A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35

Adrian Dan Stanger

Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

                           adrian@math.byu.edu

Received: 3/20/02, Revised: 6/27/02, Accepted: 6/27/02, Published: 6/28/02

Abstract

In Rademacher’s proof [Rad42] of the Ramanujan identities modulo 5 and 7, the function

L_{k}(z) = η(kz) ∑_{μ=0}^{k−1} \{\frac{1}{η (\frac{z+24μ}{k})}\}

appears. This is a modular function on Γ_{0}(k), provided that (k, 6) = 1. The basic identities

p(5n + 4) ≡ 0 mod 5,
p(7n + 5) ≡ 0 mod 7,

(as well as some generalizations to higher powers of 5 and 7) are proved by examining the coefficients of the expansions of L_{5} and L_{7} at infinity. The two identities above immediately imply the following identity:

p(35n + 19) ≡ 0 mod 35.

Rademacher asked if there was a corresponding modular identity. Such an identity is derived in this paper. Also, another identity is obtained from the expansion of L_{35} at 0.

1. Generalities

1.1 Basics

Throughout, p(n) refers to the unrestricted partition function. Euler proved the identity

∑_{n=0}^{∞} p(n)q^{n} = ∏_{m=1}^{∞} \frac{1}{1 − q^{m}} .

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002), #A09 2

We will see that this ties the partition function intimately to certain types of modular functions (and forms). Let Γ(1) = PSL_{2}(ℤ), which is the special linear group modulo its center (±I). Γ(1) is generated by the two transformations

S = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

We’ll have need also for the product T ST, which we will call W. Then in Γ(1), we have:

W = (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) .

Γ(1) acts faithfully on the upper half plane H = {z ∈ ℂ : ℑ(z) > 0} via linear fractional transformation. Throughout, we will identify a linear fractional transformation and its matrix.

We let q = e^{2πiz} throughout. In what follows, Dedekind’s function η(z) will be used, so we remind the reader of some known facts. Recall η(z) = e^{\frac{πiz}{12}} ∏_{n=1}^{∞} (1 − e^{2πinz}), and note that η(z) is periodic of period 24. It is pole-free and zero-free throughout H. It is a modular form of weight one half, and has a multiplier which is a 24^{th} root of unity in its transformation equation. The root of unity ε(a, b, c, d) is determined by the following formulas (provided c ≠ 0):

ε(a, b, c, d) = exp \{πi (\frac{a + d}{12c} + s(−d, c))\} ,

s(h, k) = ∑_{r=1}^{k−1} \frac{r}{k} (\frac{hr}{k} − [\frac{hr}{k}] − \frac{1}{2}) , (1)

where [ ] is the greatest integer function. For properties of the Dedekind s(h, k) sum see [RW41, RG72]. We will often write just ε in place of the more cumbersome ε(a, b, c, d).

Thus, for any A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1), η(Az) = ε(cz + d)^{1/2}η(z). There is a relationship

between η and p. Namely,

∑_{n=0}^{∞} p(n)q^{n} = \frac{q^{\frac{1}{24}}}{η(z)}.

Also, L_{k} has q-expansion given by:

L_{k}(z) = kq^{1+[\frac{k}{24}]} ∏_{m=1}^{∞}(1 − q^{km}) ∑_{n=0}^{∞} p(kn + Δ)q^{n},

Δ least positive solution of 24n ≡ 1 mod k. (2)

It is clear that L_{k} has all its poles located at the cusps of Γ_{0}(k). Note that when k = 35 (resp. 5, 7), Δ = 19 (resp. 4, 5). This explains the form for the identities listed earlier. That L_{k}(z) is a modular function on Γ_{0}(k) may be seen as follows. One can show that \frac{η(kz)}{η(\frac{z}{k})} is a modular function on Γ_{0}^{0}(k) using known facts about Dedekind sums (see

particularly Theorems 17 and 18 of [RW41]). The matrices (\begin{matrix}1 & 24μ \\ 0 & 1 \end{matrix}) , (0 ≤ μ ≤ k − 1)

form a set of coset representatives for Γ_{0}^{0}(k) in Γ_{0}(k). Therefore, when we sum over this set of coset representatives to obtain L_{k}(z), we obtain a modular function on Γ_{0}(k).

Let ω ∈ ℚ ⋃{i∞} be a cusp for Γ_{0}(k). We will let H_{ω} be the set of modular functions on Γ_{0}(k) having poles only at ω. It is with this space of functions that we will work. In particular we will consider functions in H_{ω}, where ω = 0 or i∞.

We mention also the Fricke involution S_{k}(z) = \frac{−1}{kz}. Let F denote the set of modular functions on Γ_{0}(k), and f ∈ F. S_{k}(z) is an operator from F to itself. Moreover, it interchanges the polar orders of f at i∞ and 0 in the appropriate uniformizing variables (i.e, the expansion of f(S_{k}(z)) at i∞ has the same order in the variable q, as the expansion of f(z) at zero has in the variable q^{\frac{1}{k}}). For the rest of section 1, let k = pQ, where p and Q are two distinct primes strictly greater than 3. Then S_{k}(z) interchanges the polar orders of f at the cusps \frac{1}{p} and \frac{1}{Q} as well.

1.2 Evaluation at 0

Pursuant to our task, we need to determine the order of L_{k} at each of the cusps. We determine the expansion about 0. To do so we subject L_{k} to the transformation S_{k} and then compute the expansion as usual.

L_{k}(S_{k}z) = η [(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) z] ∑_{μ=0}^{k−1} \{η [(\begin{matrix}24μk & −1 \\ k^{2} & 0 \end{matrix}) z]\}^{−1} .

We need to factor the inner matrix as the product of a unimodular matrix and an upper triangular matrix. Upon factoring and applying the transformation identities we obtain:

L_{k}(S_{k}z) = \frac{1}{\sqrt{k}} η (z) ∑_{μ=0}^{k−1} \frac{1}{ε(\frac{24μ}{δ}, b, \frac{k}{δ}, d) δ^{1/2} η (δ^{2}z + \frac{δy}{k})},

where δ = (μ, k), and the matrix inside factors as:

(\begin{matrix}24μ/δ & b \\ k/δ & d \end{matrix}) (\begin{matrix}δk & y \\ 0 & k/δ \end{matrix}) ,

for some integers b, d and y such that

\frac{24μ}{δ}y + \frac{bk}{δ} = −1,

\frac{24μ}{δ}d − \frac{bk}{δ} = 1.

Together, the two equations yield that y = −d (in the case μ = 0 this is clear). Thus we obtain the following expression:

L_{k}(S_{k}z) = \frac{1}{\sqrt{k}} ∏_{m=1}^{∞} (1 − q^{m}) ∑_{μ=0}^{k−1} \{ε δ^{\frac{1}{2}} e^{\frac{πiyδ}{12k}} q^{\frac{δ^{2}−1}{24}} ∏_{n=1}^{∞} (1 − e^{2πiδyn}q^{δ^{2}n})\}^{−1} . (3)

Now, since μ = 0 is the only term of the sum in which δ = k, and this yields the largest negative power of q, we see that ord_{0}(L_{k}) ≥ (1 − k^{2})/24. We will return to equation 3 later.

1.3 Evaluation at all other cusps

Since k is square-free, all the other cusps of Γ_{0}(k) occur at the points {1/l| 1 < l < k, l|k }, in other words, at \frac{1}{p} and \frac{1}{Q}. To determine the order of L_{k} at 1/l, we subject it to the transformation W^{−l} and then evaluate. Thus, we obtain

L_{k} (W^{−l}z) = η (kW^{−l}z) ∑_{μ=0}^{k−1} \{η (\frac{W^{−l}z + 24μ}{k})\}^{−1}

= η [(\begin{matrix}k/l & 1 − k/l \\ −1 & 1 \end{matrix}) (\begin{matrix}l & k/l − 1 \\ 0 & k/l \end{matrix}) z] ∑_{μ=0}^{k−1} \{η [(\begin{matrix}1 − 24lμ & 24μ \\ −kl & k \end{matrix}) z]\}^{−1} .

Now, let δ = (1 − 24μl, k/l). Then the matrix inside factors as

(\begin{matrix}(1 − 24μl)/δ & b \\ (−kl)/δ & d \end{matrix}) (\begin{matrix}δ & y \\ 0 & k/δ \end{matrix}) ,

for integers b, d and y such that

\frac{1 − 24μl}{δ} d + \frac{bk}{δ} l = 1,

\frac{1 − 24μl}{δ} y + \frac{bk}{δ} = 24μ,

\frac{−kl}{δ} y + \frac{kd}{δ} = k.

The final equation above yields −ly + d = δ. Using this fact we see that

L_{k}(W^{−l}(z)) = e^{− \frac{k+l}{12l} πi}\sqrt{l/k} η (\frac{l^{2}z − l + k}{k} ) ∑_{μ=0}^{k−1} \{ε δ^{1/2} η (\frac{δ^{2}z + δy}{k} )\}^{−1}

= e^{− \frac{k+l}{12l} πi}\sqrt{l/k} ∏_{m=1}^{∞} (1 − ρ^{(k−l)n}q^{\frac{l^{2}}{k} n}) ×

∑_{μ=0}^{k−1} \{ε δ^{1/2}ρ^{\frac{dy+k−l}{24}} q^{\frac{δ^{2}−l^{2}}{24k}} ∏_{n=1}^{∞} (1 − ρ^{δyn}q^{\frac{δ^{2}}{k} n})\}^{−1} , (ρ = e^{\frac{2πi}{k}} ) .

2. Specialization to k = 35

2.1 Obtaining an explicit polynomial identity at i∞

Now, we take k = 35 for the remainder of this paper. We see from the results above that L_{35}(z) has a zero of order 2 at i∞ in the variable q, a pole of order 51 at 0 in the variable q^{\frac{1}{35}} , is holomorphic at 1/5 in q^{\frac{1}{7}} , and has a zero of order at least 1 at 1/7 in the variable q^{\frac{1}{5}} . We refer the reader to [New57] for an explicit construction of the basis functions with which we will work, and for an account of the functions A_{9} and B_{8} (as well as other similar functions used in deriving the basis of functions at infinity). In summary, the basis functions B_{4}, B_{5}, B_{6}, and B_{7} all have poles only at i∞ in the same order as the value of their respective subscript. Similarly, A_{9} and B_{8} have poles of order 9 and 8, respectively, at i∞, and A_{9} has zeroes of orders 5, 1, and 3 at 0, 1/5 and 1/7 respectively, while B_{8} has zeros of orders 6 and 2 at 0 and 1/7 respectively.

Suffice it to say that G(z) = A_{9}(z)B_{8}^{8}(z)L_{35}(z) is pole-free in H and has order −71 at i∞, 2 at 0, 0 (or possibly greater) at 1/5, and 20 at 1/7. So we have:

Theorem 1. G(z) ∈ ℂ[B_{4}, B_{5}, B_{6}, B_{7}].

The proof follows from the work in the previous section and the fact that the B_{i} do form a polynomial basis for the set of modular functions in H_{i∞}.

Since the order at i∞ is known, the polynomial is determined, using Mathematica to calculate the coefficients. Among the possible ways of choosing an appropriate form, it was determined to use B_{7} to keep the total degree (as a polynomial in the B_{i}) as small as possible. For instance the terms matching q^{−65}, q^{−64}, q^{−63} are B_{7}^{8}B_{5}B_{4}, B_{7}^{8}B_{4}^{2}, B_{7}^{9} respectively. Call the polynomial that we obtain P_{∞}. Then L_{35}(z) = A_{9}^{−1}B_{8}^{−8}P_{∞}.

Theorem 2. Let c_{i} be the coefficient on the term of P_{∞} corresponding to q^{i}. Then the polynomial P_{∞} is determined by the coefficients in the appendix in table 1, where c_{i} is the given i^{th} coefficient multiplied by 1225 (= 35^{2}).

Corollary 3. p(35n + 19) ≡ 0 mod 35.

Remark 1. This identity also shows how the genus dramatically affects a polynomial identity associated with such an identity for the partition function. For the cases k = 5 and k = 7, where the genus is 0, Rademacher obtains a linear polynomial and a quadratic polynomial, respectively, in the respective basis function. Here, the genus is 3 and a polynomial of degree 11 in the four basis functions is obtained. In a similar manner, modular identities for the congruences for 55 and 77 may surely be constructed. It is expected that more basis functions and a higher-degree polynomial will be necessary.

2.2 An Identity at 0

We return to equation 3. Recalling the connection with the partition function p, and separating into cases depending on δ, we have :

L_{k}(S_{k}(z)) = ∑_{δ|35} \frac{q^{\frac{1−δ^{2}}{24}}}{\sqrt{k}} ∑_{μ=0}^{34} ε^{−1}δ^{− \frac{1}{2}} ∑_{n=0}^{∞} e^{\frac{(24n−1)δyπi}{12k}} p(n)q^{δ^{2}n} ∏_{m=1}^{∞}(1 − q^{m}).

When we group the outer sum into terms by δ, we obtain Gauss sums for the Jacobi-Legendre symbol of the appropriate modulus. We have the following terms:

δ = 35 : \frac{1}{35q^{51}} ∏_{m=1}^{∞}(1 − q^{m}) ∑_{n=0}^{∞} p(n)q^{1225n},

δ = 7 : −\frac{1}{7q^{2}} ∏_{m=1}^{∞}(1 − q^{m}) ∑_{n=0}^{∞} (\frac{n − 4}{5}) p(n)q^{49n},

δ = 5 : −\frac{1}{5q} ∏_{m=1}^{∞}(1 − q^{m}) ∑_{n=0}^{∞} (\frac{n − 5}{7}) p(n)q^{25n},

δ = 1 : ∏_{m=1}^{∞}(1 − q^{m}) ∑_{n=0}^{∞} (\frac{n − 19}{35}) p(n)q^{n},

where (-) indicates the Jacobi-Legendre symbol.

Now, L_{35}(S_{35}z) has order −51 (resp. 2, 1, −1) at i∞ (resp. 0, 1/5, 1/7). For the last two points we really only know that the order at each of these points is greater than or equal to the given value. Nevertheless, this is sufficient to determine that the function h(z) = B_{8}(z)L_{35}(S_{35}z) has its only pole at i∞, and the order of the pole there is 59. Thus, h(z) ∈ ℂ[B_{4}, B_{5}, B_{6}, B_{7}].

To clear fractions, multiply by 35 and then note 35L_{35}(S_{35}z) = B_{8}^{−1}P_{0}, where P_{0} is a polynomial to be determined in the four basis functions. So

q^{−51} − 5q^{−2} − 7q^{−1} + 35 + ... = P_{0}B_{8}^{−1} ∑_{n=0}^{∞} p(n)q^{n}.

This yields the following:

Theorem 4. 35L_{35}(S_{35}z) is a polynomial in the basis functions B_{4}, B_{5}, B_{6}, B_{7}, with integral coefficients given in table 2 of the appendix.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002), #A09 7

Corollary 5.

∑_{n=0}^{∞} \{p(n)q^{1225n} − 5 (\frac{n − 4}{5}) p(n)q^{49(n+1)}

−7 (\frac{n − 5}{7}) p(n)q^{25(n+2)} + 35 (\frac{n − 19}{35}) p(n)q^{n+51}\}

= q^{51}P_{0}B_{8}^{−1} ∑_{n=0}^{∞} p(n)q^{n}.

I would like to thank my doctoral advisor, Morris Newman, who suggested this problem to me, and Bruce Berndt and the referee for their helpful comments.

3. Appendix: tables

Table 1: Coefficients for P_{∞}

i coefficient i coefficient
-69  14 -33 -10569223238228513913405
-68  10487 -34 -29246449555394872646896
-67  1050000 -33 -30752043774402282243297
-66  37305717 -32 -10500761656485195685244
-65  689742738 -31  -63994584518030323125
-64  6710739396 -30  747396610368371686739
-63  66723377634 -29  41866861538739825786329
-62  306529775888 -28 -154825275874629072148111
-61  944992220254 -27 -330675045299039974754087
-60  8032835276603 -26 -330675045299039974754087
-59  41786228768706 -25 -195434688782712582767044
-58  163674979606784 -24  106519373950966952221
-57  201541582793635 -23  -4489388038461810926594
-56  1192164476376773 -22  93815803283069768407781
-55  2133885512377322 -21 -868861692593019875600320
-54  -224422822302317 -20 -780240652273282530768564
-53  9311212319487067 -19 -903341606561540466301311
-52  53104246090438488 -18 -575916410654649291605538
-51  174875061214811942 -17  -72387387334271714225
-50  59790513579741686 -16  5279005749531844180716
-49  418363890733605649 -15 -329709784193546752041430
-48  1123804424169272601 -14 -754715959977583205060136
-47  -1849756259047888932 -13 2673409023878484316383343
-46  -3961615256484350591 -12 3038512747510492176975342

continued on next page

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002), #A09 8

Table 1: continued

i coefficient i coefficient
-45  1372328949965814202 -11 1778681601887609961535840
-44 19742808525931603996 -10  16760790778989818745
-43 2177933533113105157 -9 -1894749705906830722098
-42 23516820211582241283 -8  189938163107999640927156
-41 123390318981474451231 -7 4789820626890790384933368
-40 -271841378745461882544 -6 -1532454919226226734377599
-39 -777621797568622505476 -5 -1735248899286586237586336
-38 -673143178730309518022 -4 -997109474512591275870705
-37 268511875093390021788 -3  0
-36 10323721732692166606 -2  0
-35 334287613007887666292 -1  0
-34 3963341600197861903691 0 -3000539263195134467759624

Table 2: Coefficients for P_{0}

i coefficient i coefficient
-59 1 -29 101856
-58  -5 -28 -1088550
-57  13 -27 -779536
-56  -24 -26 -868518
-55  -124 -25 -638720
-54  -167 -24  183785
-53  -100 -23  310027
-52  186 -22 -1009582
-51 82 -21 -5533
-50  -784 -20 8917982
-49  1899 -19 10010374
-48  7839 -18 6789719
-47  9103 -17  -21976
-46  5400 -16  113545
-45  -531 -15  35220
-44  2196 -14 21729982
-43  -786 -13 -703140
-42  13705 -12 -920456
-41  -1006 -11 -843856
-40  -4690 -10 -236844
-39  1882 -9  -376949
-38  -6611 -8  1365595
-37  -16039 -7 -4046123
-36  44380 -6 -12312578
-35  -84846 -5 -13942378

continued on next page

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002), #A09 9

Table 2: continued

i coefficient i coefficient
-34 -393717 -4 -9425016
-33 -427964 -3  0
-32 -282747 -2  0
-31 -17208 -1 0
-30  -53917 0 -30151150

References

[Apo95] T. M. Apostol. Introduction to Analytic Number Theory. Springer-Verlag, 1995.

[Apo97] T. M. Apostol. Modular Functions and Dirichlet Series in Number Theory, volume 41 of Graduate Texts in Mathematics. Springer-Verlag, second edition, 1997.

[New57] M. Newman. Construction and application of a class of modular functions. Proceedings of the London Mathematical Society, VII(27):334–350, July 1957.

[Rad42] H. Rademacher. The Ramanujan identities under modular substitutions. Trans. American Math. Soc., 51(3):609–636, May 1942.

[RG72] H. Rademacher and E. Grosswald. Dedekind Sums. MAA, 1972.

[RW41] H. Rademacher and A. Whiteman. Theorems on Dedekind sums. American Journal of Mathematics, 63:377–407, 1941.