Algebraic and transcendental formulas for the smallest parts function

Abstract

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the smallest parts function introduced by Andrews; $\mathrm{spt}(n)$ counts the number of smallest parts in the partitions of $n$. The generating function for $\mathrm{spt}(n)$ forms a component of a natural mock modular form of weight $3/2$ whose shadow is the Dedekind eta function. Using automorphic methods (in particular the theta lift of Bruinier and Funke), we obtain an exact formula and an algebraic formula for its values. In contrast with the case of $p(n)$, the convergence of our expression is non-trivial, and requires power savings estimates for weighted sums of Kloosterman sums for a multiplier in weight $1/2$. These are proved with spectral methods (following an argument of Goldfeld and Sarnak).

21 pages, 2 figures. Version 3 corrects an error in normalization which affects the constants in Theorem 3, Theorem 8, and Lemma 9. These changes will appear as a corrigendum to the published version

References in corpus (1)

• Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2

Cited by in corpus (7)

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• Uniform bounds for Kloosterman sums of half-integral weight with applications
• Locally harmonic Maass forms and periods of meromorphic modular forms
• Bounds for Coefficients of the $f(q)$ Mock Theta Function and Applications to Partition Ranks
• Uniform bounds for Kloosterman sums of half-integral weight, same-sign case