The analytic continuation and the order estimate of multiple Dirichlet series.
The present paper deals with multiple Dirichlet series. For \(1\leq k\leq r\), let \[ \varphi_k(s)=\sum_{n=1}^\infty{a_k(n)\over n^s} \] be a Dirichlet series with complex coefficients, being absolutely convergent for \(\sigma:=\text{ Re}\,s>\alpha_k>0\), having a meromorphic continuation to the whole complex plane except for at most a simple pole at \(s=\alpha_k\), and satisfying the order estimate \(\varphi_k(\sigma+it)=O(| t|^A)\) in any fixed strip \(\sigma_1\leq \sigma\leq \sigma_2\), where \(A\) is a non-negative constant, depending only on \(\sigma_1,\sigma_2\). The authors introduce the multiple Dirichlet series \[ \Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r)) \] \[ =\sum_{m_1=1}^\infty\sum_{m_2=1}^\infty\ldots\sum_{m_r=1}^\infty{a_1(m_1)\over m_1^s}{a_2(m_2)\over (m_1+m_2)^{s_2}}\cdot\ldots\cdot{a_r(m_r)\over (m_1+\ldots+m_r)^{s_r}} \] associated with \(\varphi_1,\ldots,\varphi_r\), where \(s_k=\sigma_k+it_k\). It is proved that \(\Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r))\) can be continued meromorphically and its possible singularities are located on subsets defined by one of the equations \[ s_j+\ldots +s_r=\alpha_j+\delta_{j+1}\alpha_{j+1}+\ldots +\delta_r\alpha_r-n, \] where \(1\leq j\leq r\), \(n\) is a non-negative integer, and \(\delta_k\) is equal to zero or one; if \(j=r\), then the equation above is to be read as \(s_r=\alpha_r-n\). The authors give some restrictions on the possible values for \(n\), depending on \(j\) and \(r\). It is also proved that \(\Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r))\) is an entire function if all \(\varphi_k(s)\) are entire. Finally, the authors prove an order estimate for \(\Phi_2((s_1,s_2);(\varphi_1,\varphi_2))\) in the region \(\sigma_1+\sigma_2>\alpha_1+\eta,\alpha_2+\eta\) for any positive \(\eta\) (except for points near the set of singularities). The method of proof relies on the classical Mellin-Barnes formula and the techniques in in [Analytic and probabilistic methods in number theory. Proceedings of the third international conference in honour of J. Kubilius, Palanga, Lithuania, 2001. Vilnius: TEV, 188--194 (2002; Zbl 1195.11119)].
scientific article; zbMATH DE number 2058869