Sums with convolution of Dirichlet characters (Q1946760)

Let \(\chi_1\) and \(\chi_2\) be primitive Dirichlet characters with conductors \(q_1\) and \(q_2\), respectively. Recently, \textit{W. D. Banks} and \textit{I. E. Shparlinski} [Manuscr. Math. 133, No. 1--2, 105--114 (2010; Zbl 1221.11172)] considered the sum \[ S_{\chi_1,\chi_2}(X):=\sum_{ab\leq X}\chi_1(a)\chi_2(b) \] and established upper bounds. In this paper the author proves more precise bounds on \(S_{\chi_1,\chi_2}\). Theorem 1. Let \(\chi_1\), \(\chi_2\), \(q_1\) and \(q_2\) as above. If \(q_1\leq q_2\) and \(T>1\) then for every \(\varepsilon >0\) one has \[ \sum_{0<xy\leq T}\chi_1(x)\chi_2(y)\ll\begin{cases} T^{2/3}(q_1q_2)^{1/9+\varepsilon} &\text{if}\;(q_1q_2)^{1/3}\leq T\leq q_1^{4/3}q_2^{1/3},\\ T^{3/4}q_2^{1/12+\varepsilon} &\text{if}\;q_1^{4/3}q_2^{1/3}\leq T, \end{cases}\tag{1} \] \[ \sum_{0<xy\leq T}\chi_1(x)\chi_2(y)\ll\begin{cases} T^{1/2}(q_1q_2)^{3/16+\varepsilon}\quad &\text{if}\;(q_1q_2)^{3/8}\leq T\leq q_1^{9/8}q_2^{3/8},\\ T^{2/3}q_2^{1/8+\varepsilon} &\text{if}\;q_1^{9/8}q_2^{3/8}\leq T, \end{cases}\tag{2} \] where the constant implied by \(\ll\) depends only on \(\varepsilon\).