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一类Tornheim型双$\bm L$函数的估计
Evaluation of a Class of Double $\bm L$-values of Tornheim's Type

田清1,丁丽萍1, 梅永刚2
TIAN Qing1,*,DING Liping1,MEI Yonggang2

1.西安建筑科技大学理学院, 西安, 陕西, 710055; 2. 西安邮电学院理学院, 西安, 陕西, 710121
1.School of Science, Xi'an University of Architecture and Technology, Xi'an, Shaanxi, 710055,P. R. China; 2.Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi, 710121, P. R. China

收稿日期: 2011-04-13
出版日期: 2013-10-25
DOI: 10.11845/sxjz.20130509

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摘要 Tornheim型双$L$函数定义如下: \begin{equation*} \mathcal{L}(k,l,d;\chi,\psi)=\sum^{\infty}_{m,n=1frac{\chi(m)\psi(m+n)}{m^kn^l(m+n)^d}, \end{equation*}其中$\chi$, $\psi$为Dirichlet特征, $k,l,d \in \mathbb{Z}$ 且 $k+d> 1$, $l+d> 1$, $k+l+d>2$. 本文给出了当 $\chi, \psi$为Dirichlet原特征, 并且满足$\chi(-1)\psi(-1)=(-1)^{k+l+d+1}$时计算 $\mathcal{L}(k,l,d; \chi,\psi)$精确结果的一种方法, 推广了[Tsumura, H., Bull. Austral. Math. Soc., 2004, 70(2): 213-221]的计算结果.
关键词 Tornheim型双$L$函数双$L$函数估值公式    
Abstract:The double $L$-series of Tornheim's type are defined as \begin{equation*} \mathcal{L}(k,l,d;\chi,\psi)=\sum^{\infty}_{m,n=1frac{\chi(m)\psi(m+n)}{m^kn^l(m+n)^d} \end{equation*} for Dirichlet characters $\chi, \psi$, where $k,l,d \in \mathbb{Z}$ with $k+d> 1$, $l+d> 1$, $k+l+d>2$. In this paper, we show that the values of $\mathcal{L}(k,l,d; \chi,\psi)$ can be evaluated for any primitive Dirichlet character $\chi, \psi$, when $\chi(-1)\psi(-1)=(-1)^{k+l+d+1}$. Our method also provides a way to calculate them explicitly. This generalizes the results of [Tsumura, H., Bull. Austral. Math. Soc., 2004, 70(2): 213-221].
Key wordsdouble $L$-value    evaluation formula
基金资助:Supported by the Youth Technology Fund of Xi'an University of Architecture and Technology (No. QN1138, No. QN1134, No. QN1135) and the Natural Science Foundation of the Education Department of Shaanxi Province of China (No. 2013JK1190).
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