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Kloosterman zeta functions for \(\text{GL}(n, \mathbb Z)\). (English) Zbl 0667.10027
Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 417-424 (1987).
The author generalizes ‘Kloostermania’ to \(\text{GL}(n, \mathbb Z)\) for \(n\geq 2\) [see also S. Friedberg, Math. Z. 196, 165–188 (1987; Zbl 0612.10020), and G. Stevens, Math. Ann. 277, 25–51 (1987; Zbl 0597.12017)], announcing his aim of achieving uniform estimates for products of classical and more general Kloosterman sums. This could be done in a way analogous to D. Goldfeld and P. Sarnak [Invent. Math. 71, 243–250 (1983; Zbl 0507.10029)].
Therefore it is necessary to establish a meromorphic continuation for the corresponding global Kloosterman \(\zeta\)-function which is not yet at hand for \(n\geq 4\). The author outlines how this could be achieved imitating the classical way of evaluating products of certain Poincaré series.
The author indicates a further approach to the desired estimates via a generalization of the methods used by Zagier in his proof of the Kuznetsov-Bruggeman sum formula.
[For the entire collection see Zbl 0657.00005.]
Reviewer: Roland Matthes

11L05 Gauss and Kloosterman sums; generalizations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M41 Other Dirichlet series and zeta functions