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**The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight. II.**
*(English)*
Zbl 1161.11336

Introduction: We introduce and investigate a family of functions called nonanalytic “pseudo-Poincaré series”. These functions are inspired by D. Niebur’s work [Trans. Am. Math. Soc. 191, 373–385 (1974; Zbl 0306.30023)] on automorphic forms and integrals of negative weight. We prove that an arbitrary Niebur modular integral (including a modular form) on the full modular group, \(\Gamma(1)\), 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-Poincaré series. We derive exact formulas, as convergent infinite series, for the Fourier coefficients of these pseudo-Poincaré 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; both involve the modified Bessel function of the first kind and generalized Kloosterman sums. In the weight range \(2/3 \leq k <1\), 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 previous paper [Acta Arith. 91, No. 4, 291–309 (1999; Zbl 0944.11014)] 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 \(\Gamma(1)\) of weight \(k\), \(0 < k < 1\).

The results established here mirror those presented in our first paper [loc. cit.], which focused on Niebur modular integrals in the weight range \(1 < k < 2\). Recall that this previous paper provided an extension of M. I. Knopp’s work [ Proc. Symp. Pure Math. 49, Part 2, 111–127 (1989; Zbl 0679.10019)] on the Fourier coefficients of modular forms of weight \(k\), \(4/3 < k < 2\). Several of the tools we employ here are the same as those used by Knopp and the author. These include Poisson summation and work pertaining to the analyticity and growth behaviour of Selberg’s Kloosterman zeta-function.

However, there are some interesting differences here as well. Our point of departure is not to invoke (nonanalytic) Poincaré series, which converge absolutely for \(\text{Re}(s) > 2-k\), but rather to explore (nonanalytic) pseudo-Poincaré series (mentioned above), which converge absolutely for \(\text{Re}(s) > k\). This favorable convergence property (recall that here \(0 < k < 1\)) is created by the subtracted “Rademacher convergence summand” present in the pseudo-Poincaré series. Ironically, it is this difference which ultimately permits us the use of the same results on Selberg’s Kloosterman zeta-function that we invoked in [loc. cit]. Another major difference here is the use of a two-variable summation formula, one which is needed to handle a certain series that arises in the period of the modular relation for our pseudo-Poincaré series. In fact, the study of this series provides us with a rediscovery of the known formulas for the Fourier coefficients of arbitrary cusp forms of weight \(k\), \(1 < k < 2\). Lastly, by summoning properties of cusp forms on the full modular group, we prove that “most” of the Niebur modular integrals studied here are actually modular forms.

The results established here mirror those presented in our first paper [loc. cit.], which focused on Niebur modular integrals in the weight range \(1 < k < 2\). Recall that this previous paper provided an extension of M. I. Knopp’s work [ Proc. Symp. Pure Math. 49, Part 2, 111–127 (1989; Zbl 0679.10019)] on the Fourier coefficients of modular forms of weight \(k\), \(4/3 < k < 2\). Several of the tools we employ here are the same as those used by Knopp and the author. These include Poisson summation and work pertaining to the analyticity and growth behaviour of Selberg’s Kloosterman zeta-function.

However, there are some interesting differences here as well. Our point of departure is not to invoke (nonanalytic) Poincaré series, which converge absolutely for \(\text{Re}(s) > 2-k\), but rather to explore (nonanalytic) pseudo-Poincaré series (mentioned above), which converge absolutely for \(\text{Re}(s) > k\). This favorable convergence property (recall that here \(0 < k < 1\)) is created by the subtracted “Rademacher convergence summand” present in the pseudo-Poincaré series. Ironically, it is this difference which ultimately permits us the use of the same results on Selberg’s Kloosterman zeta-function that we invoked in [loc. cit]. Another major difference here is the use of a two-variable summation formula, one which is needed to handle a certain series that arises in the period of the modular relation for our pseudo-Poincaré series. In fact, the study of this series provides us with a rediscovery of the known formulas for the Fourier coefficients of arbitrary cusp forms of weight \(k\), \(1 < k < 2\). Lastly, by summoning properties of cusp forms on the full modular group, we prove that “most” of the Niebur modular integrals studied here are actually modular forms.