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arXiv · 2311.13788

Non-Linear Additive Twists of $\mathrm{GL}_{3}$ Hecke Eigenvalues

Ikuya Kaneko·Wing Hong Leung
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Abstract

We bound non-linear additive twists of $\mathrm{GL}_{3}$ 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 $\mathrm{GL}_{3}$ Hardy's function, which corresponds to the cubic moment of Hardy's function studied by Ivi\'{c} (2012).

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Ikuya Kaneko, Wing Hong Leung. 2023-11-23. Non-Linear Additive Twists of $\mathrm{GL}_{3}$ Hecke Eigenvalues. https://arxiv.org/abs/2311.13788

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