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

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
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