Asymptotic behaviour of the Euler-Kronecker constant

Summary

This appendix to the beautiful paper [1] of Ihara puts it in the context of infinite global fields of our papers [2] and [3]. We study the behaviour of Euler-Kronecker constant γK when the discriminant (genus in the function field case) tends to infinity. Results of [2] easily give us good lower bounds on the ratio ${{\gamma _K } \mathord{\left/ {\vphantom {{\gamma _K } {\log \sqrt {\left| {d_K } \right|} }}} \right. \kern-\nulldelimiterspace} {\log \sqrt {\left| {d_K } \right|} }}$. In particular, for number fields, under the generalized Riemann hypothesis we prove

$$ \lim \inf \frac{{\gamma _K }} {{\log \sqrt {\left| {d_K } \right|} }} \geqslant - 0.26049.... $$

Then we produce examples of class-field towers, showing that

$$ \lim \inf \frac{{\gamma _K }} {{\log \sqrt {\left| {d_K } \right|} }} \leqslant - 0.17849.... $$

To Volodya Drinfeld with friendship and admiration.