The zeta function of the additive divisor problem and spectral decomposition of the automorphic Laplacian

A representation is obtained for the zeta function of the additive divisor problem;, by means of the spectral characteristics of the automorphic Laplacian. On the basis of this representation, the meromorphic continuability of _k(s) to the whole complex plane is proved and a power estimate of the growth of _k(s) as |s| in the critical strip 0es1 is obtained. From this, with the help of the method of complex integration, the asymptotic formula, is derived, where P_k (x) is a quadratic polynomialTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 84–116, 1984.