The Burgess bound via a trivial delta method

Keshav Aggarwal, Roman Holowinsky, Yongxiao Lin, Qingfeng Sun

Published 2018-03-01Version 1

Recently, Munshi established the following Burgess bounds ${L(1/2,g\otimes \chi)\ll_{g,\varepsilon} M^{1/2-1/8+\varepsilon}}$ and $L(1/2,\chi)\ll_{\varepsilon} M^{1/4-1/16+\varepsilon}$, for any given $\varepsilon>0$, where $g$ is a fixed Hecke cusp form for $\rm SL(2,\mathbb{Z})$, and $\chi$ is a primitive Dirichlet character modulo a prime $M$. The key to his proof was his novel $\rm GL(2)$ delta method. In this paper, we give a new proof of these Burgess bounds by using a trivial delta method.