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 $$\liminf{{\gamma\_{K}}\over\log\sqrt{| d\_{K}|}}\ge -0.26049...$$ Then we produce examples of class field towers, showing that $$\liminf{{\gamma\_{K}}\over\log\sqrt{| d\_{K}|}}\le -0.17849...$$}