Quadratic forms connected with Fourier coefficients of Maass cusp forms

Abstract

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of {\displaystyle {\sum }_{m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}}\le x}\mathrm{\lambda }(m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}})\mathrm{\Lambda }(m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}})} and {\displaystyle {\sum }_{m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}}\le x}a(m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}})\mathrm{\Lambda }(m_{\mathrm{1}}^{\mathrm{2}}+m_{\mathrm{2}}^{\mathrm{2}}+m_{\mathrm{3}}^{\mathrm{2}})}.

Key words： Circle method; Fourier coefficients of Maass cusp forms; quadratic form; exponential sum

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