Zeta invariants for Dirichlet series
The author investigates the zeta-function, for \(\Re s>s_0(A)\), defined by \[ \zeta_2(s;A)=\sum_{n \in \mathbb N_0^2}(n^TAn)^{-s}, \quad \mathbb N_0=\{1,2,\ldots\}, \] where \(A\) is a real symmetric matrix of rank 2 whose associated quadratic form is positive definite on \(\mathbb N_0\). The main result is devoted to the expansion \[ \zeta_2(s;A)=\zeta_2(0;A)+\zeta_2'(0;A)s+O(| s| ^2) \] near the point \(s=0\) with explicitly given \(\zeta_2(0;A)\) and \(\zeta_2'(0;A)\). This general result is applied to zeta-functions \[ \zeta_2(s;a,b,0)=\sum_{n,k=1}^{\infty}(an^2+bk^2)^{-s}, \quad \Re s>1, \] and \[ \zeta_1(s;a,b)=\sum_{n,k=1}^{\infty}(an+bk)^{-s}, \quad \Re s>2. \] Also some particular cases are investigated, for example, \[ \zeta_2(0;a,b,0)=\frac{1}{4}, \] \[ \zeta_2'(0;a,b,0)=\frac{1}{2}\log 2 \pi - \frac{1}{4}\log a+\frac{\pi}{12}\sqrt{\frac{a}{b}}-\log \prod_{n=1}^{\infty}\big(1-e^{-2\pi \sqrt{a/b}n}\big). \] Moreover, it is observed that \[ \zeta_2(0;0,b,c)=\zeta_1(0;b,2c) \] and \[ \zeta_2'(0;0,b,c)=\zeta_1'(0;b,2c). \] The paper continues the investigations of Zagier, Shintani, Cassou-Nogués, of the author and others.
scientific article; zbMATH DE number 5125703