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Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series. (English) Zbl 1060.11053

The double zeta-function of Barnes is defined by the meromorphic continuation of the series \[ \zeta_2(v; \beta, \omega)=\sum_{m=0}^\infty\sum_{n=0}^\infty (\beta+m+nw)^{-v}, \] where \(\beta>0\) and \(w\) is a non-zero complex number with \(| \arg w| <\pi\). The author obtains an asymptotic expansions of the above zeta-function with respect to \(\omega\), when \(| \omega| \to+\infty\) and \(| w| \to0\). For example, in the latter case it is proved that \[ \zeta_2(v; \beta, w)=\zeta(v, \beta)+\frac{\zeta(v-1, \beta)}{v-1}w^{-1}+\sum_{k=0}^{N-1}\binom{-v}{k} \zeta(v+k, \beta)\zeta(-k)w^k +O(| w| ^N) \] in the region \(\operatorname{Re} v>-N+1\) and \(0<| w| \leq1\), \(| \arg w| \leq\theta<\pi\), where the implied constant depends only on \(N\), \(v\), \(\beta\) and \(\theta\). Here \(\zeta(v)\) and \(\zeta(v, \beta)\) are the Riemann and the Hurwitz zeta-functions, respectively. As corollaries, asymptotic expansions of holomorphic Eisenstein series and double gamma-function are obtained.
Further he gives similar asymptotic expansions (with respect to \(w_1\), see below) for the following version of Shintani double zeta-functions: \[ \zeta_{SH,2}((u,v); A, W)=\sum_{m=0}^\infty\sum_{n=0}^\infty (a+m+(b+n)w_1)^{-u} (a+m+(b+n)w_2)^{-v}. \]

MSC:

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
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