The Hooley-Huxley contour method for problems in number fields. I: Arithmetic functions
Let \(K\) be a number field of degree \(n\). For an integral ideal \({\mathfrak q}\) let \(\chi\pmod{\mathfrak q}\) denote a generic narrow ideal class character \(\text{mod }{\mathfrak q}\) and \(\zeta_K(s,\chi)\) the corresponding \(L\)-function. The author studies distribution of values of coefficients of Dirichlet series \[ F(s)= \sum_{{\mathfrak a}} \frac{a({\mathfrak a})} {N({\mathfrak a})^s}, \quad \Re s>1, \] which can be represented as a product of the form \[ F(s)= F_0(s) \prod_{{\mathfrak q}} \prod_{\chi\pmod{\mathfrak q}} \zeta_K(s,\chi)^{z_\chi} (\zeta^{(n_\chi)} (s,\chi))^{m_\chi} (\log \zeta_K(s,\chi))^{r_\chi}, \] where \[ F_0(s)= \sum_{{\mathfrak a}} \frac{b({\mathfrak a})} {N({\mathfrak a})^s} \] is uniformly convergent and bounded in \(\Re s>\frac{1}{2}+ \delta\) for any \(\delta> 0\). The exponents \(z_\chi\) are complex numbers, \(n_\chi\in \mathbb{N}\), and \(m_\chi\), \(r_\chi\) are nonnegative integers that, for all but a finite number of \({\mathfrak q}\) and \(\chi\), satisfy \(z_\chi= 0\), and \(m_\chi= r_\chi= 0\). The precise formulation of the author's main result is too involved to be reproduced here in details. Generally speaking he studies sums of the form \[ A(x,\psi_0,l)= \sum_{{\mathfrak a}\in {\mathcal S} (x,\psi_0,l)} a({\mathfrak a}), \] where \({\mathcal S}(x,\psi_0,l)\) denotes the set of integral ideals with norms \(x(1-l)\leq N({\mathfrak a})\leq x(1+l)\), \(0\leq l< \frac{1}{2}\), and satisfying an additional geometric condition that the arguments of basic torsion-free Hecke characters at \(a\) are close to \(\psi_0\in (\mathbb{R}/ \mathbb{Z})^{n-1}\). Then he gives asymptotic formula for \(A(x,\psi_0,l)\) which is valid for \(l\geq x^{-5/12n+ \varepsilon}\) and fixed \(\psi_0\). Moreover, he estimates the mean square average of the corresponding remainder term. A number of interesting applications is formulated. Special attention is given to sums over Gaussian integers.
scientific article; zbMATH DE number 1271091