The distribution of Euler-Kronecker constants of quadratic fields

TL;DR

This paper studies how the Euler-Kronecker constants of quadratic fields are distributed for various discriminants, showing they follow a probabilistic model through advanced asymptotic analysis.

Contribution

It introduces a probabilistic model for the distribution of Euler-Kronecker constants in quadratic fields and validates it with asymptotic and saddle point analysis.

Findings

Distribution closely matches the probabilistic model

Asymptotic formula for the Laplace transform derived

Effective approximation over large ranges

Abstract

We investigate the distribution of large positive (and negative) values of the Euler-Kronecker constant $\gamma_{\mathbb{Q}(\sqrt D)}$ of the quadratic field $\mathbb{Q}(\sqrt{D})$ as $D$ varies over fundamental discriminants $|D|\leq x$. 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 $\gamma_{\mathbb{Q}(\sqrt D)}$ together with a careful saddle point analysis.