Non-Linear Additive Twists of ${\mathrm{G}\mathrm{L}}_3$ Hecke Eigenvalues

We bound non-linear additive twists of ${\mathrm{G}\mathrm{L}}_3$ Hecke eigenvalues, improving upon the work of Kumar-Mallesham-Singh (2022). The proof employs the DFI circle method with standard manipulations (Voronoi, Cauchy-Schwarz, lengthening, and additive reciprocity). The main novelty includes the conductor lowering mechanism, albeit sacrificing some savings to remove an analytic oscillation, followed by the iteration ad infinitum of Cauchy-Schwarz and Poisson. The resulting character sums are estimated via the work of Adolphson-Sperber (1993). As an application, we prove nontrivial bounds for the first moment of ${\mathrm{G}\mathrm{L}}_3$ Hardy’s function, which corresponds to the cubic moment of Hardy’s function studied by Ivić (2012).