# Article

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Keywords:
power-moment; SL\$_3(\mathbb Z)\$-Kloosterman sum
Summary:
Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL\$_2\$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL\$_3\$ have been considered, in which analogous GL\$_3\$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL\$_3(\mathbb Z)\$. We give formulas for the first three moments and a nontrivial bound for the fourth.
References:
[1] Adolphson, A., Sperber, S.: Exponential sums and Newton polyhedra: cohomology and estimates. Ann. Math. (2) 130 (1989), 367-406. MR 1014928 | Zbl 0723.14017
[2] Bump, D., Friedberg, S., Goldfeld, D.: Poincaré series and Kloosterman sums for SL\$(3,\mathbb Z)\$. Acta Arith. 50 (1988), 31-89. MR 0945275
[3] Goldfeld, D.: Automorphic Forms and \$L\$-Functions for the Group GL\$(n, \mathbb R)\$. With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics 99. Cambridge University Press, Cambridge (2006). MR 2254662
[4] Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17 AMS, Providence, RI (1997). MR 1474964 | Zbl 0905.11023
[5] Kloosterman, H. D.: On the representation of numbers in the form \$ax^2+by^2 +cz^2+dt^2\$. Acta Math. 49 (1927), 407-464. DOI 10.1007/BF02564120 | MR 1555249
[6] Larsen, M.: Appendix to Poincaré series and Kloosterman sums for SL\$(3,\mathbb Z)\$, in The estimation of \$SL_3(\mathbb Z)\$ Kloosterman sums. Acta Arith. 50 (1988), 86-89. MR 0945275

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