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on Martinet's constant for totally real number fields ($C_{87}$)

The statement

For a number field $K$ , let $\Delta_K$ denote the absolute value of its discriminant and let $[K:\mathbb{Q}]$ denote its degree. The root discriminant of $K$ is

\mathrm{rd}(K) := \Delta_K^{1/[K:\mathbb{Q}]}.

Martinet's constant for totally real number fields (also called the asymptotic root-discriminant infimum in the totally real signature) is

C_{87} := \liminf_{K}\ \mathrm{rd}(K),

where the $\liminf$ is over all totally real number fields $K$ ordered by degree. Equivalently, $C_{87}$ is the smallest constant $\alpha$ such that there exist infinite towers $K_1 \subset K_2 \subset \dots$ of totally real number fields with $\mathrm{rd}(K_i) \to \alpha$ from above (or $\le \alpha + o(1)$ ).

Upper bounds on $C_{87}$ come from explicit constructions of infinite tamely ramified towers of totally real number fields with controlled root discriminant, via the Golod–Shafarevich criterion ([GS1964]) applied to restricted-ramification pro- $p$ Galois groups; lower bounds come from the Odlyzko–Serre analysis of Weil's explicit formula, either unconditional or under GRH.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.