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[Figure not available: see fulltext.];, by means of the spectral characteristics of the automorphic Laplacian. On the basis of this representation, the meromorphic continuability of ζ_k(s) to the whole complex plane is proved and a power estimate of the growth of ζ_k(s) as |s|→ ∞ in the critical strip 0Řes≤1 is obtained. From this, with the help of the method of complex integration, the asymptotic formula[Figure not available: see fulltext.], is derived, where P_k (x) is a quadratic polynomial