Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem

Abstract

On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group SL(3,ℝ), which is automorphic with respect to the discrete group SL(3,ℤ). The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla. From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands. The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions σs(n)=∑d¦n,d>od5 make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as Χ → ∞ of the sum ∑n⩽Χτ3(n)τ3(n+κ), where τ3(n) is the number of solutions of the equation d1d2d3=n in natural numbers. Part II of the present work will be devoted to this binary problem. At the end of the paper properties of special functions used in Theorem 1 are discussed.