Character sums in algebraic number fields. (English) Zbl 0795.11033

Rausch, Ulrich

J. Number Theory 46, No. 2, 179-195 (1994).

Let \(K\) be an algebraic number field of degree \([K:\mathbb{Q}]= r_ 1+ 2r_ 2\). Let \(e_ p=1\) for \(p=1,\dots,r_ 1\), \(e_ p=2\) for \(p= r_ 1+1,\dots, r+1= r_ 1+r_ 2\). Put \(X= \prod_{p=1}^{r+1} x_ p^{e_ p}\), where \(x=(x_ 1,\dots, x_{r+1})\) is a vector of positive real numbers. Let \({\mathfrak q}\neq (0)\) be an integral ideal of \(K\) with norm \(N({\mathfrak q})\). The author considers sums \(F(x)= \sum f(\nu)\), where \(f\) denotes a complex-valued arithmetic function, defined on the integers of \(K\), with the property \(f(\mu)= f(\nu)\) for \(\mu\equiv \nu \pmod {\mathfrak q}\). The sum is extended over integers \(\nu\in K\) such that \(|\nu^{(p)}+ z_ p|\leq x_ p\), where \(\nu^{(p)}\) are the conjugates of \(\nu\) and \(z= (z_ 1,\dots, z_{r+1})\in \mathbb{R}^{r_ 1}\times \mathbb{C}^{r_ 2}\).
The author obtains asymptotic representations of \(F(x)\) of type \(F(x)= CX+ R(x)\), where \(C\) is a well-defined constant and \(R(x)\) satisfies inequalities of the form \[ R(x)\;\ll\;M(f,q) X^{r_ 2/(r_ 2+2)} (\log(XN({\mathfrak q}))^{2r_ 1/ (r_ 1+2)}. \]

Reviewer: E.Krätzel (Jena)

MSC:

11L40	Estimates on character sums
11R47	Other analytic theory

Keywords:

character sums; algebraic number fields; arithmetic function