On certain Dirichlet series obtained by the product of Eisenstein series and a cusp form (Q1898234)

The Dirichlet series \(D(s; \alpha, f)\) under consideration is, essentially, the generating function of \(\sigma_{2\alpha- 1} (m+ k) \rho(m)\), where \(\rho(m)\) runs over the Fourier coefficients of a Maass wave form \(f(z)\), and the \(\sigma\)-function is a divisor function in the standard notation. This series is produced from the product of \(f(z)\) with an Eisenstein-Maass series by an argument similar to that in a paper of \textit{L. A. Takhtadzhyan} and \textit{A. I. Vinogradov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 134, 84-116 (1984; Zbl 0536.10025)], where the square of an Eisenstein series played the same role as that product. The author applies the spectral theory of automorphic functions to establish an analytic continuation and estimates for \(D(s; \alpha, f)\). This information implies an estimate for a sum analogous to that in the additive divisor problem.