Double Lerch series and their functional relations
The author deals with the double Lerch series defined for \(s_1,s_2,s_3\in\mathbb{C}\), \(\text{Re}(s_1+s_2)>1\), \(\text{Re}(s_2+s_3)>1\) and \(\text{Re}(s_1+s_2+s_3)>2\) by \[ T(s_1,s_2,s_3;\alpha,\beta,\gamma):=\sum_{m,n=1}^{\infty}\frac{e^{2\pi i\alpha m}e^{2\pi i\beta n}e^{2\pi i\gamma (m+n)}}{m^{s_1}n^{s_2}(m+n)^{s_3}}, \] where \(0<\alpha,\beta,\gamma \leq 1\). Obviously, for \(\alpha=\beta=\gamma=1\), it is the well-known Tornheim double zeta function investigated by Tornheim, Mordell, Matsumoto, Tsumura and many others. Firstly, by the Mellin-Barnes method, he proved that the function \(T(s_1,s_2,s_3;\alpha,\beta,\gamma)\) can be meromorphically continued to \(\mathbb{C}^3\) with possible singularities located in the subset defined by certain equations. Next, some functional relations are given. In Proposition 3.1 he showed that the double Lerch series can be expressed as the integral of product of functions \[ \text{Li}(s;\alpha) = \sum_{n=1}^\infty \frac{e^{2\pi i \alpha n}}{n^s}. \] Using Proposition 3.1 he showed a range of functional relations for the double Lerch series in special cases, including some functional relations for alternating analogues of Tornheim double series. Moreover, by the same method, a similar functional relation is proved for so-called double Dirichlet series defined by \[ L(s_1,s_2,s_3;\chi,\varphi):=\sum_{m,n=1}^{\infty}\frac{\chi(m)\varphi(m+n)}{m^{s_1}n^{s_2}(m+n)^{s_3}}, \] where \(\chi\) and \(\varphi\) are Dirichlet characters. The author concludes the paper by computing some double L-values like \(T(0,a,b;1/2,1,1)\), when \(a+b\) is odd.
Tornheim double series, double L-values.
scientific article; zbMATH DE number 5303964