On analytic properties of a multiple \(L\)-function (Q2749820)

The author introduces the multiple \(L\)-function NEWLINE\[NEWLINE L_j(s) = \sum_{0<n_1<\cdots<n_j}\chi_1(n_1)\chi_2(n_2)\ldots \chi_j(n_j)(n_1n_2\cdots n_j)^{-s}\quad(\Re s >1),\leqno(1) NEWLINE\]NEWLINE where the \(\chi_j\) are Dirichlet characters of the same conductor \(q\). The author brings forth several results concerning the analytic continuation of \(L_j(s)\) and its order. For example, it is shown that \(L_j(s)\), defined initially by (1), admits a meromorphic continuation to the whole complex plane, its only possible poles lying on the real axis left of 1. In the domain \(0 <\sigma_0 \leq \sigma\) and \(|t|\geq 1\) one has \(L_j(s) = O(|t|^j)\), where the \(O\)-constant depends on \(\sigma_0\) and \(q\). The main tool in the proof is the Euler-Maclaurin summation formula. A comparison with other multiple zeta and \(L\)-functions is given.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00016].