The distribution of Euler–Kronecker constants of quadratic fields

We investigate the distribution of large positive (and negative) values of the Euler–Kronecker constant γQ(D) of the quadratic field Q(D) as D   varies over fundamental discriminants |D|≤x|D|≤x. We show that the distribution function of these values is very well approximated by that of an adequate probabilistic random model in a large uniform range. The main tools are an asymptotic formula for the Laplace transform of γQ(D) together with a careful saddle point analysis.