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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

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