The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, II

Abstract

The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, II by Wladimir de Azevedo Pribitkin (Princeton, NJ) 1. Introduction. We introduce and investigate a family of functions called nonanalytic "pseudo-Poincar series". These functions are inspired by Douglas Niebur's work We prove that an arbitrary Niebur modular integral (including a modular form) on the full modular group, (1), of weight k, 0 < k < 1, can be decomposed uniquely as a sum of a cusp form and a finite linear combination of (special values of) pseudo-Poincar series. We derive exact formulas, as convergent infinite series, for the Fourier coefficients of these pseudo-Poincar series. In the weight range 0 < k < 2/3, the formulas we produce for these series have precisely the same structure as the well-known expressions for negative weights found by Rademacher and Zuckerman [10]; both involve the modified Bessel function of the first kind and generalized Kloosterman sums. In the weight range 2/3 k < 1, however, the formulas we discover are not as satisfying because they contain Selberg's Kloosterman zeta-function evaluated outside of its known range of convergence. In our prequel [9] we already found expressions, which contain residues of the zeta-function just mentioned, for the Fourier coefficients of small positive powers (between 0 and 2) of the Dedekind eta-function. So our decomposition theorem implies that we possess the Fourier expansions of all Niebur modular integrals on (1) of weight k, 0 < k < 1.