Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L -functions. In this paper we prove that central values of additive twists of the L -function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k≥4{k\geq 4}) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L -functions consisting of central values of the form L⁢(f⊗χ,1/2){L(f\otimes\chi,1/2)} with χ a Dirichlet character.

中文翻译：

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尖瓣 L 函数加性扭曲的中心值

加性扭曲是与全纯尖端形式相关的重要不变量；它们对 Eichler-Shimura 同构进行编码，并包含有关自守 L 函数的信息。在本文中，我们证明了与偶数权重 k 的全纯尖峰形式 f 相关联的 L 函数的加性扭曲的中心值是渐近正态分布的。这概括了（至 k≥4{k\geq 4}）Petridis 和 Risager 最近关于模符号算术分布的突破。此外，我们给出了自守 L 函数的某些“宽”族的平均值的渐近公式，该函数由形式 L⁢(f⊗χ,1/2){L(f\otimes\chi ,1/2)} 与 χ 是狄利克雷字符。