OpenAI claims Artin’s primitive root conjecture (infinitude)

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Key facts

• Theorem 1.1 (verbatim): ‘For every admissible integer a, there are constants ca > 0 and xa ≥ 2 such that, for every real x ≥ xa, #{p prime : x < p < 2x, p ∤ a, ordp(a) = p − 1} ≥ ca x/(log x)².’ The paper: ‘The predicted counting scale is x/ log x … In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base 2.’ 92 pages, dated 4 Oct 2026; companion ‘Simultaneous primitive roots: a conditional lower bound for prime bases’ (21 pp.)
• Prior state (paper, §1.1): Hooley proved Artin’s predicted asymptotic assuming the Riemann hypothesis for the relevant Dedekind zeta functions (1967); unconditionally, Gupta and Murty gave a set of thirteen integers one of which works, and Heath-Brown ‘proved that at most two prime bases fail the infinitude assertion, so at least one of 2, 3, 5 satisfies it’ (verbatim), without identifying which
• Method: a zero-free half-plane for Hecke L-functions over cyclotomic fields from a cubic theta input, a quadratic large sieve, and ‘marked Type II estimates’ to count primes with p − 1 equal to a small cofactor times one large prime. A blog commenter (‘DH’, 9 Oct) noted that the write-up contains a zero-free region Re s > 999999/1000000 for finite-order Hecke characters of cyclotomic fields containing ζ12. No Lean scope note
• Frank Calegari (Persiflage, 8 Oct, verbatim): ‘the argument is a little bit different, but also not at all surprising’; ‘this is an interesting type of argument which I don’t find particularly surprising. But no one was going to write this paper, which depends on the quasi-Riemann hypothesis for Dirichlet L-functions, without knowing anything about general Dedekind L-functions.’
• Follow-ups: Dean Cureton and Jesse Thorner (Hexagon, 12 Oct) prove a quasi-Riemann hypothesis for every Dedekind zeta function (Re s > 31/32), building on the zero-free argument of this family (see Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)). Manjul Bhargava (Hexagon 2610.00228, 14 Oct, mixed human and AI text): ‘a positive lower density of primes have a as a primitive root’, using Cureton–Thorner’s half-plane and ‘the marked Type II estimates of OpenAI’; ‘The argument gives a lower bound rather than the exact Artin density.’ His provenance note mentions handwritten notes, ‘a conversation with Peter Sarnak’, GPT-6 and a Lean check of the deductions only
• Scientific American (8 Oct) listed Artin’s conjecture among the release’s other important number-theory results, without expert comment