Let g be a fixed Hecke cusp form for \(\mathrm SL(2,{\mathbb {Z}})\) and \(\chi \) be a primitive Dirichlet character of conductor M. The best known subconvex bound for \(L(1/2,g\otimes \chi )\) is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type bound on \(\mathrm GL(2)\). In this paper, we give a new proof of the Burgess-type bounds \({L(1/2,g\otimes \chi )\ll _{g,\varepsilon } M^{1/2-1/8+\varepsilon }}\) and \(L(1/2,\chi )\ll _{\varepsilon } M^{1/4-1/16+\varepsilon }\) that does not require the basic tools of the previous proofs and instead uses a trivial delta method.

中文翻译：

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通过简单的增量法进行的伯吉斯边界

令g为\（\ mathrm SL（2，{\ mathbb {Z}}）\）的固定Hecke尖点形式，而\（\ chi \）为导体M的原始狄利克雷特特征。\（L（1/2，g \ otimes \ chi）\）的最著名子凸边界具有Burgess强度。通过两种方法证明了边界：移位卷积和和Petersson / Kuznetsov公式分析。很自然地问，要证明\（\ mathrm GL（2）\）上的Burgess型边界，实际上需要什么输入。在本文中，我们给出了Burgess型边界\（{L（1/2，g \ otimes \ chi）\ ll _ {g，\ varepsilon} M ^ {1 / 2-1 / 8 + \ varepsilon}} \）和\（L（1/2，\ chi）\ ll _ {\ varepsilon} M ^ {1 / 4-1 / 16 + \ varepsilon} \） 不需要以前的证明的基本工具，而是使用琐碎的delta方法。