On sums of $\mathit{SL}(3,\mathbb{Z})$ Kloosterman sums

Abstract

We show that sums of the \(\mathit{SL}(3,\mathbb{Z})\) long element Kloosterman sum against a smooth weight function have cancelation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li’s generalization of the Kuznetsov formula on \(\mathit{SL}(3,\mathbb{R})\), which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin–Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion formula suitable for studying sums of the long-element \(\mathit{SL}(3,\mathbb{Z})\) Kloosterman sums.