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On Ramanujants Definition of Mock Theta Function On Ramanujan’sdefinition of mock theta function Robert C. Rhoades1 Department of Mathematics, 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 modular form 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 À Á 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 À Á À Á 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 À Á 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∞ À Á − = 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.
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