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Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions: II. The order of the Fourier coefficients of integral modular forms

  • R. A. Rankin (a1)
Abstract

Suppose that

is an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Then

where a, b, c, d are integers such that ad − bc = 1.

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* Cf. for example, Klein F. and Fricke R., Elliptische Modulfunktionen, 1 (Leipzig, 1890), 395–7.

* Salié H., “Zur Abschätzung der Fourierkoeffizienten ganzer Modulformen”, Math. Z. 36 (1931), 263–78.

Davenport H., “On certain exponential sums”, J. reine angew. Math. 169 (1932), 158–76.

Hauptkongruenzgruppe.

* Cf. Landau E., Primzahlen, 1 (Leipzig, 1909), 483–92.

This may be proved in several ways; cf., for example Hecke E., “Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I”, Math. Ann. 114 (1937), 128 (Satz 5).

* Cf. for example E. Hecke, loc. cit., Satz 7.

By the Wiener-Ikehara theorem we can deduce at once from Theorem 3 that

See Bochner S., “Ein Satz von Landau und Ikehara”, Math. Z. 37 (1933), 19.

Landau E., “Über die Anzahl der Gitterpunkte in gewissen Bereichen. II”, Nachr. Ges. Wiss. Göttingen (1915), pp. 209–43.

* | αγ, δ (n)| is not dependent on α, β.

When (m, n)>1, (m, n, N) = 1, we denote by f m n (s) the function f γ,δ(s), where γ ≡ m, δ ≡ n (mod N), (γ, δ) = 1.

* This is trivial when inline-graphic It is true also for inline-graphic since Landau's theorem can be extended to show that

for any real α > − β, where R(a, x) is the sum of the residues of inline-graphic in the strip inline-graphic

I write k′ where Landau has k to avoid confusion with the dimension − k.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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