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Sur la lacunarité des puissances de η

Published online by Cambridge University Press:  18 May 2009

Jean-Pierre Serre
Jean-Pierre Serre
Affiliation:
Collège De France, F-75231 Paris Cedex 05
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La fonction η de Dedekind est définie par

où , Im(z)>0. C'est une forme modulaire parabolique de poids 1/2. Si r est un entier, la puissance r–ième de η s'écrit;

où les coefficients pr(n) sone définis par l'identité

.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 203 - 221
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

BIBLIOGRAPHIE

1.Costello, P. J., Density Problems involving p r(n), Math. Comp. 38 (1982), 633–637.Google Scholar
2.Deligne, P. et J-P., Serre, Formes modulaires de poids 1, Ann. Sci. E.N.S. (4) 7 (1974), 507–530.Google Scholar
3.Dyson, F. J., Missed Opportunities, Bull. A.M.S. 78 (1972), 635–652.CrossRefGoogle Scholar
4.Hardy, G. H. et Wright, E. M., An Introduction to the Theory of Numbers, 3é edit. (Oxford Univ. Press, 1954).Google Scholar
5.Knopp, M. I. et Lehner, J., Gaps in the Fourier Series of Automorphic Forms, Lect. Notes in Math. 899 (1981), 360–381.Google Scholar
6.Lehmer, D. H., The Vanishing of Ramanujan's Function t(n), Duke Math. J. 14 (1947), 429–433.CrossRefGoogle Scholar
7.Li, W., Newforms and Functional Equations, Math. Ann. 212 (1975), 285–315.Google Scholar
8.Lint, J. H. van, Hecke Operators and Euler Products (Thèse, Utrecht, 1957).Google Scholar
9.Macdonald, I. G., Affine Root Systems and Dedekind's η-Function, Invent. Math. 15 (1972), 91–143.CrossRefGoogle Scholar
10.Newman, M., An Identity for the Coefficients of Certain Modular Forms, J. London Math. Soc. 30 (1955), 488–493.CrossRefGoogle Scholar
11.Newman, M., A Table of the Coefficients of the Powers of η(T), Proc. Acad. Amsterdam 59 (1956), 204–216.Google Scholar
12.Parkin, T. R. et Shanks, D., On the Distribution of Parity in the Partition Function, Math. Comp. 21 (1967), 466–480.CrossRefGoogle Scholar
13.Ramanujan, S., On Certain Arithmetical Functions, Trans. Cambridge Phil. Soc. 22 (1916), 159–184 (=Collected Papers, n° 18, 136–162).Google Scholar
14.Rankin, R., Hecke Operators on Congruence Subgroups of the Modular Group, Math. Ann. 168 (1967), 40–58.CrossRefGoogle Scholar
15.Ribet, K., Galois Representations attached to Eigenforms with Nebentypus, Lect. Notes in Math. 601 (1977), 17–52.Google Scholar
16.Schoeneberg, B., Über den Zusammenhang der Eisensteinschen Reihen und Thetareihen mit der Diskriminante der elliptischen Funktionen, Math. Ann. 126 (1953), 177–184.CrossRefGoogle Scholar
17.Serre, J-P., Divisibilité de certaines fonctions arithmétiques, L'Ens. Math. 22 (1976), 227–260.Google Scholar
18.Serre, J-P., Modular Forms of Weight One and Galois Representations, in Alg. Number Fields (Fröhlich, A. édit.) (Acad. Press 1977), 193–268.Google Scholar
19.Serre, J-P., Quelques applications du théorème de densité de Chebotarev, Publ. Math. I.H.E.S. 54 (1981), 123–201.Google Scholar
20.Shimura, G., Class Fields over Real Quadratic Fields and Hecke Operators, Ann. of Math. 95 (1972), 130–190.CrossRefGoogle Scholar
21.Shimura, G., On Modular Forms of Half Integral Weight, Ann. of Math. 97 (1973), 440–481.CrossRefGoogle Scholar
22.Swinnerton-Dyer, H. P. F., On l-adic Representations and Congruences for Coefficients of Modular Forms, Lect. Notes in Math. 350 (1973), 1–55.CrossRefGoogle Scholar
23.Waldspurger, J-L., Sur les coefficients de Fourier des formes modulaires de poids demientier, J. Math, pures et appl. 60 (1981), 375–484.Google Scholar