一类Tornheim型双L函数的估计

Evaluation of a Class of Double L-values of Tornheim's Type

Tornheim型双L函数定义如下:其中χ,φ为Dirichlet特征,k,l,d∈Z且k+d>1,l+d>1,k+l+d>2.本文给出了当χ,φ为Dirichlet原特征,并且满足χ(-1)φ(-1)=(-1)^(k+l+d+1)时计算L(k,l,d;χ,φ)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.

The double L-series of Tornheim＇s type are defined as L（k,l,d;x,ψ）=∑∞m,n=1x（m）ψ（m＋n/mknl（m＋n）d for Dirichlet characters X,ψ, where k,l,d ∈ Z with k ＋ d 〉 1, 1 ＋ d 〉 1, k ＋ l ＋ d 〉 2. In thispaper, we show that the values of L（k, l, d; X, ψ） can be evaluated for any primitive Dirichletcharacter X, ψ, when X（-1）ψ（-1） = （-1）^k＋l＋d＋1. Our method also provides a way to calculatethem explicitly. This generalizes the results of [Tsumura, H., Bull. Austral. Math. Soc., 2004, 70（2）： 213-221].