identity can be rewritten as

(1 + 3 ∑_{n=1}^{∞} σ̂(n)x^{n}) (∑_{n=0}^{∞}(−1)^{n}(2n + 1)x^{\frac{n(n+1)}{2}} ) = \frac{1 + 12 ∑_{n=1}^{∞} σ̃(n)x^{3n}}{∑_{n=0}^{∞} P_{3}(3n)x^{3n}} .  (11)

It is clear from the above identity that the right hand side is a function of x^{3} so that the same is true of the left hand side. We have thus established most of

Theorem 11. If n ∈ ℤ^{+} is congruent to either 1 or 2 mod 3, then

3 ∑_{j=0}^{\frac{j^{2}+j}{2} ≤n−1}(−1)^{j}(2j + 1)σ̂ (n − \frac{j(j + 1)}{2}) + P_{−3}(n) = 0.

Proof. Take the Cauchy product of the two series to the left of the equality in (11) and use the fact that the terms corresponding to exponents which are not congruent to 0 mod 3 vanish.

The above theorem has an amusing and interesting by-product. Let us begin by recalling that triangular numbers, those of the form \frac{j(j+1)}{2}, are always congruent to either 0 or 1 mod 3. They are never congruent to 2 mod 3. Furthermore, if n is congruent to 2 mod 3, then for any j, the difference n − \frac{j(j+1)}{2} is never congruent to 0 mod 3; so surely never congruent to 0 mod 9. It thus follows that

Corollary 3. If N is congruent to 2 mod 3, then

∑_{j=0}^{\frac{j^{2}+j}{2} ≤N−1}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) = 0.

This gives a recursion formula for σ(n) in terms of a rather thin set of values.

The above corollary leaves open the question of what happens when N is congruent to 1 mod 3. It turns out that the last formula with a correction for triangular numbers is valid in this case. This was pointed out to one of us by Frank Garvan. Garvan’s proof uses the Jacobi identity (3) and proceeds as follows. If we let f = P^{−3}, then

−x\frac{f′(x)}{f(x)} = ∑_{n=1}^{∞} \frac{3x^{n}}{1 − x^{n}} = 3 ∑_{n=1}^{∞} σ(n)x^{n}, x ∈ Δ.

Hence it follows that

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

and

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

with

a_{n} = ∑_{j=0}^{\frac{j(j+1)}{2} ≤n−1}(−1)^{j}(2j + 1)σ (n − \frac{j(j + 1)}{2}) .

Hence we conclude that if N is not a triangular number, then

∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) = 0;

while if N is a triangular number N = \frac{m(m+1)}{2}, we have

∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) = (−1)^{m+1}(2m + 1)\frac{m(m + 1)}{6}.

We therefore see that the formula of the corollary is also correct for numbers that are congruent to 1 mod 3 provided they are not triangular.

The above observations together with the last theorem therefore give Theorem 12. Assume N is congruent to 1 modulo 3. If N is not a triangular number we have

∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) = ∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ̂ (N − \frac{j(j + 1)}{2})

and in particular

∑_{j∈ℤ^{+}, j≡1 mod 3}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) =

∑_{j∈ℤ^{+}, j≡1 mod 3}(−1)^{j}(2j + 1)σ̂ (N − \frac{j(j + 1)}{2}) .

If N is a triangular number

∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ (N − \frac{j(j + 1)}{2}) = N ∑_{j=0}^{\frac{j(j+1)}{2} ≤N−1}(−1)^{j}(2j + 1)σ̂ (N − \frac{j(j + 1)}{2}) .

In the theorem, we interpret both σ(n) and σ̂(n) to be zero if n ∉ ℤ^{+}.

11.5. In the above theorems and examples we have seen that the σ-function or some variant of it enters into the discussion of partitions of the integers. That this is so should not be surprising. It follows from the fact that

\frac{xP′(x)}{P(x)} = ∑_{n=1}^{∞} \frac{nx^{n}}{1 − x^{n}} = ∑_{n=1}^{∞} σ(n)x^{n}, x ∈ Δ.

Therefore

∑_{n=1}^{∞} nP (n)x^{n} = (∑_{n=0}^{∞} P (n)x^{n}) (∑_{n=1}^{∞} σ(n)x^{n}) = ∑_{n=1}^{∞} a_{n}x^{n},

with

nP (n) = a_{n} = ∑_{j=0}^{n−1} P (j)σ(n − j).

Hence we see that the σ-function is intimately connected with the partition function. It is well known that P (n) is expressible in terms of previous values P (m), m < n. This last fact follows from the observation that PP^{−1} = 1; that is, from the identity

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

This last identity says that for all n ∈ ℤ^{+}

∑_{j∈ℤ; \frac{3j^{2}+j}{2} ≤n}(−1)^{j}P (n − \frac{3j^{2} + j}{2} ) = 0.

In a similar vein, if we work with f = P^{−1}, then,

\frac{−xf′(x)}{f(x)} = ∑_{n=1}^{∞} σ(n)x^{n}, x ∈ Δ;

so that

∑_{n=−∞}^{∞}(−1)^{n+1} \frac{3n^{2} + n}{2}x^{\frac{3n^{2}+n}{2}} = (∑_{n=1}^{∞} σ(n)x^{n}) ( ∑_{n=−∞}^{∞}(−1)^{n}x^{\frac{3n^{2}+n}{2}} ) .

Therefore if n is not pentagonal

∑_{j∈ℤ; \frac{3j^{2}+j}{2} ≤n−1}(−1)^{j}σ (n − \frac{3j^{2} + j}{2} ) = 0

and if n is pentagonal, n = \frac{3m^{2}+m}{2}, m ∈ ℤ,

∑_{j∈ℤ; \frac{3j^{2}+j}{2} ≤n−1}(−1)^{j}σ (n − \frac{3j^{2} + j}{2} ) = (−1)^{m+1} \frac{3m^{2} + m}{2}.

11.6. We present another curious property which is related to the results in Theorem 8. The identity (9) that we used there can obviously be rewritten as

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

Denoting the left hand side, as usual, by f(x) we get

\frac{xf′(x)}{f(x)} = ∑_{n=1}^{∞} (\frac{(2n − 1)x^{2n−1}}{1 − x^{2n−1}} − \frac{2nx^{2n}}{1 − x^{2n}} ) = ∑_{n=1}^{∞}(σ_{o}(n) − σ_{e}(n))x^{n},

where σ_{o}(n) and σ_{e}(n) represent the sum of the odd and even divisors of n respectively. It thus follows that

∑_{n=1}^{∞} \frac{n^{2} + n}{2}x^{\frac{n^{2}+n}{2}} = (∑_{n=1}^{∞}(σ_{o}(n) − σ_{e}(n))x^{n}) (∑_{n=0}^{∞} x^{\frac{n^{2}+n}{2}} ) = ∑_{n=1}^{∞} a_{n}x^{n},

with

a_{n} = ∑_{j=0}^{\frac{j^{2}+j}{2} ≤n−1} σ_{o} (n − \frac{j^{2} + j}{2} ) − σ_{e} (n − \frac{j^{2} + j}{2} ) .

The conclusion is therefore

Theorem 13. If N is not a triangular number then

∑_{j=0}^{\frac{j^{2}+j}{2} ≤N−1} σ_{o} (N − \frac{j^{2} + j}{2} ) − σ_{e} (N − \frac{j^{2} + j}{2} ) = 0.

11.7. Corollary 3 has an interesting consequence. It implies that σ(3N+2) ≡ 0 mod 3. However this fact follows easily without the corollary. If n is congruent to 2 mod 3, then clearly n must have at least one prime factor congruent to 2 mod 3 appearing with odd multiplicity. Denoting the factor by p and its multiplicity by 2l+1 we see that σ(p^{2l+1}) = \frac{p^{2l+2}−1}{p−1} ≡ 0 mod 3. Since the σ-function is multiplicative we are done. Once one realizes this fact, one immediately suspects that the σ-function satisfies lots of congruences. It is not hard to see that

1. σ(2N + 1) ≡ 0 mod 2 unless 2N + 1 is a square, 2. σ(3N + 2) ≡ 0 mod 3, 3. σ(4N + 3) ≡ 0 mod 4,

4. σ(6N + 5) ≡ 0 mod 6, 5. σ(8N + 7) ≡ 0 mod 8,

6. σ(12N + 11) ≡ 0 mod 12 and 7. σ(24N + 23) ≡ 0 mod 24.

It would be nice to see if these relatively simple congruences for the σ-function give some of the less simple congruences for the partition function.

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