Contemporary Mathematics Volume 00, 19xx

Ramanujan partition identities

Hershel M. Farkas and Irwin Kra

1. Introduction and summary of some results

1.1. The classical Ramanujan partition function P defined by

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

has been the subject of much study since Ramanujan made many conjectures and proved some remarkable theorems concerning congruence properties of the numbers P (n) appearing above. Ramanujan proved that^{1}

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

He also more or less conjectured (see [21] and [12, Ch. 7]) and partially proved the more general statements; one for each fixed 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^{[^{\frac{n+2}{2}}]}.

(c) If 24m ≡ 1 mod 11^{n}, then P (m) ≡ 0 mod 11^{n}. We shall refer to these three results as the (Ramanujan level n) congruences for the primes 5, 7 and 11, respectively. Formula (1) defines, of course, the value at x in the unit disc of a nonvanishing holomorphic function P.

1.2. The great interest in the function P is due in part to the combinatorial interpretation of P (n) as the number of unrestricted partitions of n ∈ ℤ^{+}. In principle, the following class of functions, the partition functions P_{N}, N ∈ ℤ, defined by

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

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0271-4132/00 $1.00 + $.25 per page

1991 MSC: 30F10, 11F12, 11F30, 14H05, 30F30, 30F35, 14H45.

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.

1The December 1997 issue of The American Mathematical Monthly contains four articles in honor of Ramanujan on the 110th anniversary of his birth. One of these [2] in the Notes section gives what may be the shortest combinatorial proof of the first of these congruences.

which are clearly related to the classical partition function, are just as interesting and have been investigated in their own right (see, for example, [1]). Further, in order to prove the above congruences for P we need information about the congruence properties of series of the form P_{N}(km + α) for various fixed integers k, N and α. For positive N, the function P_{N} can be thought of as counting the number of partitions of the integers using N sets of colored numbers.

If N is negative, then we are in a perhaps more interesting situation. For n ∈ ℤ^{+}, P_{N}(n) represents the difference between the number of even and odd partitions of n where we are using positive integers of −N distinct colors and no repetitions are permitted. In this sense a partition is called even provided it consists of an even number of parts and odd otherwise.

The case N = −1 was in fact considered by Euler; the case N = −3 by Jacobi. These are the only cases with a closed expression for the partition function. The classical partition function corresponds to N = 1 and we always write P for P_{1}.

1.3. The Ramanujan congruences are part of a vast literature on combinatorial number theory. Exposition of various topics appear, for example, in [12]. Our work in this area proceeds from function theoretic considerations involving Riemann surfaces defined by the action on the upper half plane ℍ^{2} of subgroups of the modular group, PSL(2, ℤ). The well known congruences for the primes k = 5 and 7 are established in [8] and [13] as consequences of the theta constant identities

∑_{l=0}^{4} (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 5(τ + l))}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{5})})^{\frac{1}{3}} = 25 (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 5τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ)})^{2} ,

and

∑_{l=0}^{6} (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 7(τ + l))}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{7})})^{\frac{1}{3}} =

49 (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 7τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ)})^{\frac{4}{3}} (7 (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 7τ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ)})^{\frac{4}{3}} + 1) .

In terms of the local coordinate x = exp 2πıτ at P_{∞} on the surface \overline{ℍ^{2}/Γ_{o}(k)}, the above theta identities become

∑_{m=1}^{∞} P (5m − 1)x^{m} = 5 x \frac{∏_{n=1}^{∞}(1 − x^{5n})^{5}}{∏_{n=1}^{∞}(1 − x^{n})^{6}} (2)

for k = 5 and

∑_{m=1}^{∞} P (7m − 2)x^{m} = 7 x \frac{∏_{n=1}^{∞}(1 − x^{7n})^{3}}{∏_{n=1}^{∞}(1 − x^{n})^{4}} [7 x \frac{∏_{n=1}^{∞}(1 − x^{7n})^{4}}{∏_{n=1}^{∞}(1 − x^{n})^{4}} + 1] (3)

for k = 7, respectively. The last two equations appear as early as 1919 in the papers of Ramanujan, [19]. The level one congruences for the primes 5 and 7 are immediate consequences of these identities. The proof of the level one congruence for k = 11 uses a different method which on the one hand is, in a sense to be explained, weaker than the method used above but on the other hand gives us the desired result. For this k, our proof in [6] proceeds along the following lines. We show that the Γ_{o}(11)-invariant function

∑_{l=0}^{10} (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{11})}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 11(τ + l))})^{\frac{10}{3}}

which in terms of the local coordinate x at P_{∞} is given by

11 ∏_{n=1}^{∞}(1 − x^{11n})^{−10} ∑_{m=−4}^{∞} P_{−10}(11m + 50)x^{m}

is constant (the apparent pole of order 4 is not there). From this we conclude that

11^{4}P_{−10} (\frac{m − 4}{11}) = P_{−10}(11m + 6), m ∈ ℤ^{+} ∪ \{0\}.

One easily shows that if N ≡ 10 mod 11 and n ≡ 6 mod 11, then P_{−N}(n) ≡ 0 mod 11. Taking N = −1 gives the congruence for P (11m + 6).

The purpose of this paper is to give a more or less unified treatment, based on this last described method, which gives us the three level one congruences and in fact allows us to obtain the level two congruences for the primes 5 and 7. This paper reports essentially on only known results; the reader may consult [12] for historical references and an alternate derivation of these results. Furthermore the reader can find in [1], [10], [11], and in [17], [18] techniques which are very close to the ideas we employ here. The reason for this note is that in our opinion, our methods yield slightly more general results than other known methods and our approach explains the congruences in a very simple unified manner \underline{from} a function theoretic point of view. The congruences are essentially consequences of the fact that certain procedures produce constant functions on compact Riemann surfaces. A similar unified proof, based on function and number theoretic arguments in [14] and [15], of the three level one Ramanujan congruences appeared in [16]^{2}. The methods and results of that paper should be contrasted with those of the current manuscript.

In the above papers where the ideas are similar to ours, the function theory tends to be hidden and mysterious. We believe that we expose it here in a manner that non-experts in number theory can appreciate. In this way we feel that we are opening this area to non experts. Furthermore, the principal actor in our investigations is the theta constant with rational characteristics as opposed to the η-function which appears in almost all the other presentations. It seems to us that since the theta function is a more fundamental object which appears in many other places, its occurrence in this theory should be highlighted rather than suppressed. However, our work on partition identities involves only the z-derivative of the theta function with integral odd characteristic evaluated at z = 0 (that is, the classical η-function). In previous publications [5], [3], [4] we have also seen that certain identities when expressed in terms of theta constants tend to be simpler than when expressed in other forms. Finally, the theta function used here is the “one variable” function; however the theta function has a several variables generalization which plays a central role in the theory of compact Riemann surfaces of genus greater than one . It is hoped that ultimately some of the ideas used here will be generalized to the several variable case.

2We learned of this paper only after much of the work reported here was completed.

2. Construction of meromorphic functions

In [5], we discussed a procedure for the construction of meromorphic functions and differentials on the Riemann surfaces ℍ^{2}/Γ(k) and the calculation of their divisors. Here k ∈ ℤ^{+} is arbitrary. The technique, using theta functions, considered in that paper is useful for other subgroups of Γ = PSL(2, ℤ). In particular, it works nicely for the subgroups^{3} G(k), Γ_{o}(k) and Γ^{o}(k) as well as for the intersection of the last two listed groups which we denote by Γ(k, k). Let us now fix for the rest of this paper, a positive prime k.

The idea for constructing interesting functions is to start with a fixed Riemann surface and a meromorphic function on the surface, average the function over a group (not always a group of automorphisms of the surface) and thus obtain a meromorphic function on the quotient of the original surface by the group. This is not a new idea, however the reason it works so well in our case is that we have fairly good control over the divisors of the functions in question. In fact, in many cases we are constructing in this way constant functions and we can evaluate the constant.

We begin with the meromorphic Γ(k)-invariant function

\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, kτ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{k})},

whose divisor on \overline{ℍ^{2}/Γ(k)} is

(\frac{P_{\frac{1}{k}} P_{\frac{2}{k}} ...P_{\frac{k−1}{2k}}}{A (P_{\frac{1}{k}} P_{\frac{2}{k}} ...P_{\frac{k−1}{2k}} )})^{\frac{k^{2}−1}{8}} .

In the above, the point P_{x} is the puncture on the Riemann surface ℍ^{2}/Γ(k) represented by the point x on the (extended) real axis and A is the motion τ ↦ \frac{−1}{τ}. Thus our function is holomorphic and nonzero on the surface ℍ^{2}/Γ(k) and has a meromorphic extension to its natural compactification \overline{ℍ^{2}/Γ(k)}. The above function is not, in general, Γ(k, k)-invariant. However,

(\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, kτ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{k})})^{8}

3These families of groups are defined in [7], where we also introduced the notation used here.

is and its divisor on \overline{ℍ^{2}/Γ(k, k)} is (\frac{P_{∞}}{P_{0}} )^{k^{2}−1}. Our final observation is that

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

is Γ_{o}(k)-invariant and its divisor on \overline{ℍ^{2}/Γ_{o}(k)} is (\frac{P_{∞}}{P_{0}} )^{k−1}. We are now faced with the following question: what is the least positive integer N for which

(\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, kτ)}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ}{k})})^{\frac{8N}{k^{2}−1}} (4)

is a meromorphic function on ℍ^{2}/Γ(k, k)? Another way of phrasing this question is:

what is the least N ∈ ℤ^{+} for which the divisor (\frac{P_{∞}}{P_{0}} )^{N} is principal on \overline{ℍ^{2}/Γ(k, k)}?

Our strategy is now as follows: having found a minimal N described above, that we call β(Γ(k, k)), then all N for which (4) defines a Γ(k, k)-invariant function are multiples of this minimal one. We have shown in [6] that β(Γ(k, k)) is a factor of the maximum of 1 and \frac{k^{2}−1}{24}. We conjecture that β(Γ(k, k)) = β(k), this maximum. We have established the conjecture for k ≤ 17. Hence we define for each N ∈ ℤ^{+} the function

F_{k,−N}(τ) = \frac{1}{k} ∑_{l=0}^{k−1} (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, k(τ + l))}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{k})})^{− \frac{α(k)N}{3}} ,

where

α(k) = \{ \begin{matrix}8 for k = 2 \\ 3 for k = 3 \\ 1 for k ≥ 5 \end{matrix} .

We then easily see that F_{k,−N} is a meromorphic function on \overline{ℍ^{2}/Γ_{o}(k)} with a pole of order at most [\frac{β(k)N}{k}] at P_{∞} that is regular elsewhere. Notice that the above construction imposes severe restrictions on the function F_{k,−N}. Namely, if the genus of ℍ^{2}/Γ_{o}(k) is positive, then any N with β(k)N < 2k will give rise to a constant function automatically while if the genus of ℍ^{2}/Γ_{o}(k) is zero any N with β(k)N < k produces a constant.

We need to recall some standard facts (whose proofs can be found, for example in [8]). In terms of the local coordinate x = exp(2πıτ) at P_{∞} on \overline{ℍ^{2}/Γ_{o}(k)}, we have

θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ) = −2πx^{\frac{1}{8}} ∏_{n=1}^{∞}(1 − x^{n})^{3},

and for all l ∈ ℤ,

θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ + l) = (exp \{\frac{πıl}{4} \}) θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ).

It thus follows that

(\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{k})}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, k(τ + l))})^{\frac{8N}{k^{2}−1}} = \frac{(ϵ_{l}x^{\frac{1}{k}})^{−N} ∏_{n=1}^{∞}(1 − (ϵ_{l}x^{\frac{1}{k}})^{n})^{\frac{24N}{k^{2}−1}}}{∏_{n=1}^{∞}(1 − x^{nk})^{\frac{24N}{k^{2}−1}}} ,

where ϵ_{l} = exp (\frac{2πıl}{k}). Hence in terms of the above local coordinate

∑_{l=0}^{k−1} (\frac{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, \frac{τ+l}{k})}{θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, k(τ + l))})^{\frac{8N}{k^{2}−1}} =

∑_{n=0}^{∞} P_{\frac{24N}{k^{2}−1}} (n)x^{kn} (∑_{l=0}^{k−1}(ϵ_{l}x^{\frac{1}{k}})^{−N} ∏_{n=1}^{∞}(1 − (ϵ_{l}x^{\frac{1}{k}})^{n})^{\frac{24N}{k^{2}−1}} )

= k (∑_{n=0}^{∞} P_{\frac{24N}{k^{2}−1}} (n)x^{kn}) ( ∑_{m=−[\frac{N}{k}]}^{∞} P_{\frac{24N}{1−k^{2}}} (mk + N)x^{m}) . (5)

All of our results concerning the functions F_{k,−N} and when they are constant (in particular, all our proofs of the level one Ramanujan congruences) now flow from the above calculations.

Theorem 1. Let let k be a positive prime and let N ∈ ℤ^{+}. Then F_{k,−N} is a meromorphic function on \overline{ℍ^{2}/Γ_{o}(k)} with a pole at P_{∞} of order at most [\frac{β(k)N}{k} ].

Proof. Our definition of β(k) guarantees that (^{\frac{θ′[ \begin{matrix}1 \\ 1 \end{matrix} ](0, \frac{τ}{k})}{θ′[ \begin{matrix}1 \\ 1 \end{matrix} ](0,kτ)}} )^{\frac{8Nβ(k)}{k^{2}−1}} is a

meromorphic function on \overline{ℍ^{2}/Γ(k, k)} with divisor (\frac{P_{0}}{P_{∞}} )^{β(k)N}. We average this function over the group (viewed as a subgroup of PSL(2, ℤ_{k})) generated by the τ ↦ τ + 1 which gives us a function on \overline{ℍ^{2}/Γ_{o}(k)}. The statement about the order of this function at P_{∞} is clear as soon as one realizes that if x is a local coordinate at P_{∞} on \overline{ℍ^{2}/Γ(k, k)} then x^{k} is a local coordinate at P_{∞} on \overline{ℍ^{2}/Γ_{o}(k)}. Obviously formula (5) tells us what ord_{∞}F_{k,−N} is.

In terms of the correct local coordinate x at P_{∞} on \overline{ℍ^{2}/Γ_{o}(k)}, the function F_{k,−N}(τ) is given by (5). We have already remarked that 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 at P_{∞}. The condition that the function be regular at P_{∞} is that the negative Laurent coefficients in (5) vanish. This is our next

Theorem 2. Let k and N be as in the previous theorem. The following three 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.

(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 kP_{−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.

Remark 1. The list of [\frac{β(k)N}{k} ] arguments for the function P_{−α(k)N}

− [\frac{β(k)N}{k} ] k + β(k)N, − ([\frac{β(k)N}{k} ] + 1) k + β(k)N..., −k + β(k)N,

can also be written as

R (\frac{β(k)N}{k} ) , R (\frac{β(k)N}{k} ) + k, ..., β(k)N − k,

where R (\frac{n}{k}) is the remainder of dividing k into n. This list is to be considered as the empty set for integers N with 1 ≤ β(k)N < k.

Remark 2. It is a corollary of the last theorem that

P_{−α(k)N} (R (\frac{β(k)N}{k} )) = 0 ⟺ ord_{∞}F_{k,−N} > − [\frac{(β(k)}{k}] .

This observation provides a reformulation of Lehmer’s question on the vanishing of coefficients of the Fourier series expansion of the discriminant function from the elliptic theory (see our discussion in [8]).

3. The level one congruences

Theorem 3. For k and N satisfying one (hence all three) of the conditions of the Theorem 2,

\frac{∑_{m=0}^{∞} P_{−α(k)N}(km + β(k)N)x^{m}}{∑_{m=0}^{∞} P_{−α(k)N}(m)x^{km}} = P_{−α(k)N}(β(k)N).

Corollary 1. Let m ∈ ℤ^{+} ∪ {0}. For k and N as in the theorem^{4},

P_{−α(k)N}(km + β(k)N) = P_{−α(k)N}(β(k)N) P_{−α(k)N} (\frac{m}{k}) .

We can now prove the level one congruences for the primes 5, 7 and 11. We use (see, for example, [12, Ch. 4]) the known results that

β(Γ(5, 5)) = 1, β(Γ(7, 7)) = 2, β(Γ(11, 11)) = 5.

We give function theoretic proofs, without relying on number theoretic arguments, of these three equalities in [6]. Of course it is trivial that β(Γ(5, 5)) = 1 because \overline{ℍ^{2}/Γ(5, 5)} is conformally equivalent to the Riemann sphere. We enumerate the constant functions we need for our results. (A more complete list of constant functions is found in the appendix §5.4.)

4We use the convention that P_{N} (n) = 0 if n ∉ ℤ^{+} ∪ {0}.

(1) F_{5,−4}(τ) is constant and thus it follows that

∑_{m=0}^{∞} P_{−4}(5m + 4)x^{m} = P_{−4}(4) ∑_{m=0}^{∞} P_{−4}(m)x^{5m} = −5 ∑_{m=0}^{∞} P_{−4}(m)x^{5m};

so that, in particular, P_{−4}(5m + 4) ≡ 0 mod 5.

(2) F_{7,−3}(τ) is constant and thus

∑_{m=0}^{∞} P_{−3}(7m + 6)x^{m} = P_{−3}(6) ∑_{m=0}^{∞} P_{−3}(m)x^{7m} = −7 ∑_{m=0}^{∞} P_{−3}(m)x^{7m};

so P_{−3}(7m + 6) ≡ 0 mod 7.

(3) F_{7,−6}(τ) is constant since P_{−6}(5) = 0. We thus have

∑_{m=0}^{∞} P_{−6}(7m + 12)x^{m} = P_{−6}(12) ∑_{m=0}^{∞} P_{−6}(m)x^{7m} = 7^{2} ∑_{m=0}^{∞} P_{−6}(m)x^{7m}

and P_{−6}(7m + 5) ≡ 0 mod 7^{2}.

(4) F_{11,−10}(τ) is constant since P_{−10}(6) = P_{−10}(17) = P_{−10}(28) = P_{−10}(39) = 0. Therefore

∑_{m=0}^{∞} P_{−10}(11m + 50)x^{m} = P_{−10}(50) ∑_{m=0}^{∞} P_{−10}(m)x^{11m} = 11^{4} ∑_{m=0}^{∞} P_{−10}(m)x^{11m}

and P_{−10}(11m + 6) ≡ 0 mod 11^{4}.

The above congruences are immediate consequences of the theorem at the beginning of this section. The remarkable fact is that they allow us to conclude

Theorem 4. For all m ∈ ℤ^{+} ∪ {0}, we have

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

Proof. Clearly for each N ∈ ℤ, P = P^{−N+1}P^{N}; so that

P(x) = ∏_{n=1}^{∞} \frac{(1 − x^{n})^{N−1}}{(1 − x^{n})^{N}} .

The Frobenius automorphism tells us that for each positive prime k,

(1 − x^{n})^{k} ≡ 1 − x^{kn} mod k.

Taking the Cauchy product of the appropriate power series leads to the identity

P (n) ≡ ∑_{j=0}^{n} P (\frac{j}{k} ) P_{1−k}(n − j) mod k,

for all primes k. Since P (\frac{j}{k}) = 0 unless j is a multiple of k, we can replace the previous identity by

P (n) ≡ \begin{matrix}[ & \frac{n}{k} & ] \\ ∑ \\ l=0 \end{matrix} P (l)P_{1−k}(n − kl) mod k.

If we now take respectively for the primes k = 5, 7 and 11, and for n = 5m + 4, 7m + 5, and 11m + 6, we obtain the congruences of the theorem from the constant functions described in (1), (3) and (4) above. It is important to observe that even though the congruence obtained for P_{−6}(7m + 5) was mod 7^{2} and the congruence for P_{−10}(11m + 6) was mod 11^{4}, our result is only mod the appropriate k, since we are relying on the Frobenius automorphism.

Remark 3. Many of the ideas of this paper can be traced back to Ramanujan. For example, the congruences

P_{−4}(5m + 4) ≡ 0 mod 5 and P_{−6}(7m + 5) ≡ 0 mod 7,

used above were proved in [19] using the Euler and Jacobi identities.

It should be apparent to the reader that we have not enumerated all the possible constant functions F_{k,−N} and have therefore neglected many other possible congruences. Our aim to show how to get the level one congruences in a unified and simple way motivated our presentation. In the event there are readers who would like to see the extent of the myriad of possibilities afforded by these ideas we include in an appendix at the end of this paper some more possibilities. There are still many unanswered questions which we shall be discussing in [6].

4. The level two congruences for the primes 5 and 7

Equation (2) is much stronger than the level one congruence for the prime 5. It is equivalent to

∑_{m=0}^{∞} P (5m + 4)x^{m+1} = 5 x ∑_{m=0}^{∞} P_{−5}(m)x^{5m} ∑_{m=0}^{∞} P_{6}(m)x^{m}

and implies the recurrence formula

P (5m + 4) = 5 \begin{matrix}[ & \frac{m}{5} & ] \\ ∑ \\ j=0 \end{matrix} P_{−5}(j)P_{6}(m − 5j). (6)

Starting with the obvious relation P^{6} = PP^{5}, we conclude from the Frobenius automorphism that P_{6}(5m + 4) ≡ 0 mod 5. (This is one of the k = 5, N = 1 congruences of Theorem 7.) Hence replacing m by 5m + 4 in (6) tells us that 5^{2}|P (25m + 24). This is the level two congruence for the prime 5. Similarly, (3) can be rewritten as

∑_{m=0}^{∞} P (7m + 5)x^{m} =

7 ∑_{m=0}^{∞} P_{−3}(m)x^{7m} ∑_{m=0}^{∞} P_{4}(m)x^{m} + 7^{2} x ∑_{m=0}^{∞} P_{−7}(m)x^{7m} ∑_{m=0}^{∞} P_{8}(m)x^{m};

that implies the recurrence formula

P (7m + 5) = 7 \begin{matrix}[ & \frac{m}{7} & ] \\ ∑ \\ j=0 \end{matrix} P_{−3}(j)P_{4}(m − 7j) + 7^{2} \begin{aligned}[ & \frac{m−1}{7} & ] \\ & ∑ \\ & j=0 \end{aligned} P_{−7}(j)P_{8}(m − 1 − 7j). (7)

In the previous section we have seen that P_{−3}(7m + 6) ≡ 0 mod 7. This implies (using P^{4} = P^{−3}P^{7}) that P_{4}(7m + 6) ≡ 0 mod 7. (This is one of the k = 7, N = 0, congruences of Theorem 7.) Replacing m by 7m+6 in (7), we obtain the level two congruence for the prime 7. We have established

Theorem 5. For all m ∈ ℤ^{+} ∪ {0},
P (5^{2}m + 24) ≡ 0 mod 5^{2} and P (7^{2}m + 47) ≡ 0 mod 7^{2}.

5. Two appendices

We discuss in the first appendix (first subsection) a family of subgroups of the modular group that is useful for our work, and we list in the second (last two subsections) more consequences of Theorem 3. Both appendices will be important in future reports.

5.1. The groups Γ(k, k) are special cases of a more general family of subgroups of PSL(2, ℤ). Let p and q be two positive integers. We define

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

We let A_{q} = [ \begin{matrix}0 & −1 \\ q & 0 \end{matrix} ]. Our first observation is that for every motion γ ∈

Γ(p, q), A_{q} ∘ γ ∘ A_{q}^{−1} ∈ Γ_{o}(pq) as is easily seen from the fact that A_{q} conjugates

[ \begin{matrix}a & b \\ c & d \end{matrix} ] to [ \begin{matrix}d & −\frac{c}{q} \\ −qb & a \end{matrix} ]. Hence Γ(p, q) ≅ Γ_{o}(pq). It follows that

[Γ_{o}(q) : Γ(p, q)] = \frac{μ(Γ_{o}(pq))}{μ(Γ_{o}(q))} = p ∏_{P|p,P|̸q} (1 + \frac{1}{P} ) .

Fix a prime k, and let m and n ∈ ℤ^{+} ∪ {0}. Then

[Γ_{o}(k^{n}) : Γ(k^{m}, k^{n})] = \{ \begin{matrix}k^{m} if n > 0 or m = 0 \\ k^{m−1}(1 + k) if n = 0 and m > 0 \end{matrix} .

In particular, if n > 0, then the k^{m} motions

B^{l}; l = 0, ..., k^{m} − 1,

provide representatives for a complete set of left or right Γ(k^{m}, k^{n})-cosets for Γ_{o}(k^{n}), and if

P_{∞}, P_{0}, ..., P_{r},

is a list of punctures on ℍ^{2}/Γ_{o}(k^{n}), then

P_{∞}, P_{B^{l}(0)}, ..., P_{B^{l}(r)}, l = 0, 1, ..., k^{m} − 1,

is the corresponding list on ℍ^{2}/Γ(k^{m}, k^{n}). In particular,

P_{∞}, P_{0}, ..., P_{k^{m}−1},

are the punctures on ℍ^{2}/Γ(k^{m}, k). In the last two lists, there are duplications. We know that the number of punctures on ℍ^{2}/Γ_{o}(k^{n}) is given by

ν_{∞}(Γ_{o}(k^{n})) = \{ \begin{matrix}2k^{\frac{n−1}{2}} if n is odd \\ k^{\frac{n}{2} −1}(k + 1) if n is even \end{matrix} .

Thus ℍ^{2}/Γ(9, 3) has 6 punctures; they are

P_{∞}, P_{0}, P_{3}, P_{6}, P_{1} = P_{4} = P_{7}, P_{2} = P_{5} = P_{8}.

The motion C = [ \begin{matrix}13 & −9 \\ 3 & −2 \end{matrix} ] ∈ Γ(9, 3) maps 1 to 4, for example.

If n = 0, then P_{∞} is the only puncture on ℍ^{2}/Γ; the punctures on ℍ^{2}/Γ^{o}(k^{m}) are described in [20, Ch. I]. If m = 1, we can take

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

as right coset representatives for Γ/Γ^{o}(k).

A second case of interest is (p, q) = 1. In this case,

[Γ_{o}(q) : Γ(p, q)] = p ∏_{P|p} (1 + \frac{1}{P} ) = μ(Γ_{o}(p)).

The ideas encountered here will be explored further in [6] (see also [9]); in particular, in our study of the higher level Ramanujan congruences.

5.2. We start with a well known fact. Its proof is reproduced for the reader’s convenience.

Proposition 1. [16], [12, Th. 8 of Ch. 7] Let k be a positive prime and let N, s and t ∈ ℤ. Assume that N ≡ s mod k. If for all n ∈ ℤ^{+} with n ≡ t mod k, P_{−s}(n) ≡ 0 mod k, then for all such n, P_{−N}(n) ≡ 0 mod k.

Proof.

∑_{n=0}^{∞} P_{−N}(n)x^{n} = ∏_{n=1}^{∞}(1 − x^{n})^{N} = ∏_{n=1}^{∞}(1 − x^{n})^{Qk+s} = ∏_{n=1}^{∞}(1 − x^{n})^{Qk}(1 − x^{n})^{s}

≡ ∏_{n=1}^{∞}(1 − x^{kn})^{Q} (1 − x^{n})^{s} = ∑_{n=0}^{∞} P_{−Q}(n)x^{kn} ∑_{n=0}^{∞} P_{−s}(n)x^{n} mod k

(the next to last equality is a consequence of the Frobenius automorphism). Hence from the last equality

∑_{n=0}^{∞} P_{−N}(n)x^{n} ≡ ∑_{n=0}^{∞} P_{−Q} (\frac{n}{k}) x^{n} ∑_{n=0}^{∞} P_{−s}(n)x^{n} mod k;

comparing coefficients in this last equation yields

P_{−N}(n) ≡ ∑_{j=0}^{n} P_{−Q} (\frac{j}{k} ) P_{−s}(n − j) = \begin{matrix}[ & \frac{n}{k} & ] \\ ∑ \\ j=0 \end{matrix} P_{−Q}(j) P_{−s}(n − kj) mod k.

The last equality is the consequence of the fact that only those j’s that are multiples of k contribute to the sum.

5.3. We are interested, because some of the cases are needed in this work and others will be used in the future, in determining for which primes k and which positive integers N is F_{k,−N} constant. When F_{k,−N} is a constant function we need to know (to get good applications) when does k|P_{−α(k)N}(N).

For k = 2, 3, 5, β(Γ(k, k)) = 1, hence for N ∈ ℤ^{+}, Theorem 2 yields a function with a single pole at P_{∞} of order ≤ [\frac{N}{k} ]. Similarly, for k ≥ 5, β(k) = \frac{k^{2}−1}{24} and hence for every N ∈ ℤ^{+}, we have a function with a pole of order at most [\frac{(k^{2}−1)N}{24k} ] (the results, in these cases, apply to the partition functions P_{−α(k)N}). There are two reasons why a particular function F_{k,−N} reduces to be a constant. The first, trivial, reason is because a function of degree ≤ [\frac{β(k)N}{k} ] does not exist on

the surface \overline{ℍ^{2}/Γ_{o}(k)}; the second, computational, reason is that the combinatorial number theory of the partition of function P_{−α(k)N} forces enough Laurent series coefficients to vanish.

We have not made an exhaustive study of when we obtain the constant functions. We summarize some of our conclusions below; more details and more special cases will be found in our monograph [6]. We have included here many of the most interesting cases.

(a) As mentioned earlier, for k = 2, 3 and 5, β(Γ(k, k)) = 1. Hence in these cases for N = 1, 2, ..., k − 1, F_{k,−N} is constant since it is regular at P_{∞}.

(a1) For k = 2, only F_{2,−1} is constant. This follows from the fact that P_{−8N}(1) = −8N (used for odd N) and P_{−8N}(0) = 1 (used for even N).

(a2) For k = 3, only F_{3,−1} and F_{3,−2} are constant. This follows from the fact that P_{−3N}(1) = −3N (used for N ≡ 1 mod 3), P_{−3N}(2) = \frac{9}{2}N(N − 1) (used for N ≡ 2 mod 3) and P_{−3N}(0) = 1 (used for N ≡ 0 mod 3).

(a3) For k = 5, the functions F_{5,−N} for N = 1, 2, 3, 4, are constant for trivial reasons. 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 will also use below that P_{−8}(8) = −5^{3} and P_{−14}(14) = 5^{6}.

(b) For k = 7, we have β(Γ(7, 7)) = 2 and p(\overline{ℍ^{2}/Γ_{o}(7)})^{5} = 0. The functions F_{7,−N}, N = 1, 2, 3, are constant for trivial reasons. The functions F_{7,−6} and F_{7,−10} are constant as a result of the calculations 0 = P_{−6}(5) = P_{−10}(6) = P_{−10}(13). We also will need to know that P_{−6}(12) = 7^{2} and P_{−10}(20) = 7^{4}. There are no other constant functions F_{7,−N} as a consequence of the tabulation of P_{N}(n) in [15].

(c) For k = 11, β(Γ(11, 11)) = 5 and p(\overline{ℍ^{2}/Γ_{o}(11)}) = 1. Hence for N = 1, 2, 3, 4, the functions F_{11,−N} are constant for trivial reasons. The calculations

0 = P_{−6}(8) = P_{−6}(19), 0 = P_{−8}(7) = P_{−8}(18) = P_{−8}(29),
0 = P_{−10}(6) = P_{−10}(17) = P_{−10}(28) = P_{−10}(39),
0 = P_{−26}(9) = P_{−26}(20) = ... = P_{−26}(120),

show that F_{11,−N} are constant functions for N = 6, 8, 10 and 26. We also have

P_{−6}(30) = 11^{2}, P_{−8}(40) = −11^{3}, P_{−10}(50) = 11^{4} and P_{−26}(130) = 11^{12}.

(d) For k = 13 (β(Γ(13, 13)) = 7 and p(\overline{ℍ^{2}/Γ_{o}(13)}) = 0) there are two constant functions F_{13,−1} (for trivial reasons) and F_{13,−3} (because P_{−3}(8) = 0 but P_{−3}(21) = 13).

(e) For k = 17 (β(Γ(17, 17)) = 12 and p(\overline{ℍ^{2}/Γ_{o}(17)}) = 1) there are three constant functions F_{17,−1}, F_{17,−2} (for trivial reasons) and F_{17,−3} (because P_{−3}(2) = 0 = P_{−3}(19) but P_{−3}(36) = 17).

(f) For k = 19 (β(19) = 15, β(Γ(19, 19)) is a divisor of 15 and p(\overline{ℍ^{2}/Γ_{o}(19)}) = 1) there are three constant functions F_{19,−1}, F_{19,−2} (for trivial reasons) and F_{19,−3} (because P_{−3}(7) = 0 = P_{−3}(26) but P_{−3}(45) = −19). So is F_{19,−10} (with P_{10}(150) = 19^{4}), but F_{19,−18} is not constant.

(g) For k = 23 (β(23) = 22 and p(\overline{ℍ^{2}/Γ_{o}(23)}) = 2) there are three constant functions F_{23,−1}, F_{23,−2} (for trivial reasons) and F_{23,−3} (because P_{−3}(20) = 0 = P_{−3}(43) but P_{−3}(66) = −23).

(h) For k = 29 (β(29) = 35 and p(\overline{ℍ^{2}/Γ_{o}(19)}) = 1) there are three constant functions F_{29,−1} (for trivial reasons), F_{29,−2} (because P_{−2}(12) = 0 = P_{−2}(41) but P_{−2}(70) = −1) and F_{29,−3} (because P_{−3}(18) = 0 = P_{−3}(47) = P_{−3}(76) but P_{−3}(105) = 29).

5We denote the genus of the closed surface X by p(X).

(i) It is easy to determine at this point all the primes k for which F_{k,−N} is constant for N = 1, 2 and 3; for example, F_{67,−1} is constant.

5.4. From Theorem 3, its corollary and the calculations of the previous paragraph, we conclude the following congruences for arbitrary m ∈ ℤ^{+} ∪ {0}.

(a) For k = 2, 3, 5, and N = 1, 2, ..., k − 1, we have

P_{\frac{24N}{1−k^{2}}} (km + N) = P_{\frac{24N}{1−k^{2}}} (N) P_{\frac{24N}{1−k^{2}}} (\frac{m}{k}) .

Hence (after evaluating appropriate coefficients P_{\frac{24N}{1−k^{2}}} (N)) :

(a1) (k = 2)

P_{−8}(2m + 1) = −2^{3}P_{−8} (\frac{m}{2}) .

(a2) (k = 3)

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

(a3) (k = 5) Here we also have nontrivial congruences for N = 8 and 14.

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}) , P_{−14}(5m + 4) = 5^{6}P_{−4} (\frac{m − 2}{5}) .

(b) (The case k = 7.) For N = 1, 2, 3, 6 and 10, and all m ∈ ℤ^{+}, we have

P_{−N}(7m + 2N) = P_{−N}(2N)P_{−N} (\frac{m}{7});

that is,

P_{−1}(7m+2) = −P_{−1} (\frac{m}{7}) , P_{−2}(7m+4) = P_{−2} (\frac{m}{7}) , P_{−3}(7m+6) = −7P_{−3} (\frac{m}{7}) ,

P_{−6}(7m + 5) = 7^{2}P_{−6} (\frac{m − 1}{7}) , P_{−10}(7m + 6) = 7^{4}P_{−10} (\frac{m − 2}{7}) .

(c) (The case k = 11.) For N = 1, 2, 3, 4, 6, 8, 10 and 26, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(11m + 5N) = P_{−N}(5N) P_{−N} (\frac{m}{11});

that is,

P_{−1}(11m + 5) = P_{−1} (\frac{m}{11}) , P_{−2}(11m + 10) = P_{−2} (\frac{m}{11}) ,

P_{−3}(11m + 4) = −11P_{−3} (\frac{m − 1}{11}) , P_{−4}(11m + 9) = −11P_{−4} (\frac{m − 1}{11}) ,

P_{−6}(11m + 8) = 11^{2}P_{−6} (\frac{m − 2}{11}) , P_{−8}(11m + 7) = −11^{3}P_{−8} (\frac{m − 3}{11}) ,

P_{−10}(11m + 6) = 11^{4}P_{−10} (\frac{m − 4}{11}) , P_{−26}(11m + 9) = 11^{12}P_{−26} (\frac{m − 11}{11}) .

14 HERSHEL M. FARKAS AND IRWIN KRA

(d) (The case k = 13.) For N = 1 and 3, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(13m + 7N) = P_{−N}(7N) P_{−N} (\frac{m}{13});

that is,

P_{−1}(13m + 7) = P_{−1} (\frac{m}{13}) , P_{−3}(13m + 8) = 13P_{−3} (\frac{m − 1}{13}) .

(e) (The case k = 17.) For N = 1, 2 and 3, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(17m + 12N) = P_{−N}(12N) P_{−N} (\frac{m}{17});

that is,

P_{−1}(17m + 12) = −P_{−1} (\frac{m}{17}) , P_{−2}(17m + 7) = −P_{−2} (\frac{m − 1}{17}) ,

P_{−3}(17m + 2) = 17P_{−3} (\frac{m − 2}{17}) .

(f) (The case k = 19.) For N = 1, 2, 3 and 10, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(19m + 15N) = P_{−N}(15N)P_{−N} (\frac{m}{19});

that is,

P_{−1}(19m + 15) = −P_{−1} (\frac{m}{19}) , P_{−2}(19m + 11) = P_{−2} (\frac{m − 1}{19}) ,

P_{−3}(19m + 7) = 19P_{−3} (\frac{m − 2}{19}) ,

P_{−10}(19m + 17) = 19^{4}P_{−10} (\frac{m − 7}{19}) .

(g) (The case k = 23.) For N = 1, 2 and 3, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(23m + 22N) = P_{−N}(22N)P_{−N} (\frac{m}{23});

that is,

P_{−1}(23m + 22) = P_{−1} (\frac{m}{23}) , P_{−2}(23m + 21) = P_{−2} (\frac{m − 1}{23}) ,

P_{−3}(23m + 20) = −23P_{−3} (\frac{m − 2}{23}) .

(h) (The case k = 29.) For N = 1, 2 and 3, and all m ∈ ℤ^{+} ∪ {0}, we have

P_{−N}(29m + 35N) = P_{−N}(35N)P_{−N} (\frac{m}{29});

that is,

P_{−1}(29m + 35) = −P_{−1} (\frac{m}{29}) , P_{−2}(29m + 12) = P_{−2} (\frac{m − 2}{29}) ,

P_{−3}(29m + 18) = 29P_{−3} (\frac{m − 3}{29}) .

(i) There are many more similar identities for P_{−N} with N = 1, 2 and 3 involving primes k ≥ 31; for example^{6},

P_{−1}(67m + 187) = −P_{−1} (\frac{m}{67}) .

Remark 4. The above equalities carry much information. For example, the statement P_{−4}(5m + 4) = −5P_{−4} (\frac{m}{5}) tells us that

P_{−4}(5m + 4) ≡ 0 mod 5 for all m ∈ ℤ^{+} ∪ {0}

and also that

P_{−4}(5m + 4) = 0 for all m ∈ ℤ^{+} ∪ {0} ≢ 0 mod 5;

in particular,

0 = P_{−4}(25m + 9) = P_{−4}(25m + 14) =
P_{−4}(25m + 19) = P_{−4}(25m + 24) for all m ∈ ℤ^{+} ∪ {0}.

We rewrite our basic equality as

P_{−4}(5^{2}m + 4) = −5P_{−4}(m)

and define for each n ∈ ℤ^{+} ∪ {0},

l_{n} = 4(5^{2n} + ... + 5^{2} + 1) = \frac{5^{2n+2} − 1}{6}.

Then we conclude that

P_{−4}(5^{2n}m + l_{n−1} = (−5)^{n}P_{−4}(m)

and P_{−4}(5^{2n+1}m + l_{n}) = (−5)^{n+1}P_{−4} (\frac{m}{5}) for all m ∈ ℤ^{+} ∪ \{0\}.

The proof is by induction on n. We have established the first claim for n = 1 and the second for n = 0. The induction steps read

P_{−4}(5^{2n+2}m + l_{n}) = P_{−4}(5^{2}(5^{2n}m + l_{n−1}) + 4) = −5P_{−4}(5^{2n}m + l_{n−1}) =
(−5)^{n+1}P_{−4}(m)

and

P_{−4}(5^{2n+3}m + l_{n+1}) = P_{−4}(5^{2}(5^{2n+1}m + l_{n}) + 4) = −5P_{−4}(5^{2n+1}m + l_{n}) =

(−5)^{n+2}P_{−4} (\frac{m}{5}) .

The next theorem summarizes some of the equalities we have obtained.

Theorem 6. For all n ∈ ℤ^{+} and all m ∈ ℤ^{+} ∪ {0},

(k = 2)

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

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

(k = 3)

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

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

6Such an identity certainly exists whenever N < \frac{48k}{k^{2}−1} .

P_{−6} (3^{2n}m + \frac{3^{2n} − 1}{4}) = 3^{2n}P_{−6}(m)

and P_{−6} (3^{2n+1}m + \frac{3^{2n+2} − 1}{4}) = 3^{2(n+1)}P_{−6} (\frac{m}{3}) ,

(k = 5)

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}) ,

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})

(k = 7)

P_{−1} (7^{2n}m + \frac{7^{2n} − 1}{24}) = (−1)^{n}P_{−1}(m)

and P_{−1} (7^{2n+1}m + \frac{7^{2n+2} − 1}{24}) = (−1)^{n+1}P_{−1} (\frac{m}{7}) ,

P_{−2} (7^{2n}m + \frac{7^{2n} − 1}{12}) = P_{−2}(m) and P_{−2} (7^{2n+1}m + \frac{7^{2n+2} − 1}{12}) = P_{−2} (\frac{m}{7}) ,

P_{−3} (7^{2n}m + \frac{7^{2n} − 1}{8}) = (−7)^{n}P_{−3}(m)

and P_{−3} (7^{2n+1}m + \frac{7^{2n+2} − 1}{8}) = (−7)^{n+1}P_{−3} (\frac{m}{7}) ,

P_{−6} (7^{2n}m + \frac{7^{2n} − 1}{4}) = 7^{2n}P_{−6}(m)

and P_{−6} (7^{2n+1}m + \frac{7^{2n+2} − 1}{4}) = 7^{2(n+1)}P_{−6} (\frac{m}{7}) ,

P_{−10} (7^{2n}m + 5\frac{7^{2n} − 1}{4}) = 7^{4n}P_{−10}(m)

and P_{−10} (7^{2n+1}m + 5\frac{7^{2n+2} − 1}{4}) = 7^{4(n+1)}P_{−10} (\frac{m}{7}) ,

(k = 11)

P_{−1} (11^{2n}m + \frac{11^{2n} − 1}{24}) = P_{−1}(m)

and P_{−1} (11^{2n+1}m + \frac{11^{2n+2} − 1}{24}) = P_{−1} (\frac{m}{11}) ,

P_{−2} (11^{2n}m + \frac{11^{2n} − 1}{12}) = P_{−2}(m)

and P_{−2} (11^{2n+1}m + \frac{11^{2n+2} − 1}{12}) = P_{−2} (\frac{m}{11}) ,

P_{−3} (11^{2n}m + \frac{11^{2n} − 1}{8}) = (−11)^{n}P_{−3}(m)

and P_{−3} (11^{2n+1}m + \frac{11^{2n+2} − 1}{8}) = (−11)^{n+1}P_{−3} (\frac{m}{11}) ,

P_{−4} (11^{2n}m + \frac{11^{2n} − 1}{6}) = (−11)^{n}P_{−4}(m)

and P_{−4} (11^{2n+1}m + \frac{11^{2n+2} − 1}{6}) = (−11)^{n+1}P_{−4} (\frac{m}{11}) ,

P_{−6} (11^{2n}m + \frac{11^{2n} − 1}{4}) = 11^{2n}P_{−6}(m)

and P_{−6} (11^{2n+1}m + \frac{11^{2n+2} − 1}{4}) = 11^{2(n+1)}P_{−6} (\frac{m}{11}) ,

P_{−8} (11^{2n}m + \frac{11^{2n} − 1}{3}) = (−11)^{3n}P_{−8}(m)

and P_{−8} (11^{2n+1}m + \frac{11^{2n+2} − 1}{3}) = (−11)^{3(n+1)}P_{−8} (\frac{m}{11}) ,

P_{−10} (11^{2n}m + 5\frac{11^{2n} − 1}{12}) = 11^{4n}P_{−10}(m)

and P_{−10} (11^{2n+1}m + 5\frac{11^{2n+2} − 1}{12}) = 11^{4(n+1)}P_{−10} (\frac{m}{11}) ,

P_{−26} (11^{2n}m + 13\frac{11^{2n} − 1}{12}) = 11^{12n}P_{−26}(m)

and P_{−26} (11^{2n+1}m + 13\frac{11^{2n+2} − 1}{12}) = 11^{12(n+1)}P_{−26} (\frac{m}{11}) ,

(k = 13)

P_{−1} (13^{2n}m + \frac{13^{2n} − 1}{24}) = P_{−1}(m)

and P_{−1} (13^{2n+1}m + \frac{13^{2n+2} − 1}{24}) = P_{−1} (\frac{m}{13}) ,

P_{−3} (13^{2n}m + \frac{13^{2n} − 1}{8}) = 13^{n}P_{−3}(m)

and P_{−3} (13^{2n+1}m + \frac{13^{2n+2} − 1}{8}) = 13^{n+1}P_{−3} (\frac{m}{13}) ,

(k = 17)

P_{−1} (17^{2n}m + \frac{17^{2n} − 1}{24}) = (−1)^{n}P_{−1}(m)

and P_{−1} (17^{2n+1}m + \frac{17^{2n+2} − 1}{24}) = (−1)^{n+1}P_{−1} (\frac{m}{17}) ,

P_{−2} (17^{2n}m + \frac{17^{2n} − 1}{12}) = (−1)^{n}P_{−2}(m)

and P_{−2} (17^{2n+1}m + \frac{17^{2n+2} − 1}{12}) = (−1)^{n+1}P_{−2} (\frac{m}{17}) ,

P_{−3} (17^{2n}m + \frac{17^{2n} − 1}{8}) = 17^{n}P_{−3}(m)

and P_{−3} (17^{2n+1}m + \frac{17^{2n+2} − 1}{8}) = 17^{n+1}P_{−3} (\frac{m}{17}) ,

(k = 19)

P_{−1} (19^{2n}m + \frac{19^{2n} − 1}{24}) = (−1)^{n}P_{−1}(m)

and P_{−1} (19^{2n+1}m + \frac{19^{2n+2} − 1}{24}) = (−1)^{n+1}P_{−1} (\frac{m}{19}) ,

P_{−2} (19^{2n}m + \frac{19^{2n} − 1}{12}) = P_{−2}(m)

and P_{−2} (19^{2n+1}m + \frac{19^{2n+2} − 1}{12}) = P_{−2} (\frac{m}{19}) ,

P_{−3} (19^{2n}m + \frac{19^{2n} − 1}{8}) = 19^{n}P_{−3}(m)

and P_{−3} (19^{2n+1}m + \frac{19^{2n+2} − 1}{8}) = 19^{n+1}P_{−3} (\frac{m}{19}) ,

P_{−10} (19^{2n}m + 5\frac{19^{2n} − 1}{12}) = 19^{4n}P_{−10}(m)

and P_{−10} (19^{2n+1}m + 5\frac{19^{2n+2} − 1}{12}) = 19^{4(n+1)}P_{−10} (\frac{m}{19}) .

Remark 5. In contrast (see, for example, [9]), the results for positive N analogous to the equalities in the last theorem are congruences; nontrivial only for some of the cases.

Remark 6. We compare our results with those of Newman [15]. The main result of [15] may be restated as follows: Restrict N to 2, 4, 6, 8, 10, 14, and 26. Let k > 3 be a prime such that

N(k + 1) ≡ 0 mod 24.

Then

P_{−N} (km + \frac{N(k^{2} − 1)}{24}) = (−k)^{\frac{1}{2} N−1}P_{−N} (\frac{m}{k}) ,

for all m ∈ ℤ ∪ {0}.

Our work applies also to the primes 2 and 3. In addition, neither of the two lists of congruences is a proper subset of the other. For example, the congruences for P_{−14}(5m + 4) and P_{−10}(7m + 6) are on our list but not on Newman’s; whereas the congruences for P_{−4}(17m + 14) and P_{−8}(17m + 11) are on Newman’s list but not ours. Since we have a necessary and sufficient condition for a function F_{k,−N} to be constant, any congruence on Newman’s list can easily be added to ours by computing a finite set of partition coefficients.

The above calculations together with Proposition 1 yield immediately the following

Theorem 7. For all N ∈ ℤ and all m ∈ ℤ^{+} ∪ {0},
P_{2N}(2m + 1) ≡ 0 mod 2,
P_{3N}(3m + 1) ≡ 0 ≡ P_{3N}(3m + 2) mod 3,
P_{5N+2}(5m + 3) ≡ 0 ≡ P_{5N+1}(5m + 4) mod 5,
P_{7N+4}(7m + 6) ≡ 0 ≡ P_{7N+1}(7m + 5) mod 7,
P_{11N+8}(11m + 4) ≡ 0 ≡ P_{11N+7}(11m + 9) ≡ P_{11N+5}(11m + 8) mod 11,
P_{11N+3}(11m + 7) ≡ 0 ≡ P_{11N+1}(11m + 6) ≡ P_{11N+7}(11m + 9) mod 11,
P_{13N+10}(13m + 8) ≡ 0 mod 13,
P_{17N+14}(17m + 2) ≡ 0 mod 17

and

P_{19N+16}(19m + 7) ≡ 0 ≡ P_{19N+9}(19m + 17) mod 19.

The mysterious patterns described above are clearly in need of further investigation and explanation. As remarked earlier, the last theorem contains, of course, the three Ramanujan level one congruences for the primes 5, 7 and 11.

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The Hebrew University of Jerusalem, Jerusalem 91904, Israel and State University of New York at Stony Brook, Stony Brook, NY 11794, USA