Extreme Euler–Kronecker constants of quadratic fields

Determine whether the extreme values of the Euler–Kronecker constants of quadratic fields, as the fundamental discriminant ranges over values satisfying |D|\le x, are asymptotic to \log\log x+\log\log\log x+O(1) for both signs.

Background

For a quadratic field K=\mathbb{Q}(\sqrt{D}) with fundamental discriminant D, the Euler–Kronecker constant is expressed in terms of the logarithmic derivative of the associated quadratic Dirichlet L-function. The paper recalls unconditional and conditional lower bounds showing that, for infinitely many discriminants, both positive and negative extreme values reach at least the order \log\log |D|, with an additional \log\log\log |D| term under GRH.

The recalled conjecture concerns the optimal order of the maximum of \pm\gamma_{\mathbb{Q}(\sqrt{D})} over fundamental discriminants with |D|\le x. The paper proves conditional lower bounds of the conjectured order under GRH, but does not establish the corresponding upper bounds or the full asymptotic characterization.