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Eisenstein
series part of the primitive representations for even-rank
quadratic forms
Ben Kane and Luhao Xue |
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Vol. 17 (2024), No. 5, 771–794
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Abstract |
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We first investigate the relationship between the number of primitive representations of by quadratic forms and the number of nonprimitive ones. Recall that the generating function for the number of representations is a modular form, which naturally splits into an Eisenstein series, giving the main asymptotic contribution, and a cusp form, contributing an error term. We hence obtain a theorem to deal with the Eisenstein series part with quadratic Dirichlet character when deriving the formula for the number of primitive representations of an integer by even-rank quadratic forms from the number of nonprimitive ones. Formulas for special cases are given as examples. |
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Keywords
modular forms, Fourier coefficients, quadratic forms,
Eisenstein series, Möbius inversion
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Mathematical Subject Classification
Primary: 11F11, 11F27, 11E20
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Milestones
Received: 31 March 2023
Revised: 30 June 2023
Accepted: 3 July 2023
Published: 21 November 2024
Communicated by Ken Ono
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Authors |
| Ben Kane | |
| Department of Mathematics University of Hong Kong Pokfulam Hong Kong |
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| Luhao Xue | |
| Department of Mathematics The University of Hong Kong Pokfulam Hong Kong |
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| © 2024 MSP (Mathematical Sciences Publishers). |
