Bound for the sum involving the Jacobi symbol in \(\mathbb Z[i]\) (Q1038620)

In this paper a nontrivial estimate of a certain sum involving the Jacobi symbol in \(\mathbb{Z}[i]\) is given which is a generalization of a character sum estimate of \textit{D. R. Heath-Brown} [Acta Arith. 72, 235--275 (1995; Zbl 0828.11040)]. The main result is stated as follows: Theorem 1. Let \(M,N\) be positive integers and \(\{a_n\}\) be an arbitrary complex sequence. Then we have \[ \sideset\and{^*}\to\sum_{N(m)\leq M}\left|\;\sideset\and{^*}\to\sum_{N(m)\leq N} a_n\left[\frac nm\right]\right|^2\ll_\varepsilon (MN)^\varepsilon (M+N)\sideset\and{^*}\to\sum_{N(m)\leq N} | a_n|^2 \] for any \(\varepsilon >0\). \(\sideset\and{^*}\to\sum\) denotes restriction of of the range of summation to odd squarefree integers, \([\cdot]\) denotes the Jacobi symbol in \(\mathbb Z[i]\), and \(N(n)\) the norm of \(n\) in \(\mathbb Z[i]\). The proof is along the lines of Heath-Brown, however several difficulties occurring here have to be resolved.