Taylor Dupuy

math.NT — Number Theory

Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents

Contributed by Taylor Dupuy

This report was created on 23 October 2026 from six AI research reports generated with the frontier-dump skill on Dinesh Thakur's conjectures in function field arithmetic. Taylor Dupuy initiated those queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The six reports were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We survey the current status of several conjectures of Dinesh Thakur and his coauthors on the arithmetic of $\mathbb{F}_q[t]$: the degree hypotheses for power sums, the characterisation of zeta-like multizeta values in genus zero and in higher genus, the relations among multizeta values, the distribution of zeros of Goss zeta functions, and the automata-theoretic transcendence questions for special values. For each conjecture we give the precise statement and its status in the literature. The six computational research dumps report several results and refutations. Among them, the recursion predicted by Thakur's second power-sum hypothesis fails for an explicit family of exponents at thirteen non-prime values of $q$ (a computer-assisted result); the shuffle relations, even with Frobenius and Carlitz's Eulerian evaluations, do not generate all relations among multizeta values; a closed formula for a $v$-adic valuation difference conjectured by Thakur holds only after a sign change (and, for $d\ge4$, halving), and the corrected formula is proved, with computer assistance, for degrees $d\le 11$; and the $v$-adic Newton polygon of a Goss zeta function over $\mathbb{F}_2[t]$ at a prime of degree two has a slope of multiplicity one, which appears to contradict a result claimed in a 2023 preprint (this conflict needs human verification). The shuffle result and an assembly of published theorems describing all relations are elementary, and we claim no novelty for them. We also write out a proof of a theorem of Wen and Yao on $v$-adic arithmetic gamma values. Computational evidence is labelled as such throughout, and the open problems are listed. This manuscript is AI-generated by Claude Code (model claude-opus-5-5) from AI-generated research dumps; its proofs received only AI review, and no human has checked them.