Asymptotic behaviour of the Euler-Kronecker constant

Abstract

This appendix to the beautiful paper of Ihara puts it in the context of infinite global fields of our papers. We study the behaviour of Euler--Kronecker constant $\gamma \_K$ when the discriminant (respectively, the genus) tends to infinity. Results of our paper easily give us good lower bounds on the ratio $\gamma \_K/\log \sqrt{|d\_K|}$. In particular, for number fields, under the generalized Riemann hypothesis we prove $$\mathrm{liminf}\frac{\gamma \_K}{\log \sqrt{|d\_K|}}\ge -\mathrm{0.26049...}$$ Then we produce examples of class field towers, showing that $$\mathrm{liminf}\frac{\gamma \_K}{\log \sqrt{|d\_K|}}\le -\mathrm{0.17849...}$$}