Browse › Number theory › 029 Primitive roots for every admissible integer base
Simultaneous primitive roots: a conditional lower bound for prime bases
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Abstract
For every fixed finite set of distinct positive primes, we prove that at least $cx/(\log x)^2$ primes in $(x,2x)$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Dean Cureton and Jesse Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions (hexagon:2610.00200v2, 2026-10-12). Extends the strategy of the primitive-root paper's analytic core to arbitrary number fields and claims nonvanishing for Re(s)>31/32, strengthening the zero-free input used by both papers in this family. [AI-generated per source] [family-level]
- Boyang Hu, A seven-eighths zero-free half-plane for unitary Weil L-functions (hexagon:2610.00216v1, 2026-10-13). Builds on the primitive-root and quasi-Riemann manuscripts to claim nonvanishing for Re(s)>7/8 for unitary Hecke characters over arbitrary number fields and then unitary Weil representations, extending the analytic input shared by this family. [AI-generated per source] [family-level]
Formalization
- 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)
- Pieter Moree, Artin's primitive root conjecture – a survey (arXiv:math/0412262, v2 2012). A survey of results on Artin's conjecture and its generalizations, useful context for the conditional and unconditional results (Hooley, Matthews, Gupta–Murty, Heath-Brown) recalled in the introduction.
Elsewhere
- whataifound: Registry entry
- Emergent Mind: family 029
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026,
author = {{OpenAI}},
title = {{Simultaneous primitive roots: a conditional lower bound for prime bases}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf}{OAI:Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026}},
year = {2026}
}Family 029: Primitive roots for every admissible integer base
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least $c_a x/(\log x)^2$ primes in every sufficiently large interval $(x,2x)$ have primitive root a, with $c_a\gt 0$.
Other papers in this family:
- Primitive roots for every admissible integer base · October 4, 2026