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arXiv · 2209.09479

Sub-Weyl strength bounds for twisted $GL(2)$ short character sums

Aritra Ghosh·Kummari Mallesham
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Abstract

Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, λ_{f}(n) \, χ(n),$$ where $λ_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $χ$ is a primitive character of conductor $p^{r}$. In this article we prove a sub-Weyl strength bounds for $S(N)$. Indeed, we obtain $$S(N) \ll \, N^{\frac{5}{9}} \ p^{\frac{13r}{45}},$$ provided that $ p^{13r/20} \leq N \leq p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\geq \left(p^{r}\right)^{\frac{2}{3}-\frac{1}{60}}$.

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Aritra Ghosh, Kummari Mallesham. 2022-09-20. Sub-Weyl strength bounds for twisted $GL(2)$ short character sums. https://arxiv.org/abs/2209.09479

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