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The \(f(q)\) mock theta function conjecture and partition ranks. (English) Zbl 1135.11057

As it well known, Ramanujan defined several mock theta functions in his last letter to Hardy (January, 1920). Despite the passage of 88 years, the properties of these functions have remained somewhat mysterious up to the present day. In this important paper the authors prove a conjecture made by George Andrews more than 40 years ago concerning the Fourier expansion of
\[ f(q)= 1+\sum^\infty_{n=1} {q^{(n^2)}\over (1+ q)^2\cdots (1+ q^n)^2},\tag{1} \]
one of Ramanujan’s “third-order” mock theta functions.
The transformation properties under the inversion \(z\to -1/2\) were first treated by G. N. Watson in [J. Lond. Math. Soc. 11, 55–80 (1936; Zbl 0013.11502)]. In [Trans. Am. Math. Soc. 72, 474–500 (1952; Zbl 0047.27902)], an article based upon her doctoral dissertaion, L. A. Dragonette applied Watson’s results to obtain approximate transformation formulae for \(f(q)\) under all elements of \(\text{SL}(2,\mathbb{Z})\), and she used these, together with the circle method, to determine the first \(\sqrt{n}\) terms in the asymptotic expansion of \(\alpha(n)\), the \(n\)th Fourier coefficient of \(f(q)\), with a remainder term of the order of growth \(\sqrt{n}\log n\), as \(n\to\infty\). George Andrews continued Dragonette’s work in his Ph.D. thesis (mid-1960’s; like Dragonette’s, directed by Rademacher), reducing the estimate of the order of growth of the remainder to \(O(n^\varepsilon)\), arbitrary \(\varepsilon> 0\). Andrews achieved this striking improvement by deriving more precise transformation formulae of \(f(q)\) under \(\text{SL}(2,\mathbb{Z})\) through application of Poisson summation. The results of Dragonette and Andrews bring to mind the growth of \(p(n)\), the partition function of number theory. Indeed, the order of growth in \(n\) of \(p(n)\) is \(\exp(\pi\sqrt{2n/3})\), while that of \(\alpha(n)\) is the square root of the latter expression, that is, \(\exp(\pi\sqrt{n/6})\).
The paper under review completes the work of Dragonette and Andrews, establishing an exact convergent series formula for \(\alpha(n)\) and thereby proving the conjecture of Andrews, to which the authors have, quite reasonably, appended Dragonette’s name. The series in question is: \[ \alpha(n)= \pi(24n-1)^{-1/4} \sum^\infty_{k=1} (-1)^{[{k+1\over 2}]}A_{2k}(n- \rho(k))\times I_{1/2}(\pi\sqrt{24n-1}/12k),\tag{2} \] where \(I_{1/2}\) is the modified Bessel function of the first kind, \(A_k(n)\) is the usual Kloosterman sum occurring in Rademacher’s convergent series representation of \(p(n)\) (i.e., the summands of \(A_k(n)\) include the multiplier system of the Dedekind eta-function as factors), and \(p(k)= k/2\) for even \(k\); \(0\) for odd \(k\). The expression (2) is strikingly similar to Rademacher’s series for \(p(n)\), in which \(I_{3/2}(\pi\sqrt{24n-1}/6k)\) occurs (in place of \(I_{1/2}(\pi\sqrt{24n-1}/12k))\) and \(A_k(n)\) replaces the more complicated \(A_{2k}(n- \rho(k))\) of (2).
Having proved (2), the authors likewise obtain the long-sought exact formulas for \(N_e(n)\) and \(N_0(n)\), where these arithmetic functions are, respectively, the number of partitions of \(n\) of even and odd rank. (The rank of a partition is its largest part minus the number of its parts.) This follows from the identities \(p(n)= N_e(n)+ N_0(n)\), \(\alpha(n)= N_e(n)- N_0(n)\), the first one obvious and the second well known.
The proof that Bringmann and Ono adduce to prove (2) is remarkable for its depth and ingenuity, not surprisingly in light of the widely-recognized difficulty of the problem they have solved. It entails work of S. P. Zwegers [Contemp. Math. 291, 269–277 (2001; Zbl 1044.11029)], which recasts Watson’s modular transformation properties (of \(f(q)\)) as the transformation formulae of a real analytic three-dimensional vector-valued modular form (weight \({1\over 2}\)). The proof involves as well Maass forms and the Serre-Stark basis theorem for holomorphic modular terms of weight \({1\over 2}\). Especially interesting is the application (§5) of a weight-changing differential operator which transforms weight \(k\) Maass forms to weight \(2-k\) holomorphic modular forms, and therefore can be understood as an antilinear analogue of the Bol differential operator.

MSC:

11P82 Analytic theory of partitions
05A17 Combinatorial aspects of partitions of integers
11F37 Forms of half-integer weight; nonholomorphic modular forms
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References:

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