Asymptotic behaviour of the Euler-Kronecker constant

Details Asymptotic behaviour of the Euler-Kronecker constant Asymptotic behaviour of the Euler-Kronecker constant

2005-03-17

arxiv.org/abs/math/0503347v2

Mathematics

Number Theory

Scientific paper

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 $$\liminf{{\gamma\_{K}}\over\log\sqrt{| d\_{K}|}}\le -0.17849...$$}

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