A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35

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1 A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35 Adrian Dan Stanger Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA 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 } 1 1 L (z =η(z η ( z+24µ appears. This is a modular function on Γ 0 (, provided that (, 6 = 1. The basic identities p(5n +4 0mod5, p(7n +5 0mod7, (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 mod35. Rademacher ased 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 Generalities 1.1 Basics Throughout, p(n refers to the unrestricted partition function. Euler proved the identity p(nq n = 1 1 q m.

2 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 ( 0 1 S =, T = 1 0 ( We ll have need also for the product TST, which we will call W. Then in Γ(1, we have: ( 1 0 W =. 1 1 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, Dedeind s function η(z will be used, so we remind the reader of some nown facts. Recall η(z =e πiz 12 n=1 (1 e2πinz, 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: ( } a + d ε(a, b, c, d = exp πi + s( d, c, 12c s(h, =. 1 r=1 ( r hr [ ] hr 1, (1 2 where [] isthegreatest integer function. For properties of the Dedeind s(h, sum see [RW41, RG72]. We will ( often write just ε in place of the more cumbersome ε(a, b, c, d. a b Thus, for any A = Γ(1, η(az =ε(cz + d c d 1/2 η(z. There is a relationship between η and p. Namely, p(nq n = q 1 24 η(z. Also, L has q-expansion given by: L (z =q 1+[ 24] (1 q m p(n + q n, least positive solution of 24n 1mod. (2 It is clear that L has all its poles located at the cusps of Γ 0 (. Note that when = 35 (resp. 5, 7, = 19 (resp. 4, 5. This explains the form for the identities listed earlier. That L (z is a modular function on Γ 0 ( may be seen as follows. One can show

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 3 that η(z η( is a modular function on z Γ0 0( using nown facts about Dedeind sums (see ( 1 24µ particularly Theorems 17 and 18 of [RW41]. The matrices, (0 µ form a set of coset representatives for Γ 0 0( inγ 0 (. Therefore, when we sum over this set of coset representatives to obtain L (z, we obtain a modular function on Γ 0 (. Let ω Q i } be a cusp for Γ 0 (. We will let H ω be the set of modular functions on Γ 0 ( having poles only at ω. It is with this space of functions that we will wor. In particular we will consider functions in H ω, where ω =0ori. We mention also the Frice involution S (z = 1. Let F denote the set of modular z functions on Γ 0 (, and f F.S (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 (z at i has the same order in the variable q, as the expansion of f(z at zero has in the variable q 1. For the rest of section 1, let = pq, where p and Q are two distinct primes strictly greater than 3. Then S (z interchanges the polar orders of f at the cusps 1 and 1 as well. p Q 1.2 Evaluation at 0 Pursuant to our tas, we need to determine the order of L at each of the cusps. We determine the expansion about 0. To do so we subject L to the transformation S and then compute the expansion as usual. L (S z=η [( ] z 1 0 η [( 1 24µ 1 z]} 2. 0 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 (S z= 1 1 η (z ε( 24µ,b,,d δ δ δ1/2 η ( δ 2 z + δy where δ =(µ,, and the matrix inside factors as: ( ( 24µ/δ b δ y, /δ d 0 /δ for some integers b, d and y such that 1, 24µ δ y + b δ = 1,

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A µ δ d b 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 (S z= 1 1 (1 q m εδ 1 2 e πiyδ 12 δ q n=1 ( } 1 1 e 2πiδyn q δ2 n. (3 Now, since µ = 0 is the only term of the sum in which δ =, and this yields the largest negative power of q, we see that ord 0 (L (1 2 /24. We will return to equation 3 later. 1.3 Evaluation at all other cusps Since is square-free, all the other cusps of Γ 0 ( occur at the points 1/l 1 <l<,l }, in other words, at 1 p and 1 Q. To determine the order of L at 1/l, we subject it to the transformation W l and then evaluate. Thus, we obtain = η ( L W l z = η ( W l z 1 ( } W l 1 z +24µ η ( ] l /l 1 1 [( 1 24lµ 24µ z η 0 /l l [( /l 1 /l 1 1 z]} 1. Now, let δ =(1 24µl, /l. Then the matrix inside factors as ( ( (1 24µl/δ b δ y, ( l/δ d 0 /δ for integers b, d and y such that 1 24µl d + b δ δ l =1, 1 24µl y + b δ δ =24µ, l y + d δ δ =. The final equation above yields ly + d = δ. Using this fact we see that L (W l (z = e +l 12l πi l/ η ( l 2 z l + 1 εδ 1/2 η ( δ 2 z + δy } 1

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 5 = e +l 12l πi l/ (1 ρ ( ln q l2 n 1 εδ 1/2 ρ dy+ l 24 q δ2 l 2 24 (1 } 1 ρ δyn q δ2 n, n=1 ( ρ = e 2πi. 2. Specialization to = Obtaining an explicit polynomial identity at i Now, we tae = 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 1 35, is holomorphic at 1/5 inq 1 7, and has a zero of order at least 1 at 1/7 in the variable q 1 5. We refer the reader to [New57] for an explicit construction of the basis functions with which we will wor, 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 (zb 8 8(zL 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 wor 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 nown, 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 eep 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 B7B 8 5 B 4, B7B 8 4, 2 B7 9 respectively. Call the polynomial that we obtain P. Then L 35 (z =A 1 9 B8 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 (= Corollary 3. p(35n mod35. Remar 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 = 5

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 6 and = 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 (S (z = δ 35 q 1 δ ε 1 δ 1 2 e (24n 1δyπi 12 p(nq δ2 n (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: 1 δ =35: (1 q m p(nq 1225n, 35q 51 δ =7: 1 ( n 4 (1 q m p(nq 49n, 7q 2 5 δ =5: 1 ( n 5 (1 q m p(nq 25n, 5q 7 ( n 19 δ =1: (1 q m p(nq n, 35 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 now 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 (zl 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=b8 1 P 0, where P 0 is a polynomial to be determined in the four basis functions. So This yields the following: q 51 5q 2 7q = P 0 B 1 8 p(nq n. 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.

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 7 Corollary 5. ( n p(nq 1225n 5 p(nq 25(n = q 51 P 0 B 1 8 ( n 4 p(nq 49(n+1 5 ( } n 19 p(nq n p(nq n. I would lie to than 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 continued on next page

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 8 Table 1: continued i coefficient i coefficient Table 2: Coefficients for P 0 i coefficient i coefficient continued on next page

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 1 (2002, #A09 9 Table 2: continued i coefficient i coefficient References [Apo95] T. M. Apostol. Introduction to Analytic Number Theory. Springer-Verlag, [Apo97] T. M. Apostol. Modular Functions and Dirichlet Series in Number Theory, volume 41 of Graduate Texts in Mathematics. Springer-Verlag, second edition, [New57] M. Newman. Construction and application of a class of modular functions. Proceedings of the London Mathematical Society, VII(27: , July [Rad42] H. Rademacher. The Ramanujan identities under modular substitutions. Trans. American Math. Soc., 51(3: , May [RG72] H. Rademacher and E. Grosswald. Dedeind Sums. MAA, [RW41] H. Rademacher and A. Whiteman. Theorems on Dedeind sums. American Journal of Mathematics, 63: , 1941.