Eisenstein series part of the primitive representations for even-rank quadratic forms

Kane, Ben; Xue, Luhao

doi:10.2140/involve.2024.17.771

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Eisenstein series part of the primitive representations for even-rank quadratic forms

Ben Kane and Luhao Xue

Vol. 17 (2024), No. 5, 771–794

DOI: 10.2140/involve.2024.17.771

Abstract

We first investigate the relationship between the number of primitive representations of $n$ 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 $n$ by even-rank quadratic forms from the number of nonprimitive ones. Formulas for special cases are given as examples.

Keywords

modular forms, Fourier coefficients, quadratic forms, Eisenstein series, Möbius inversion

Mathematical Subject Classification

Primary: 11F11, 11F27, 11E20

Milestones

Received: 31 March 2023

Revised: 30 June 2023

Accepted: 3 July 2023

Published: 21 November 2024

Communicated by Ken Ono

Authors

Ben Kane
Department of Mathematics University of Hong Kong Pokfulam Hong Kong

Luhao Xue
Department of Mathematics The University of Hong Kong Pokfulam Hong Kong

Vol. 17, No. 5, 2024