Martinet's constant for totally real number fields

Upper bound, last-listed row

| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion. The totally real example [HMR2019, §3.3.3] is a *degree-$12$* field $\mathrm{K}$ with $\mathrm{rd}\_{\mathrm{K}} < 770.6432$, cut at a single prime above $13$ of norm $13$, giving $\mathrm{rd}\_{\mathrm{K}\_S^{[1]}} = \mathrm{rd}\_{\mathrm{K}}\cdot 13^{\frac{1}{12}(1-\frac{1}{2})} < 857.5662\dots$ — a saving of a factor $13^{1/24}$. Current record. |text edited 2026-08-23 — bound unchanged

Lower bound, last-listed row

| $215.333\dots$ | [Odl1990] | GRH-conditional Odlyzko–Serre lower bound in the totally real signature. |tracked since 2026-08-14 — no movement seen yet

Checked against material for its cited source

• reference entry citing [Mar2006] — the claim we checked: that a work can be located matching the author, title and year listed in [Mar2006]: John S. Martin, “Improved root-discriminant bounds via ramification-restricted class-field towers”, 2006. The entry’s separate attribution to “Martin [25]” in [HMR2019] is not checked here. What we read could not settle it: no work matching the entry's author, title and year was located in this search. Two Cornell mathematics department records and [HMR2019]'s reference list give a 2006 Cornell Ph.D. thesis by Jason Martin the title Building Infinite Ray-Class Towers with Specific Signatures and Small Bounded Root Discriminant, and the Mathematics Genealogy Project gives a variant of it; the year matches, the given name and title do not, and the thesis itself was not read. None of Crossref's top five matches for the entry's title is that title. source link (read in the reference list of [HMR2019], arXiv version 1 (14 Jan 2019), p. 22, from the page image, and in two Cornell mathematics department records and the Mathematics Genealogy Project record of the thesis; the thesis itself was not read). Selected: chosen because something already looked wrong.
• Another row, not one shown above: bound row $1058.565$ line 17 citing [Mar1978] — the claim we checked: that the cited 1978 paper gives the upper bound 1058.565 through Martinet's construction of infinite 2-class field towers of totally real number fields, as the row's comment says. What we read could not settle it: the paper supports the construction, an infinite 2-class field tower over the totally real field Q(√2, √(3·5·7·23·29)) (Proposition 4.1 and Example 4.2, p. 69), and gives the bound 4√70035 = 1058.565066…, printed as 1058,565…; the row's 1058.565 is that bound rounded to three decimals, but read as an exact upper bound it is about 0.0000665 below the paper's, and the paper cannot say which the row means. source link (read in the cited journal edition, Inventiones Mathematicae 44 (1978), pp. 65–73, all nine pages, from the Göttingen Digitisation Centre's page images). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. 1 reference entry here was also checked; those are not bound rows and are not counted against that figure. A row not listed here has NOT been checked.