On Ramanujan’sdefinition of mock

Robert C. Rhoades1

Department of , Stanford University, Stanford, CA 94305

† Edited by George E. Andrews, Pennsylvania State University, University Park, PA, and approved April 1, 2013 (received for review January 17, 2013) In his famous “deathbed” letter, Ramanujan “defined” the notion recently (4), Griffin et al. proved that there is no weakly holo- of a mock theta function and offered some examples of functions morphic M such that fðqÞ − MðqÞ is bounded ra- he believed satisfied his definition. Very recently, Griffinetal. dially toward all roots of unity. Until the results of that paper established for the first time that Ramanujan’s mock theta func- were announced, it was not known whether any of Ramanujan’s tions actually satisfy his own definition. On the other hand, mock theta functions satisfied his own definition. Zwegers’ 2002 doctoral thesis [Zwegers S (2002) Mock theta Despite the lack of a definition, Ramanujan’s mock theta func- functions. PhD thesis (Univ Utrecht, Utrecht, The Netherlands)] tions were shown to possess many striking properties. For ex- showed that all of Ramanujan’s examples are holomorphic parts ample, Ramanujan himself related certain sums of mock theta of harmonic Maass forms. This has led to an alternate definition functions to modular forms. As an example, he claimed that of a mock theta function. This paper shows that Ramanujan’s definition of mock theta function is not equivalent to the modern 2ϕð−qÞ − fðqÞ = bðqÞ; [1.1] definition. where fðqÞ and bðqÞ are as above, and ϕðqÞ := n his famous “deathbed” letter (1), Ramanujan introduced the P∞ n2 q is one of his third-order mock theta Inotion of a mock theta function. The following version of n=0ð1 + q2Þð1 + q4Þ⋯ð1 + q2nÞ Ramanujan’sdefinition follows Andrews and Hickerson (2) and functions. Identities of this flavor are often referred to as Zwegers (3). “mock theta conjectures” [see the survey of Gordon and Mc- Intosh (5) or the work of Andrews and Garvan (6)]. Many Ramanujan’sDefinition. A mock theta function is a function f of examples of such identities proved themselves very difficult the complex variable q,defined by a q-series of a particular type to establish. The most significant were proved in Hickerson’s (Ramanujan calls this the Eulerian form), which converges for works (7, 8). jqj < 1 and satisfies the following conditions: Other striking properties are the Hecke-type series found by Andrews (9). As an example, consider Ramanujan’s fifth-order (1) infinitely many roots of unity are exponential singularities; mock theta function (2) for every root of unity ξ, there is a theta function ϑξðqÞ, X∞ 2 such that the difference fðqÞ − ϑξðqÞ is bounded as q → ξ qn f ðqÞ := : radially; and 0 ð1 + qÞ⋯ð1 + qnÞ n = 0 (3) f is not the sum of two functions, one of which is a theta function and the other a function that is bounded radially Andrews proved that toward all roots of unity. P ∞ X∞ X Remark: ’ αnðn+1Þ=2βnðn−1Þ=2 Ramanujan s theta function ∈Z , n j 1 nð5n+1Þ−j2 4n+2 n ∏ 1 − q · f q = −1 q2 1 − q : with jαβj < 1, is more general than the notion of theta function ð Þ 0ð Þ ð Þ n = 1 = −∞ considered here. As in refs. 3 and 4, we define a theta function as j jjj≤n a weakly holomorphic modular form. Consequentially, these objects will have asymptotics that resemble those discussed by Note the resemblance to Ramanujan in his deathbed letter. Moreover, all of the examples ∞ X∞ X fi ϑξ constructed in his letter satisfy his de nition with each ðqÞ 2 − j 1 + − 2 + ∏ − n − 2n 1 = − 2 nð3n 1Þ j − 2n 1 a weakly holomorphic modular form [see the work of Griffin ð1 q Þ 1 q ð 1Þ q 1 q = et al. (4) and the discussion therein]. n 1 n = 0 jjj≤n Ramanujan gave 17 examples of functions he believed satisfied these properties. The most famous is (see equation 5.15 of ref. 9, for example). These hints of structure and many others led to Dyson’s (10, q q4 p 20) statement in 1987: fðqÞ := 1 + + + 2 + 2 + 2 2 ð1 qÞ ð1 qÞ ð1 q Þ Somehow it should be possible to build them (the mock theta func- q9 tions) into a coherent group-theoretic structure, analogous to the + + ⋯: structure of modular forms which Hecke built around the old theta- + 2 + 2 2 + 3 2 ð1 qÞ ð1 q Þ ð1 q Þ functions of Jacobi. This remains a challenge for the future.

Clearly limq→ξfðqÞ = Oð1Þ when ξ is an odd order root of unity. This group-theoretic structure was discovered by Zwegers. (Here and throughout the remainder of the paper, all limits are Zwegers’ 2001 PhD thesis (11) was a breakthrough in the study radial limits.) In his letter, Ramanujan claimed that at all prim- of the mock theta functions. As a result of his thesis, it is known itive, even 2k roots of unity, say ξ, that all of Ramanujan’s examples are essentially the holomorphic lim fðqÞ − ð−1Þk bðqÞ = Oð1Þ; →ξ q Author contributions: R.C.R. wrote the paper. The author declares no conflict of interest. = ∏∞ ð1 − qnÞ where bðqÞ = 2 is,uptoapowerofq, a weakly hol- † n 1ð1+qnÞ This Direct Submission article had a prearranged editor. omorphic modular form. Watson (1) proved this claim. Very 1E-mail: [email protected].

7592–7594 | PNAS | May 7, 2013 | vol. 110 | no. 19 www.pnas.org/cgi/doi/10.1073/pnas.1301046110 Downloaded by guest on October 1, 2021 part of weight 1=2 weak harmonic Maass forms whose non- Moreover, it was proved by Bringmann (20) that SðqÞ is essen- holomorphic parts are period integrals of weight 3=2 unary tially a mock modular form with shadow theta functions. ∞ X This has led to a huge number of results that are out of reach 2 1 n 12 n otherwise. Perhaps the most astonishing breakthroughs are the ηðqÞ = q24 ∏ ð1 − q Þ = q24 ; n works of Bringmann and Ono (12, 13), which establish congru- n = 1 n∈Z fi ence properties and exact formulas for the coef cients of mock · theta functions. Zwegers’ thesis also led to a simpler and more where · is the Kronecker symbol. conceptual proof of the mock theta conjectures [see Folsom’s Our theorem makes use of a different mock theta function work (14)]. with shadow proportional to ηðqÞ. Zwegers’ construction results in an alternative definition of Theorem 1.1. ’ fi “mock theta function.” However, that definition has seemingly Ramanujan sde nition of a mock theta function fi nothing to do with Ramanujan’sdefinition. Theorem 1.1 shows is not equivalent to the modern de nition of a mock theta these definitions cannot be equivalent. A number of questions function. Proof: fi that will hopefully lead to a reconciliation of the two definitions De ne the two q-series are raised in 2. Questions and Remarks. ! X n 1 1 q n 3n2−n 1. Modern Definition of Mock Theta Function := − + − 2 + n V1ðqÞ 2 3 ð 1Þ q ð1 q Þ ‡ ðqÞ∞ 12 1 − qn Following Zagier (15) , we offer the following definition of a n≠0 ð Þ mock theta function. ! ∞ X n 1 1 nq n+1 1 nðn−1Þ Modern Definition. A mock theta function is a q-series := − + − 2 ; P∞ V2ðqÞ 2 n 1 ð 1Þ q = n λ ∈ Q ðqÞ∞ 12 1 − q HðqÞ n=0anq such that there existsP a rational number n = 1 = + n and a unary theta function gðzÞ n∈Q bnq of weight k,such = λ + * = ∏∞ − n that hðzÞ q HðqÞ g ðzÞ is a nonholomorphic modular form where ðqÞ∞ n= ð1 q Þ. 1 2 of weight 2 − k,where Remark: In the notation of ref. 21, V1ðqÞ = ðqÞ∞v1ðqÞ and = − 2 + X V2ðqÞ ðqÞ∞ðv2ðqÞ v3ðqÞÞ. * 2−k −n We show that either V1ðqÞ is a mock theta function according g ðzÞ = ck n bnΓðk − 1; 4πnyÞq + to the modern definition, but not Ramanujan’sdefinition, or n∈Q V2ðqÞ is a mock theta function according to Ramanujan’sdefi- R nition, but not the modern definition. Γ ; = ∞ w−1 −u with ðw tÞ t u e du, the incomplete Gamma function First, it follows from theorem 1.2 (1) of ref. 21 that V1ðqÞ is and ck a constant that depends only on k. The function g is called a mock theta function according to the modern definition with the shadow. η ‡ shadow proportional to ðqÞ. As remarked in Dabholkar, et al. , the condition that the Assume that V1ðqÞ is mock theta function according to Ram- shadow be a unary theta function forces the weight k to be either anujan’sdefinition as well. In this case, we prove that V ðqÞ is = = ’ = = 2 1 2or3 2. All of Ramanujan s examples have k 3 2. a mock theta function according to Ramanujan’sdefinition, but There are two particularly elegant examples of mock theta that it is not a mock theta function according to the modern = functions with shadow proportional to weight 1 2 unary theta definition. It was proved by Andrews (22) that functions. The work of Bringmann and Lovejoy (8, 16) studies the series X∞ − = n+1 − 2 − 2 2⋯ − n 2: X∞ 1 nðn+1Þ V1ðqÞ V2ðqÞ q ð1 qÞ 1 q ð1 q Þ q2 fðqÞ = 1 + 2 : n = 0 + ⋯ + n−1 + n 2 n = 1 ð1 qÞ ð1 q Þð1 q Þ The right-hand side of this identity is Oð1Þ for all roots of unity q, because the sum is finite. Therefore, V and V have all of the fi 1 2 The coef cients of this series are related to the rank of an same singularities at roots of unity. Because V1ðqÞ is a mock overpartition. They prove that this is a mock theta function theta function according to Ramanujan’sdefinition for every Θ = Pwith shadow proportional to the unary theta function ðqÞ root of unity ξ, there exists a theta function ϑξ such that n2 n∈Zq . This series is particularly interesting because of V1ðqÞ − ϑξðqÞ is bounded as q → ξ radially. Therefore, MATHEMATICS its relation with the class number generating function of V2ðqÞ − ϑξðqÞ is also bound as q → ξ radially. Moreover, because Zagier (18), which is also a mock theta function with the the exponential singularities of V1ðqÞ and V2ðqÞ are identical and same shadow. V1ðqÞ was assumed to be a mock theta function according to The second elegant example of a mock theta function with Ramanujan, it follows that there is no single theta function that = shadow proportional to a weight 1 2 unary theta function is the “cuts out” all of the singularities of V2ðqÞ. generating function for the number of smallest parts of a parti- To complete the proof it suffices to prove that V2ðqÞ is not tion, denoted spt. The value sptðnÞ is defined as the number of a mock theta function according to the modern definition. 1 appearances of the smallest parts in the partitions of n. Andrews However, by theorem 1.2 of ref. 21, 2 V2ðqÞ is a mock theta ðqÞ∞ (19) showed that 3 function with shadow proportional to η ðqÞ. Therefore, V2ðqÞ is X∞ X∞ a mixed-mock modular form; that is, it is the product of a mock qn SðqÞ := sptðnÞqn = : theta function and a modular form. Because the product of two − n 2 − n+1 − n+2 ⋯ n = 0 n = 1 ð1 q Þ ð1 q Þð1 q Þ harmonic Maass forms is a only when both forms being multiplied are actually modular forms or one of them is a constant, it is clear that V2ðqÞ is not the holomorphic ‡ Dabholkar A, Murthy S, Zagier D, Quantum Black Holes, Wall Crossing, and Mock Mod- part of a harmonic Maass form and thus not a mock theta ular Forms, preprint. function according to the modern definition.

Rhoades PNAS | May 7, 2013 | vol. 110 | no. 19 | 7593 Downloaded by guest on October 1, 2021 2. Questions and Remarks ; = − 1 + 1 ; ; = Since g3ðw qÞ w wð1 − wÞ Rðw qÞ where Rðw qÞ P 2 The work of Griffin et al. (4) as well as this text leaves many ∞ qn ’ n = 0 wq;q w−1q;q is Dyson s partition rank generating function, questions to be answered. ð Þnð Þn it is clear that Sðg3ð−1; qÞÞ is finite. It should be true that for ζ ≥ > 2.1. The proof of our main theorem uses a mixed-mock theta any root of unity and integers A and B with B 0 and A 0, the ‡ ζ B; A ζ B; A fi function (section 7.3 of Dabholkar, et al. ). Are all mixed-mock sets Sðg3ð q q ÞÞ and Sðg2ð q q ÞÞ are nite. It would be in- theta functions mock theta functions according to Ramanujan’s teresting to determine these sets in general. definition? Remark: It should be noted that for some roots of unity the theta functions occasionally need to be modified by a power of q 2.2. The proof of Theorem 1.1 uses an example of a weight 3=2 to get the radial limits to match. mock theta function. Does there exist a weight 1=2 counter 2.4. example to the equivalence of Ramanujan’sdefinition and the Is V1ðqÞ, given above, a mock theta function according to ’ fi modern definition? Ramanujan sde nition? The results of ref. 4 show that there is no weakly holomorphic modular form M such that V1ðqÞ − MðqÞ is 2.3. Let F be a mock theta function according to Ramanujan’s bounded radially toward all roots of unity. Is there a set of weakly ϑ ξ definition. Let SðFÞ = fϑξgξ be the set of nonzero weakly hol- holomorphic modular forms f ξðqÞgξ,where runs over all roots omorphic modular forms from condition 2 of Ramanujan’s of unity, such that V1ðqÞ − ϑξðqÞ is bounded as q → ξ radially? definition. Define the Gordon–McIntosh (5) universal mock theta functions by 2.5. There exists a modular form MðqÞ such that X∞ nðn+1Þ f0ðqÞ + 2ψðqÞ = MðqÞ; ; = q g3ðw qÞ − and P ; 1 ; ∞ 1 + + = ðw qÞn+1ðw q qÞn+1 ψ = 2 ðn 1Þðn 2Þ + ⋯ + n n 0 where ðqÞ n=0 q ð1 qÞ ð1 q Þ is one of Rama- ∞ X 1 nðn+1Þ nujan’s fifth-order mock theta functions (ref. 5, p. 104). Moreover, ð−q; qÞ q2 ; = n ; [2.1] f ðqÞ and 2ψðqÞ have singularities at disjoint sets of roots of g2ðw qÞ −1 0 ðw; qÞ + ðw q; qÞ + n = 0 n 1 n 1 unity. As a result, in the notation above, the set Sðf0Þ contains only one nontrivial modular form. This appears to be typical of ; := ∏n−1 − j fi where ðx qÞn j=0 ð1 xq Þ. the fth-order mock theta functions of Ramanujan.

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