An approximated functional equation for \(L(s,\chi_ 1)L(s,\chi_ 2)\) (Q1363274)

This note announces two results involving primitive characters \(\chi_1,\chi_2\) modulo \(k_1\) and \(k_2\), respectively. Let \(s=\sigma+it, xy=(t/2\pi)^2k_1k_2, \rho(s,\chi_j)=L(s,\chi_j)/L(1-s,\bar{\chi}_j) (j = 1,2)\) and let \[ \begin{multlined} R(s,x,\chi_1,\chi_2) := L(s,\chi_1)L(s,\chi_2)\\ -\sum_{mn\leq x}\chi_1(m)\chi_2(n)(mn)^{-s}- \rho(s,\chi_1)\rho(s,\chi_2) \sum_{mn\leq y}\chi_1(m)\chi_2(n)(mn)^{s-1}\end{multlined} \] denote the error term in the approximate functional equation for \(L(s,\chi_1)L(s,\chi_2)\). The main result is as follows. If \(0\leq\sigma\leq 1\), \(t>t_0>0\), \(A,B\) are two coprime natural numbers, \(A\leq t^{\varepsilon}\), \(B\leq t^{\varepsilon}\) and \(k_1k_2\leq t^{1/9-2\varepsilon}\), then \[ R\left(s,{tk_1\over2\pi}{A\over B},\chi_1,\chi_2\right)\ll_\varepsilon t^{1/3-\sigma+2\varepsilon}\left((k_1k_2)^{2-\sigma}+(k_1k_2)^{1+\sigma}\right). \]