Goldfeld, Dorian 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 PoincarĂ© 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 Cited in 1 ReviewCited in 4 Documents 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 Keywords:Kloosterman sums; hyper-Kloosterman sums; meromorphic continuation; global Kloosterman \(\zeta\)-function Citations:Zbl 0618.10024; Zbl 0608.12015; Zbl 0657.00005; Zbl 0612.10020; Zbl 0597.12017; Zbl 0507.10029 PDFBibTeX XML