Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j

Abstract : Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get an convergent series expansion of the Fourier coefficients and we show that this expansion allows to find again these estimates. Our results improve previous ones by K. Mahler and O. Hermann. In particular, we show that Fourier coefficients of j are smaller than their asymptotically equivalent given by Petersson and Rademacher
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https://hal-lara.archives-ouvertes.fr/hal-02102113
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Nicolas Brisebarre, Georges Philibert. Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. [Research Report] LIP RR 2003-50, Laboratoire de l'informatique du parallélisme. 2003, 2+28p. ⟨hal-02102113⟩

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