Convolution Dirichlet Series and a Kronecker Limit Formula for Second-Order Eisenstein Series
Convolution Dirichlet Series and a Kronecker Limit Formula for Second-Order Eisenstein Series
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卷积狄利克雷级数和二阶爱森斯坦级数的克罗内克极限公式
DOI：
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发表时间：
2004
影响因子：
0.8
通讯作者：
C. O’Sullivan
中科院分区：
文献类型：
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作者：
J. Jorgenson;C. O’Sullivan
本文导出了二阶爱森斯坦级数的Kronecker极限公式的解析方面和傅里叶方面。设Γ为作用于双曲上半空间H上的第一类Fuchsian群，使得商Γ\H具有有限体积但非紧化。与Γ\H的每个顶点相关联，存在一个经典研究的一阶非全纯爱森斯坦级数E(s, z)，该级数由广义狄利克雷级数定义，该级数收敛于Re(s) > 1。爱森斯坦级数E（s, z）允许在s = 1处有一个简单极点的亚纯延拓。经典地，Kronecker的极限公式是研究常数项1 (z)在s = 1时E（s, z）的Laurent展开式。许多作者最近研究了所谓的二阶艾森斯坦级数E *(年代,z),这是由扭曲的狄利克雷级数定义系列E(由给定的时期,z)尖端形成f。在我们目前的工作,我们研究一个模拟的克罗内克极限公式的设置二级艾森斯坦级数E *(年代,z),这意味着我们确定常数项2的Laurent扩张(z) E *(年代,z)在第一次,这也是在s = 1。为了开始我们的研究，我们证明了与一阶Kronecker极限函数1相关的傅里叶系数的边界。然后，我们定义了两个用m∈n表示的卷积狄利克雷级数族，它们是由傅立叶系数1和权二尖形式f构成的。我们证明了对于所有m，并承认一个亚纯延拓，并且在s = 1时是全纯的。把我们的注意力转向二阶克罗内克极限函数2，我们首先将2表示为各种微分方程的解。然后我们得到了它的完整傅立叶展开式的尖点形式f，一阶Kronecker极限函数1的傅立叶系数，以及卷积Dirichlet级数的特殊值(1)和(1)。最后，我们证明了特殊值(1)和(1)的一个界，从而暗示了2的傅里叶系数的一个界。我们的分析导致了关于全纯投影算子的一些自然问题，我们通过检查一些数值例子来结束本文并提出了未来研究的问题。
Abstract In this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studied first-order non-holomorphic Eisenstein series E(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein series E(s, z) admits a meromorphic continuation with a simple pole at s = 1. Classically, Kronecker’s limit formula is the study of the constant term 1 (z) in the Laurent expansion of E(s, z) at s = 1. A number of authors recently have studied what is known as the second-order Eisenstein series E*(s, z), which is formed by twisting the Dirichlet series that defines the series E(s, z) by periods of a given cusp form f. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein series E* (s, z), meaning we determine the constant term 2(z) in the Laurent expansion of E*(s, z) at its first pole, which is also at s = 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function 1. We then define two families of convolution Dirichlet series, denoted by and with m ∈ ℕ, which are formed by using the Fourier coefficients of 1 and the weight two cusp form f. We prove that for all m, and admit a meromorphic continuation and are holomorphic at s = 1. Turning our attention to the second-order Kronecker limit function 2, we first express 2 as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form f, the Fourier coefficients of the first-order Kronecker limit function 1, and special values (1) and (1) of the convolution Dirichlet series. Finally, we prove a bound for the special values (1) and (1) which then implies a bound for the Fourier coefficients of 2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.