The author presents a new variant of the classical circle method. The basic idea is to replace \(\chi(x)\), the characteristic function of \( [0,1] \) with \(\widetilde{\chi}(x)\), precisely defined in the text. The latter function has the property that the mean square integral of \(\chi(x) - \widetilde{\chi}(x)\) over \( [0,1] \) is small under certain assumptions. This is used then to prove the bound (\(N \geq 1, m \geq 1\)) \[ \sum_{n=1}^\infty \overline {a(n)}a(n+m)\exp\left(-{2\pi(2n+m)\over N}\right) \ll_\varepsilon N^{k-{1\over 2}+\varepsilon}m^\alpha.(1) \] Here \(a(n)\) is the \(n\)th Fourier coefficient of a holomorphic cusp form of weight \(k\) for the full modular group, and \(\alpha (\leq 5/28)\) is the constant such that \(t_j(n) \ll_\varepsilon n^{\alpha+\varepsilon}\) holds, where \(t_j(n)\) is the eigenvalue for the \(j\)th Maass wave form. The bound (1) (without the dependence on \(m\)) was mentioned by D. Goldfeld [Astérisque 61, 95-107 (1979; Zbl 0401.10034)]. Further possibilities for applications of the author’s method are indicated at the end of the paper.
For the entire collection see [Zbl 0910.00038].