Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields

For a number field K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf{K}}}$$\end{document}, the Euler–Kronecker constant γK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{{\textbf{K}}}$$\end{document} associated to K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf{K}}}$$\end{document} is an arithmetic invariant the size and nature of which is linked to some of the deepest questions in number theory. This theme was given impetus by Ihara who obtained bounds, both unconditional as well as under GRH for Dedekind zeta functions. In this note, we study the analogous constants associated to the narrow ray class fields of an imaginary quadratic field. Our goal for studying such families is twofold. First to show that for such families, the conditional bounds obtained by Ihara can be improved on the average, again under GRH for Dedekind zeta functions. Further, our family of number fields are non-abelian while such average bounds have earlier been studied for cyclotomic fields. The technical part of our work is to make the dependence of these upper bounds on the ambient number field explicit. Such explicit dependence is essential to further our objective.