On sums of hyper-Kloosterman sums (Q6133252)

A conjecture of \textit{Yu. V. Linnik} [Proc. Int. Congr. Math., Stockholm 1962, 270--284 (1963; Zbl 0116.03604)] on the average of the classical Kloosterman sums \[ \mathrm{Kl}_2(m,n,c)=\sum_{\substack{x,y\,(\mathrm{mod}\,c)\\ xy\equiv 1\,(\mathrm{mod}\,c)}}e\left(\frac{mx+ny}{c}\right),\quad e(x):=\exp(2\pi i x), \] states that there should be good cancellation among the terms of an average over the moduli \(c\). As one of very few examples where we can detect cancellation in an average over the moduli of an exponential sum, \textit{N. V. Kuznecov} [Mat. Sb., N. Ser. 111 (153), 334--383 (1980; Zbl 0427.10016)] (with an English translation in [Math. USSR, Sb. 39, 299--342 (1981; Zbl 0461.10017)]) solved Linnik's problem by relating the spectral and Bruhat expansions of Fourier coefficients of Poincaré series over \(\mathrm{SL}(2,\mathbb{Z})\) to obtain a pair of formulas of the shape \begin{align*} & \sum_{c=1}^\infty\frac{\mathrm{Kl}_2(m,n,c)}{c}F_1\left(\frac{\sqrt{mn}}{c}\right)+\text{trivial term}\\ = & \sum_{\phi}\frac{\lambda_\phi(m)\lambda_\phi(n)}{L(1,\mathrm{Ad}^2\phi)}F_2(\mu_\phi)+\text{continuous term}, \end{align*} where the sum on the right is over a basis of (minimal-weight) Maass cusp forms for \(\mathrm{SL}(2,\mathbb{Z})\) with Hecke eigenvalues \(\lambda_\phi(m)\) and spectral parameters \(\mu_\phi\), and the (mostly) arbitrary test functions \(F_1\) and \(F_2\) are related by Bessel or inverse-Bessel transforms (giving two formulas). In the paper under review, the author applies \(\mathrm{GL}(3)\) automorphic forms to obtain a Kuznetsov-type formula for the first hyper-Kloosterman sums, which is given by \[ \mathrm{Kl}_3(m_1,m_2,m_3,c)=\sum_{\substack{x_1,x_2,x_3\,(\mathrm{mod}\,c)\\ x_1x_2x_3\equiv 1\,(\mathrm{mod}\,c)}}e\left(\frac{m_1x_1+m_2x_2+m_3x_3}{c}\right). \] Considering his result, the author resolves a long-standing problem of \textit{D. Bump}, \textit{S. Friedberg} and \textit{D. Goldfeld} [Acta Arith. 50, No. 1, 31--89 (1988; Zbl 0647.10020)], where they determined explicitly the Bruhat decomposition of Fourier coefficients of an \(\mathrm{SL}(3,\mathbb{Z})\) Poincaré series. The author also considers a related question asking whether an arbitrary `nice' function can be expanded into Bessel functions. He extends his answer of the long-element Bessel function to this question to the Bessel functions attached to the other Weyl elements. Finally, he gives Mathematica code for some of the computations in the paper.