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

Abstract

Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients ${\lambda }_f(n)$. Let $\chi$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $\epsilon >0$, $$\sum _{|n|\ll N}{\lambda }_f(n)\chi (n){\ll }_{f,\epsilon }{N}^{3/4}{p}^{1/6}(pN{)}^{\epsilon }.$$ We can see an improvement of the range $N>p^{3/4}$ to the range $N>p^{2/3}$ and we get a bound for $S_{f,\chi }(N)$ without going into the $L$-function.

arXiv commentFinal verson, appeared in The Ramanujan Journal (2022)

References in corpus (6)

• The Weyl bound for Dirichlet $L$-functions of cube-free conductor
• Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets
• A generalized cubic moment and the Petersson formula for newforms
• Burgess-like subconvex bounds for $G{L}_2\times G{L}_1$
• Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli
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Cited by in corpus (1)

• Bounds for short character sums for $GL(2)\times GL(3)$ twists