Analytic continuation of \(\zeta _ 3(s,k)\) in the critical strip. Arithmetical part. (Russian) Zbl 0643.10024

Vinogradov, A. I.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 162, 43-76 (1987).

This work is the detailed program to obtain the meromorphic continuation for the Dirichlet series \[ \zeta_ 3(s,k)=\sum^{\infty}_{n=1}(n(n+k))^{-s/2}\tau_ 3(n)\tau_ 3(n+k), \] where \(\tau_ 3(n)=\sum_{d_ 1d_ 2d_ 3=n}1\). The main idea is to consider the quantities \(\tau_ 3(n)\) as an averaging of the Fourier coefficients of the Eisenstein series for SL(3,\({\mathbb{Z}})\). This is by no means a simple way; there are a lot of problems to integrate and to sum the Selberg-Linnik zeta-function. For these purposes the sum formulae for the Kloosterman sums are used.

Reviewer: N.V.Kuznetsov

MSC:

11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11L03	Trigonometric and exponential sums (general theory)
11F27	Theta series; Weil representation; theta correspondences
11M35	Hurwitz and Lerch zeta functions

Keywords:

three-dimensional hyperbolic space; additive divisor problem; sum of Kloosterman sums; meromorphic continuation; Dirichlet series; Selberg- Linnik zeta-function