On exponential sums involving the ideal counting function in quadratic number fields (Q1849691)

Let \(K\) be a quadratic number field with discriminant \(D\) and \(\chi\) an arbitrary primitive Dirichlet character modulo \(k>1\). Further let \(F_\chi(n)= \sum_{d|n}\chi(d)\). The author considers exponential sums involving the arithmetical function \(F_\chi(n)\), namely \[ R\bigl(x; \tfrac hq\bigr)= \sum_{n\leq x} F_\chi(n) e^{2\pi ihn/q}, \] where \(h\), \(q\) are co-prime integers with \(q\geq 1\). The remainder term in an asymptotic expansion of this function can be described as \[ P \bigl(x; \tfrac hq\bigr)= R\bigl(x; \tfrac hq\bigr)- x \;\underset {s=1} {\text{Res}} F_\chi \bigl(s; \tfrac hq\bigr)- F_\chi\bigl(0; \tfrac hq\bigr). \] Here \(F_\chi (s; \frac hq)\) means the corresponding generating function \[ F_\chi \bigl(s; \tfrac hq \bigr)= \sum_{n=1}^\infty F_\chi(n) e^{2\pi ihn/q} n^{-s}. \] Now the asymptotic behaviour of \(P(x; \frac hp)\) is studied. A nontrivial upper bound and mean square formulas are given. Furthermore, a sum formula of Voronoi type is developed, that is a representation of \[ \sum_{a< n\leq b} F_\chi(n) e^{2\pi ihn/q} f(n) \] by a convergent infinite series. This generalizes the main result of Chapter 1 in the book of \textit{M. Jutila} [Lecture on a method in the theory of exponential sums, Tata Institute of Fundamental Research, Springer (1987; Zbl 0671.10031)].