# Revista Matemática Iberoamericana

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**Volume 35, Issue 7, 2019, pp. 2093–2118**

**DOI: 10.4171/rmi/1113**

Published online: 2019-07-22

The arithmetic Kuznetsov formula on GL(3), I: The Whittaker case

Jack Buttcane^{[1]}(1) The University of Maine, Orono, USA

The original formulae of Kuznetsov for SL(2, $\mathbb Z$) allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for SL(3, Z), and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of GL(2). In this paper, we demonstrate a linear combination of the spherical and weight-one SL(3, $\mathbb Z$) Kuznetsov formulae that isolates one particular integral kernel, which is the spherical GL(3) Whittaker function. Using the known inversion formula of Wallach, we give the first arithmetic Kuznetsov formula for SL(3, $\mathbb Z$) and use it to study smooth averages and the Kloosterman zeta function attached to this particular choice of signs.

*Keywords: *GL(3), Kuznetsov, Kloosterman sums, Whittaker functions, Kloosterman zeta function, exponential sums

Buttcane Jack: The arithmetic Kuznetsov formula on GL(3), I: The Whittaker case. *Rev. Mat. Iberoam.* 35 (2019), 2093-2118. doi: 10.4171/rmi/1113