#### Vol. 16, No. 3, 2022

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The arithmetic Kuznetsov formula on GL(3), II: The general case

### Jack Buttcane

Vol. 16 (2022), No. 3, 567–646
##### Abstract

We obtain the last of the standard Kuznetsov formulas for $\mathrm{SL}\left(3,ℤ\right)$. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach’s Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach’s theorem for $\mathrm{SL}\left(3\right)$. As applications, we again consider the Kloosterman zeta functions and smooth sums of Kloosterman sums. The new Kloosterman zeta functions pose the same difficulties as we saw with the positive-sign case, but for the negative-sign case, we obtain some analytic continuation of the unweighted zeta function and give a sort of reflection formula that exactly demonstrates the obstruction to further continuation arising from the Kloosterman sums whose moduli are far apart. The completion of the remaining sign cases means this work now both supersedes the author’s thesis and completes the work started in the original paper of Bump, Friedberg and Goldfeld.

##### Keywords
analytic number theory, Kloosterman sums, Kuznetsov formula, $\mathrm{GL}(3)$, harmonic analysis on symmetric spaces, Bessel functions
##### Mathematical Subject Classification 2010
Primary: 11L05
Secondary: 11F55, 11F72