2026/08/03 by Huimin Zheng
Mathematics · #math.NT #msc:11R06 #msc:11F03 #msc:11M41
12 pages. Status report; companion paper proves the two n2-family conjectures (submitted simultaneously). Code: https://doi.org/10.5281/zenodo.21711884
arxiv created 2026/08/03 · arxiv updated 2026/08/04
This note archives the status of Samart's Table-6 conjecture n4(81)=40M7, M7:=L'(g7,0), where g7(τ)=η(τ)3η(7τ)3 is the newform of S3(Γ0(7),χ-7) and n4(s):=4m(x4+y4+z4+1+s1/4xyz). The conjecture is the cleanest of Samart's open interior-point entries (discriminant -7, class number 1, a single L-value), and it was dropped as a theorem target in the companion paper, where the two n2-family conjectures were proved. We record what is proved and precisely where those methods fail. First, a complete proof of the L-value side (P1): at the CM point τ2=(7+√(-7))/4 the Eisenstein--Kronecker expression underlying Samart's formula evaluates exactly to EK4(τ2)=40M7, via lattice sums over the ring of integers of ℚ(√(-7)) and the principal ideal (\varpi), with an exact cancellation of the parasitic ζK(2)-terms. Second, two quantitative obstructions to the remaining half n4(81)=EK4(τ2): the critical image of the n4-family is a two-dimensional astroid disc containing the parameter c=3 in its interior (in contrast to the one-dimensional slit [0,64] of the n2-family), so no continuation path can approach the CM point; and Samart's U-series converges on all of the upper half-plane but leaves the geometric sheet of the holomorphic Mahler measure everywhere below Im τ=1/√(2), so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: n4(81)-40M7=+0.0586706795972872..., five orders of magnitude above the integration error floor, so the identity as literally stated is refuted; a closed form for the true value n4(81) remains open and appears to require regulator/monodromy machinery.