Estimates for character sums in number fields
Let \(\rho_ 1,...,\rho_ r\) be (continuous) complex finite-dimensional representations of the Weil group of an algebraic number field K of finite degree over the rationals. Let \[ L(s,\rho_ j)=\sum_{{\mathfrak a}}c({\mathfrak a},\rho_ j)(N{\mathfrak a})^{-s},\quad 1\leq j\leq r, \] be the Artin-Weil L-function associated to \(\rho_ j\). Here \({\mathfrak a}\) ranges over the integral ideals of K. The purpose of the present paper is to derive asymptotic estimates for the expression \[ A(x;\rho_ 1,...,\rho_ r)=\sum_{N{\mathfrak a}\leq x}\prod^{r}_{j=1}c({\mathfrak a},\rho_ j) \] as \(x\to \infty\) with effective numerical constants in the error terms. Let us cite one of the main results: There is a polynomial \(P_{\rho}(t)\) and there are numbers \(a_ j<1, b_ j, d_ j\) for \(j=1,...,r\), depending on the location of exceptional Siegel zeros, such that \[ A(x;\rho_ 1,...,\rho_ r)=x\cdot P_{\rho}(\log x)+\sum^{r}_{j=1}d_ j\cdot x^{a_ j}\cdot (\log x)^{b_ j}+R(x;\rho_ 1,...,\rho_ r), \] where \[ R(x;\rho_ 1,...,\rho_ r)\ll x\cdot \exp (-c_ 0(\rho,K)\cdot \sqrt{\log x}). \] Making assumptions about \(\rho_ j\) or \(L(s,\rho_ j)\) it is possible to improve the error term. The proofs of the results follow the pattern of classical analytic number theory.
complex finite-dimensional repesentations of Weil group of algebraic number field
effective numerical constants in the error terms
scientific article; zbMATH DE number 4027615
scientific article