Contributions to the theory of Ramanujan's function τ(<i>n</i>) and similar arithmetical functions

Rankin, R. A.

doi:10.1017/s0305004100021101

Cited by 216 publications

(50 citation statements)

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“…See, for example, Rankin [12], Selberg [14], Peterson [ll], as well as [18], Lemma 3.3, and [19], p. 95. Anyway, D(s,f, g) can be continued to a meromorphic function on the whole plane.…”

Section: The Relation Between the Special Values And The Petersson Inmentioning

confidence: 99%

The special values of the zeta functions associated with cusp forms

Shimura¹

2002

Collected Papers

187

124

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“…See, for example, Rankin [12], Selberg [14], Peterson [ll], as well as [18], Lemma 3.3, and [19], p. 95. Anyway, D(s,f, g) can be continued to a meromorphic function on the whole plane.…”

Section: The Relation Between the Special Values And The Petersson Inmentioning

confidence: 99%

The special values of the zeta functions associated with cusp forms

Shimura¹

2002

Collected Papers

187

124

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“…For modular functions which tend to zero at the cusp (cuspidal functions), such integrals may be computed using the Rankin-Selberg method, in terms of the constant Fourier mode of the cuspidal function [23,24]. Modular graph functions, however, have polynomial growth at the cusp, just as real-analytic Eisenstein series do, and their integration requires regularization.…”

Section: Introductionmentioning

confidence: 99%

Integral of two-loop modular graph functions

D’Hoker¹

2019

J. High Energ. Phys.

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The integral of an arbitrary two-loop modular graph function over the fundamental domain for SL(2, Z) in the upper half plane is evaluated using recent results on the Poincaré series for these functions.Recent investigations of the transcendentality properties of the genus-one amplitude in [27] directly motivate the problems addressed in the present paper.The organization of the remainder of this paper is as follows. In section 2 we shall briefly review the definition and basic properties of modular graph functions in terms of their Kronecker-Eisenstein and Poincaré series representations. The main results of the paper are presented in Theorems 3.1 and 3.2 of section 3, respectively giving the (regularized) integrals for two-loop modular graphs functions of arbitrary odd and even weight. We conclude in section 4 with a discussion of open problems and speculations regarding the integrals of higher loop modular graph functions.

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“…All of these estimates follow from well-known results. The estimate in (5.1) can be proved using the techniques of Rankin [45] and Selberg [51]. Proposition 2.3 of Rudnick and Sarnak [48]…”

Section: Averages Of the Fourier Coefficients λ F (N)mentioning

confidence: 98%