Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
Let \(p(n)\) be the number of all partitions of natural number \(n\). It is well known the asymptotic formula for \(p(n)\) obtained by Hardy and Ramanujan expressing \(p(n)\) through certain exponential function and Rademacher's ``exact'' formula expressing \(p(n)\) as a Kloosterman sum and a modified Bessel function of the first kind. The authors obtains new formula for \(p(n)\), which expresses \(p(n)\) as a finite sum of singular moduli for a weak Maass form. The form is constructed using Dedekind's eta-function and quasimodular Eisenstein series. The sum of special values of the weak Maass form defines a ``trace'' \(T(n)\). In this paper, a formula it is obtained, where \(p(n)\) is expressed by this trace. Also it is provided an algorithm for computing \(p(n)\).
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