Primitive roots for every admissible integer base

Follow-up works

• Manjul Bhargava, Every nonsquare integer not equal to −1 is a primitive root modulo a positive lower density of primes (hexagon:2610.00228v1, 2026-10-14). Upgrades the conclusion to a positive lower density of primes, assuming a zero-free half-plane $\Re s>1-1/N$ for all Dedekind zeta functions (supplied, per the abstract, by the Cureton–Thorner result); it uses this paper's marked replacement proposition (Lemma 10.2, Proposition 10.3), the proof of Lemma 12.1 and Sections 11–12; the provenance statement reports contributions by GPT-6 and a Lean check of the deductions only, with the cited theorems kept as explicit hypotheses.
• Manjul Bhargava, Rational points on elliptic curves that are primitive modulo a positive proportion of primes (hexagon:2610.00294v1, 2026-10-17). Proves a Lang–Trotter-type positive lower density for $P=(2,2)$ on $y^2=x^3-2x$, citing this paper (Lemma 10.2, Proposition 10.3, Section 12, at a pinned commit) for its marked Type II input; the provenance statement reports GPT-6 contributions and partial Lean formalization.
• Dean Cureton and Jesse Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions (hexagon:2610.00200v2, 2026-10-12). Claims that every Dedekind zeta function is zero-free in $\Re s>31/32$, following the strategy of this paper's zero-free theorem for finite-order Hecke $L$-functions over cyclotomic fields containing $\mu_{12}$ (cited as reference 30); the abstract states the results were formalized in Lean using AxiomProver. [AI-generated per source]
• Boyang Hu, A seven-eighths zero-free half-plane for unitary Weil L-functions (hexagon:2610.00216v1, 2026-10-13). Claims a zero-free half-plane $\Re s>7/8$ for $L$-functions of unitary representations of the Weil group; the provenance statement says the work grew out of AI-assisted investigations of this paper's Hecke zero-free arguments, and the paper cites it as reference 17. [AI-generated per source]
• OpenAI, Deterministic Polynomial Factorization over Prime Fields (OpenAI Math Release preprint, 2026-10-04). Another release manuscript, which the VibeMathed entry for it describes as depending on this paper's Theorem 1.2 (uniform Hecke zero-free strip over cyclotomic fields). [AI-generated per source]

Comments and discussion

• Persiflage (galoisrepresentations.org), A Brief Digest of Some New OpenAI Results (blog, 2026-10-08). Item 4 discusses this paper: Hooley's argument would need quasi-RH for Dedekind $L$-functions, so the paper takes a different route, which the post calls "a little bit different, but also not at all surprising"; in the comments (2026-10-09) DH points to the paper's more general quasi-RH for Hecke characters of cyclotomic fields, and Persiflage replies that Hooley's argument "needs full GRH for these".
• Almost Sure (@Almost_Sure), post on X (2026-10-14). Contrasts this paper with a linked work "claiming a positive lower density of primes", calling that claim, if true, "significantly stronger" than this result.
• Go Big or Go Home (@AmericaGetBig), post on X (2026-10-14). Notes that Hooley obtained both infinitude and the exact density on GRH in 1967 and asks: "Does the QRH route recover the full density, or just infinitude?"
• Sendtoshi Profimoto (@profimoto), post on X (2026-10-09). Part of a thread summarizing the method as building primes with $p-1=2\cdot r\cdot Q$, $Q$ a prime above $x^{0.9}$, "plus a uniform zero-free strip for Hecke L-functions of cyclotomic fields", and suggesting the engine does not need an integer base.

Formalization

• Yao Xu (cadamcat), artin-primitive-roots-lean4 (GitHub, 2026). An independent Lean 4 project that follows this paper, with Lean theorems for the controlled-predecessor construction, uniform splitting and the Hecke zero-free region, and a main theorem stating that the set of primes with primitive root $a$ is infinite; the README says it was developed with AI assistance, that it is kernel-checked using only the standard axioms, and that it depends on pinned external libraries including openai/math.
• Axiom Math, Dean Cureton and Jesse Thorner, Dedekind Zeta Zero-Free Region (Lean 4 repository, October 2026). Reports formalizations of the Dedekind zeta nonvanishing theorem for Re(s)>31/32 and its finite-order Hecke corollary, covering a stronger version of the family's analytic input rather than either primitive-root counting theorem. [AI-generated per source] [family-level]

Possibly relevant literature (found by the agent, not checked)

• M. Ram Murty and Sunil Naik, A non-abelian large sieve and Artin's primitive root conjecture (arXiv:2608.21573, 2026-08-21). Derives Artin's conjecture for a fixed base from a conjectural non-abelian large sieve inequality, an alternative conditional route to the infinitude statement in Theorem 1.1.
• Pieter Moree, Artin's primitive root conjecture – a survey (arXiv:math/0412262, v2 2012). A survey of the history and results behind Artin's conjecture, covering the conditional and unconditional results summarized in the paper's historical section (Hooley, Gupta–Murty, Heath-Brown, averages over the base).