DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS. George E. Andrews and Ken Ono. February 17, Introduction and Statement of Results

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1 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS George E. Andrews and Ken Ono February 7, Introduction and Statement of Results Dedekind s eta function ηz, defined by the infinite product ηz :=q /24 q n, where q := e 2πiz, is one of most fundamental modular forms. For example, Ramanujan s z function, the unique normalized weight 2 cusp form on SL 2 Z, is given by z =η 24 z. However, more is true. For instance, every modular form on SL 2 Z is a rational function in the eta-function since the classical Eisenstein series E 4 z ande 6 z satisfy E 4 z = E 6 z = 504 d n d n d 3 q n = η6 z η 8 2z + 256η6 2z η 8 z, d 5 q n = η24 z η 2 2z 480η2 2z 6896 η2 2zη 8 4z η24 4z η 8 z η 2 2z. Therefore, every modular form on SL 2 Z is essentially an algebraic expression in infinite products. 99 Mathematics Subject Classification. Primary F20; Secondary P8. Both authors thank the National Science Foundation for its generous research support. The second author thanks the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation for their support. Typeset by AMS-TEX

2 2 GEORGE E. ANDREWS AND KEN ONO In a different direction, Borcherds [B] has shown that many suitable modular forms have Fourier expansions which are infinite products whose exponents are Fourier coefficients of certain corresponding modular forms. A special case of his work implies that E 6 z = 504q 6632q 2 = q 504 q = q n cn2 where the exponents cn 2 are coefficients of a specific modular form fz = cnq n. In addition to their role in the theory of modular forms, such infinite products play a crucial role in q-series and the theory of partitions. Therefore, it is natural to investigate the modularity and the combinatorial properties of truncated products of such series. Note. Empty products shall equal throughout. With this as motivation, we make a first step by examining the q-series obtained by summing the differences between η24z and /η24z and all of their truncated products. Specifically, we study the two q-series 2 3 η24z = q 23 +3q 47 +6q 7 + q q 24 q 48 q 24n η24z q q 24 q 48 q 24n = q 25 2q 49 q 73 Note: A similar construction was studied by the first author in [A2] in connection with an identity appearing in Ramanujan s Lost Notebook involving a Mock theta type function. At first glance, there is no reason to suspect that the q-series in 2 and 3 are easily described in terms of modular forms. However, we show that these series are indeed modular forms up to a simple q-series. In addition, we give two interesting applications of these results to partitions. Recall that a partition of a non-negative integer N is any nonincreasing sequence of positive integers whose sum is N. If pn denotes the number of partitions of N, thenwe have 4 η24z = N=0 pnq 24N = q q 24n = q + q 23 +2q 47 +3q 7 +5q Similarly, let d e N resp.d o N denote the number of partitions of N into an even resp. odd number of distinct parts. Euler s Pentagonal Number Theorem asserts that the Fourier expansion of η24z S /2 Γ 0 576,χis 5 η24z = d e N d o N q 24N+ = q q 25 q 49 + = N=0 χnq n2

3 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS 3 where if n ± mod 2, 6 χn := if n ±5 mod 2, 0 otherwise. To state our results, let Dq be the q-series defined by 7 Dq := 2 + q n q = n 2 + dnq n = 2 + q +2q2 +2q 3 +3q 4 +, where dn denotes the number of positive divisors of a positive integer n. Theorem. The following q-series identity is true: η24z =2 η24z 2Dq q q 24 q 48 q 24n As a corollary, we obtain the following partition theoretic result. η24z. Corollary 2. If n is a positive integer, then let a n N denote the number of partitions of N into parts not exceeding n. For every positive integer N we have N +pn = N pn ndn+ N a n N. Corollary 2 is equivalent to the observation used by Erdös [E] in the beginning of his study of the asymptotics of pn byelementarymeans. Now we consider the q-series defined in 3. Theorem 3. If Rz S 3/2 Γ 0 576,χ is the cusp form with Fourier expansion Rz := χnnq n2 = q 5q 25 7q 49 +q 2 +, then Rz =2 η24z q q 24 q 48 q 24n 2Dqη24z. As with Theorem, Theorem 3 has an interesting partition theoretic consequence which is similar in flavor to Euler s Pentagonal Number Theorem 5. We require some notation.

4 4 GEORGE E. ANDREWS AND KEN ONO If π is a partition, then let m π denote its largest part. Moreover, if N is an integer, then let S e N resp. S o N denote the set of partitions of N into an even resp. odd number of distinct parts. Using this notation, define the two partition functions A e N anda o N by 8 9 A e N := A o N := Corollary 4. If N is a positive integer, then A e N A o N = π S e N π S o N m π, = k Z k d N 3k 2 + k/2 + k 3k 0 otherwise. m π k 3k if N = 3k2 +k 2 with k 0, if N = 3k2 k 2 with k, In 2 we prove Theorem and Corollary 2, and in 3 we prove Theorem 3 and Corollary 4. We conclude by raising two natural questions which arise from these results. Question. Under what conditions does a modular form fz given by an infinite product have the property that the sum of all the differences between fz and its truncated products is a modular form modulo a function such as 2Dqfz? Question 2. Is there a direct combinatorial proof of Corollary 4? 2. The proof of Theorem and Corollary 2 We begin with an equivalent form of the conjectured identity which is simple to obtain. Proposition 2.. Let P q and Dq be the q-series defined by P q := Dq := q n = pnq n =+q +2q 2 +3q 3 +5q 4 +, q n q = n The identity in Theorem is true if and only if P q dnq n = q +2q 2 +2q 3 +3q 4 +. q q 2 q n = P qdq.

5 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS 5 Proof. This claim follows easily from Theorem by first multiplying both sides of the conjectured identity in Theorem by q, replacing q by q /24 and then by using the fact that Dq = 2 + Dq. The next Lemma provides a convenient form of the left hand side of the conjectured identity in Proposition 2.. Lemma 2.2. The following identity is true P q q q 2 q n = nq n q q 2 q n. Proof. It is easy to see that P q q q 2 q n is the generating function for the number of partitions of N with at least one part exceeding n. Therefore, each partition π of an positive integer N is counted with multiplicity m π in the telescoping sum. On the other hand, the generating function for the number of partitions of π of N with largest part m π = n counted with multiplicity n is easily seen to be The proof follows immediately. nq n q q 2 q n. In view of Proposition 2. and Lemma 2.2, to prove Theorem it suffices to prove that 0 nq n q q 2 q n = P qdq. For completeness we include a proof of 0. However, Uchimura [U] originally proved 0 and Dilcher [D] generalized it. To prove this identity, we require the following identity.

6 6 GEORGE E. ANDREWS AND KEN ONO Proposition 2.3. For z, q < we have F z :=+ z n q n q q 2 q n = zq n. Proof. This is a classical result of Euler. In particular, it follows by letting t = zq in [Cor. 2.2, A]. Proof of Theorem. Since the q-series F z is it follows that F z = F = Therefore, in view of 0 it suffices to prove that nz n q n q q 2 q n, nq n q q 2 q n. F = P qdq. However, we may apply the product rule to F z using its infinite product form from Proposition 2.3, and we find that F z = = q n zq n m= zqm m= zqm q n zq n. Therefore, we have that This is, and so the proof is complete. F = P qdq. Proof of Corollary 2. Here we use the identity of Theorem in the form 2 P q q q 2 q n = P qdq.

7 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS 7 If m is a non-negative integer, then the series 3 P q q q 2 q m = qm+ + n=m+2 b m nq n for some sequence of integers b m n. Now let N be a positive integer, then it is easy to see that the coefficient of N on the right hand side of 2 is N pn ndn. Therefore, to prove the corollary it suffices to show that the coefficient of q N hand side of of 2 is on the left 4 N +pn N a n N. By letting m = N as in 3, it is easy to see that the coefficient of q N on the left hand side of 2 is the coefficient of q N in N P q Claim 4 follows from the obvious fact that This completes the proof. a n Nq n = N=0. q q 2 q n q q 2 q n. 3. The proof of Theorem 3 and Corollary 4 In this section we prove Theorem 3 and Corollary 4. The proof of Theorem 3 is more involved than the proof of Theorem. We begin with a reformulation of Theorem 3.

8 8 GEORGE E. ANDREWS AND KEN ONO Proposition 3.. Let Eq be the q-series defined by Eq := q n = q q 2 + q 5 +. The identity in Theorem is true if and only if Eq q q 2 q n = = EqDq+ k 3kq 3k2 +k/2 + k 3k q 3k2 k/2. k=0 k= Proof. Begin by noticing the following identities: and Dq = 2 + Dq, 5 χnnq n2 = q k 6k q 24k3k /2 + q k 6k +q 24k3k+/2. k= k=0 With these identities, one obtains the reformulation by dividing both sides of the identity in Theorem 3 by q and then replacing q by q /24. As in 2, we reformulate the q-series appearing in the left hand side of the alleged identity in Proposition 3.. Lemma 3.2. The following identity is true: Eq q q 2 q n = nq n q q 2 q n. Proof. As in 5, it is easy to see that Eq = q n = d e n d o n q n.

9 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS 9 Since Eq is the generating function for the number of partitions into an even number of distinct parts minus the number into an odd number of distinct parts, it is easy to see that Eq q q 2 q n is the generating function whose coefficient of q N is the number of partitions of N into an even number of distinct parts where at least one part exceeds n minus the number of partitions of N into an odd number of distinct parts where at least one part exceeds n. Consequently, each partition π into distinct parts is counted with multiplicity n π m π where n π is the number of parts in π. This implies the claimed identity. By Lemma 3.2, Theorem 3 is equivalent to the truth of the following identity: nq n q q 2 q n = 6 = EqDq+ k 3kq 3k2 +k/2 + k=0 The following Proposition is simple. Proposition 3.3. If Gz is the q-series defined by Gz =+ q q 2 q n z n q n, k 3k q 3k2 k/2. k= then Theorem 3 is equivalent to the truth of the following identity qg qg = = EqDq+ k 3kq 3k2 +k/2 + k=0 Proof. It is easy to see that nq n q q 2 q n = = q q n q q 2 q n +q = q + q q n q q 2 q n +q = qg + qg. k 3k q 3k2 k/2. k= n q n q q 2 q n nq n q q 2 q n

10 0 GEORGE E. ANDREWS AND KEN ONO The claim follows from 6. Now we obtain a reformulation of the series Gz. Lemma 3.4. The series Gz satisfies the following identity Gz =Eq zq + q n. q q 2 q n zq n+ Proof. This identity is a corollary of Heine s first transformation [Cor. 2.3, A]. In particular, let a = q, c =0,b= q and t = zq in the transformation. Proposition 3.5. The series G is given by G = q q Eq. Proof. By letting z = in Lemma 3.4, we find that G = Eq q + q n q q 2 q n+ = Eq q + q n+ q q q 2 q n+ = Eq q q + q n q q q 2 q n = q q n 7 Eq q q 2 q n By Proposition 2.3, we see that q n + q q 2 q n = E q. Therefore, 7 reduces to This completes the proof. G = q Eq +E q = q q Eq. We record the following obvious Proposition for convenience.

11 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS Proposition 3.6. The series G satisfies the following identity G = nq n q q 2 q n. We now recall an identity due to L. J. Rogers see [p. 29, Ex. 0, A]. Proposition 3.7. If Hz is the series Hz :=z z 2 q z 2 q 2 z 2 q n z 2n+3 q n, then Hz =z + n q 3n2 n/2 z 6n + q 3n2 +n/2 z 6n+. Proof. To obtain this identity, simply let x = z 2 in [p. 29, Ex. 0, A] then multiply both sides of the identity by z. Now we prove a crucial Lemma regarding Hz. Lemma 3.8. The series H satisfies the following identities: H = + n 6n q 3n2 n/2 +6n +q 3n2 +n/2 = 4 2qG + 5Eq+2 n q n q q 2 q n q j q j. j= Proof. The first claim regarding H follows trivially from Proposition 3.7. Now we use the definition of Hz in Proposition 3.7 to give another formulation of H. We begin by computing H z. We find by the product rule that H z = z 2 q z 2 q 2 z 2 q n 2n +3z 2n+2 q n n z 2n+3 q n z 2 q z 2 q 2 z 2 q n 2zq j z 2 q. j j=

12 2 GEORGE E. ANDREWS AND KEN ONO Therefore, we find that H = q q 2 q n 2n +3q n n q n q q 2 q n 2q j q j j= = 2 q q 2 q n n q n 5 q q 2 q n q n +2 = 2q +2 n q n q q 2 q n q j q j j= q q 2 q n nq n 5q q q 2 q n q n n q n q q 2 q n q j q. j However, by the definition of G and Proposition 3.6, we find that H = 2qG 5qG +2 j= n q n q q 2 q n j= q j q j. The claimed identity follows from Proposition 3.5. Here is the last crucial Lemma that we require. Lemma 3.9. The following identity is true: n q n q q 2 q n q j = Eq EqDq. qj j= Proof. Let Iz be the series in the following identity Iz := zq n zq zq 2 zq n = zq n.

13 DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS 3 It is easy to see, by differentiating both expressions above, that I z = Therefore, we find that I = = q n zq zq 2 zq n = n zq n zq zq 2 zq n q j zq j zq n q j zq. j j= q n q q 2 q n + = EqDq. By the definition of Gz, we now find that j= n q n q q 2 q n q j q j n q n q q 2 q n q j = qg EqDq. qj Proposition 3.5 completes the proof. Proof of Theorem 3. By Proposition 3.3 and Proposition 3.5, it suffices to show that j= +Eq qg EqDq = = k 3kq 3k2 +k/2 + k=0 However, by Lemma 3.9 this is equivalent to 2+2Eq qg + = k 3kq 3k2 +k/2 + k=0 j= k 3k q 3k2 k/2. k= n q n q q 2 q n q j q j j= k 3k q 3k2 k/2. k=

14 4 GEORGE E. ANDREWS AND KEN ONO However, by Lemma 3.8 we find that this is equivalent to 2 H 2 Eq = k 3kq 3k2 +k/2 + k=0 k 3k q 3k2 k/2. Lemma 3.8 and Euler s Pentagonal Number Theorem 5 completes the proof. k= Proof of Corollary 4. The combinatorial interpretation of Lemma 3.2 together with Theorem 3 immediately implies the Corollary. References [A] G. E. Andrews, The theory of partitions, Cambridge University Press, 984. [A2] G. E. Andrews, Ramanujan s Lost Notebook V: Euler s partition identity, Advances in Math , [B] R. E. Borcherds, Automorphic forms on O s+2,2 R and infinite products, Inventiones Mathematicae , [D] K. Dilcher, Some q-series identities related to divisor functions, Discrete Mathematics , [E] P. Erdös, On an elementary proof of some asymptotic formulas in the theory of partitions, Annals of Mathematics , [K] N. Koblitz, Introduction to elliptic curves and modular forms, vol. GTM 97, Springer-Verlag, New York, 984. [Mi] T. Miyake, Modular forms, Springer-Verlag, New York, 989. [U] K. Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combinatorial Theory A, 3 98, Department of Mathematics, Penn State University, University Park, Pa address: andrews@math.psu.edu Department of Mathematics, Penn State University, University Park, Pa address: ono@math.psu.edu Department of Mathematics, University of Wisconsin, Madison, Wisconsin address: ono@math.wisc.edu