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On the divisibility of r2(n)

Published online by Cambridge University Press:  18 May 2009

E. J. Scourfield
E. J. Scourfield
Affiliation:
Department of Mathematics, Westfield College, London NW3 7ST
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During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Let

be a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such that

as x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 109 - 111
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

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