A multiple character sum and a multiple \(L\)-function
The main result of the paper is an asymptotic formula for the Riesz mean \[ {1\over k!}\sum_{n\leq x}(x-n)^kh_j(n),\leqno(1) \] where \[ h_j(n) = \sum_{m_1m_2\cdots m_{j-1}| n, m_1<\cdots <m_{j-1}<n/(m_1\cdots m_{j-1})} \chi_1(m_1)\cdots\chi_{j-1}(m_{j-1}) \chi_j\bigl({n\over m_1\cdots m_{j-1}}\bigr), \] with \(\chi_i\;(i = 1,\cdots, j)\) Dirichlet characters to the same modulus \(q\). The sum is connected to the multiple \(L\)-function \[ L_j(s) = \sum_{0<n_1<\cdots<n_j} {\chi_1(n_1)\cdots \chi_j(n_j)\over (n_1\cdots n_j)^s} = \sum_{n=1}^\infty {h_j(n)\over n^s}\quad(\Re s > 1). \] The asymptotics for (1) (the reader is referred to the paper for details) are derived by the Perron inversion formula in the form suitable for the evaluation of Riesz means. The main term comes from the poles of the analytic continuation of \(L_j(s)\). However, the lack of the functional equation for \(L_j(s)\) makes its estimation difficult. The paper ends with some explicit examples of the asymptotic formulas for (1) when \(j = 3\) and \(j = 4\).
scientific article; zbMATH DE number 1889274