The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II
This is the continuation of the author's earlier work [Math. Z. 223, 435-461 (1996; Zbl 0865.11062)] on the analogue of the binary additive divisor problem for Fourier coefficients of cusp forms (holomorphic and non-holomorphic). The author gives now a detailed proof of the bound (stated already in Part I of the paper) \[ \sum_{\kappa_j\leq K}|c_j|^2\exp(\pi\kappa_j) + \int_{- K}^K|c(u)|^2\exp(\pi|u|) \text{ d}u \ll_{\varepsilon,\kappa} K^\varepsilon, \] which is important in treating the additive problem for Fourier coefficients of cusp forms. The above result is valid for \(K \geq 1\), and the notation is the same as in Part I: \[ c_j = (u_j(z),u^2(z)),\quad c(t) = \Bigl(E(z, {\textstyle{1\over 2}}+it),u^2(z)\Bigr), \] where \((f,g)\) is the Petersson inner product of \(f\) and \(g\), \(u(z)\) is a Maaas wave form of certain fixed index \(\nu\), \(\kappa = \kappa_{\nu}\), and \(E(z,s)\) is the non-holomorphic Eisenstein series. The intricate proof, based on three lemmas, involves techniques such as the use of Bessel functions, the hypergeometric function and the spectral large sieve inequality.
scientific article; zbMATH DE number 1277931