Journal of the Ramanujan Mathematical Society

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Effective Lower and Upper Bounds for the Fourier Coefficients of Powers of the Modular Invariant j


Affiliations
1 LArAl, Universite J. Monnet, 23, rue du Dr P. Michelon, F-42023 Saint-Etienne Cedex, France and Arenaire, LIP, Ecole Normale Superieure de Lyon, 46, Allee d’Italie, F-69364 Lyon Cedex 07, France
2 LArAl, Universite Jean Monnet, 23, rue du Dr P. Michelon, F-42023 Saint-Etienne Cedex, France
     

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Using an elementary approach, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j . Moreover, a straightforward adaptation of an old result of Rademacher yields a convergent series expansion of these Fourier coefficients and we show that this expansion allows to find a weaker version of these estimates in the general case and sharper ones in the case of j . Our results improve on previous ones by K. Mahler and O. Herrmann. In particular, we show that the Fourier coefficients of j are smaller than their asymptotically equivalent given by Petersson and Rademacher.
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  • Effective Lower and Upper Bounds for the Fourier Coefficients of Powers of the Modular Invariant j

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Authors

Nicolas Brisebarre
LArAl, Universite J. Monnet, 23, rue du Dr P. Michelon, F-42023 Saint-Etienne Cedex, France and Arenaire, LIP, Ecole Normale Superieure de Lyon, 46, Allee d’Italie, F-69364 Lyon Cedex 07, France
Georges Philibert
LArAl, Universite Jean Monnet, 23, rue du Dr P. Michelon, F-42023 Saint-Etienne Cedex, France

Abstract


Using an elementary approach, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j . Moreover, a straightforward adaptation of an old result of Rademacher yields a convergent series expansion of these Fourier coefficients and we show that this expansion allows to find a weaker version of these estimates in the general case and sharper ones in the case of j . Our results improve on previous ones by K. Mahler and O. Herrmann. In particular, we show that the Fourier coefficients of j are smaller than their asymptotically equivalent given by Petersson and Rademacher.