partition function, rank generating function, Maass form, weakly holomorphic modular form
Introduction and statement of results. Recent works have illustrated that the Fourier coefficients of harmonic weak Maass forms of weight 1/2 contain a wealth of number-theoretic and combinatorial information. After these works, it is known that many enigmatic q-series (the “mock theta functions” of Ramanujan, and certain rank-generating functions from the theory of partitions, for example) arise naturally as the “holomorphic parts” of such forms. See, for example, Bringmann and Ono [5, 6], Bringmann, Ono, and Rhoades , Zwegers , Bringmann and Lovejoy , Lovejoy and Osburn , or see the survey paper  for an overview. As another striking example, Bruinier and Ono  show that the coefficients of the holomorphic parts of weight 1/2 Maass forms determine the fields of definition of certain Heegner divisors in the Jacobians of modular curves, which in turn determine the vanishing or non-vanishing of derivatives of modular L-functions
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c Instytut Matematyczny PAN, 2008
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Ahlgren, Scott and Treneer, Stephanie, "Rank Generating Functions as Weakly Holomorphic Modular Forms" (2008). Mathematics. 108.
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