Sums with convolutions of Dirichlet characters
Let \(\chi_1\) and \(\chi_2\) be primitive Dirichlet characters with conductors \(q_1\) and \(q_2\), respectively, and let \[ S_{\chi_1,\chi_2}(X):=\sum_{ab\leq X}\chi_1(a)\chi_2(b). \] The authors prove that if \(X\geq q_2^{\frac 23}\geq q_1^{\frac 23}\) and \(\log X=q_2^{o(1)}\), then \[ \left| S_{\chi_1,\chi_2}(X)\right|\leq X^{\frac {13}{18}}q_1^{\frac{2}{27}}q_2^{\frac 19+o(1)}. \] If \(X\geq q_2^{\frac 34}\geq q_1^{\frac 34} \) and \(\log X=q_2^{o(1)}\) then \[ \left| S_{\chi_1,\chi_2}(X)\right|\leq X^{\frac {5}{8}}q_1^{\frac{3}{32}}q_2^{\frac {3}{16}+o(1)}. \]
scientific article; zbMATH DE number 5782781