Theory of Eisenstein series for the group $SL(3,\mathbf R)$ and its application to a binary problem. I. Fourier expansion of the highest Eisenstein series

A. I. Vinogradov   ; L. A. Takhtadzhyan

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions, Tome 76 (1978), pp. 5-52

Résumé

On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group $SL(3,\mathbf R)$, which is automorphic with respect to the discrete group $SL(3,\mathbf Z)$. 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 $\sigma_s(n)=\sum_{d|n,d>0}d^s$ make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as $X\to\infty$ of the sum $\sum_{n\leqslant X}\tau_3(n)\tau_3(n+k)$, where $\tau_3(n)$ is the number of solutions of the equation $d_1d_2d_3=n$ 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.