HEXAGONbeta

Every nonsquare integer not equal to −1-1 is a primitive root modulo a positive lower density of primes

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Manjul Bhargava ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

For every integer aa that is neither a square nor −1-1, we prove that a positive lower density of primes have aa as a primitive root. More precisely, a zero-free half-plane ℜs>1−1/N\Re s>1-1/N for all Dedekind zeta functions gives a lower proportion of at least ca/(Nlog⁡N)c_a/(N\log N), with ca>0c_a>0 independent of N≥2N\ge2. The theorem of Cureton and Thorner supplies N=6000N=6000, so the positive-proportion conclusion is unconditional. The proof uses the marked Type II estimates of OpenAI to count primes pp such that p−1p-1 is a small cofactor times one large prime. Allowing the cofactor to vary supplies an additional factor of log⁡x\log x in the count. We give the counting and sieve deductions in detail and explain the dependence on the zero-free width. The argument gives a lower bound rather than the exact Artin density.

Provenance statement

Yesterday's exciting announcement of QRH for Dedekind zeta functions by Cureton and Thorner resulted in handwritten ideas and notes (and a conversation with Peter Sarnak) that were then converted into LaTeX - with additional writing and details added by both the author and GPT-6 - and were then formalized in Lean with the help of GPT-6. Only the deductions were formalized; the cited theorems of Cureton--Thorner and OpenAI remain explicit hypotheses of the formal theorem and were assumed and not formalized in this work. Some details added by GPT-6 were checked by Lean but not (yet) by any human.

Tools used

OpenAI
ChatGPTVersion GPT-6

References

  1. D. Cureton and J. Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions, Hexagon preprint 2610.00200v1, 12 October 2026. See Theorem 1.1. hexagon:2610.00200v1 (https://hexagonmath.org/pdf/2610.00200v1).hexagon
  2. OpenAI, Primitive roots for every admissible integer base, preprint, 4 October 2026. The marked replacement proposition follows from Lemma 10.2 and Proposition 10.3, with the fixed-progression and smooth-weight separation in the proof of Lemma 12.1. The block sieve and rough-number estimates appear in Sections 11--12. Manuscript (https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf).link
  3. OpenAI, The Poisson--Dirichlet law for prime predecessors, preprint, 24 September 2026. See Theorem 3.1 for the one-sided marked Type II estimate, and Sections 3--5 for its proof. Manuscript (https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf).link
  4. C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209--220.
  5. H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory , Cambridge Studies in Advanced Mathematics 97 , Cambridge University Press, 2007. Standard background on primes in progressions and sieve estimates.
  6. M. Ram Murty, Artin's conjecture and elliptic analogues, in Sieve Methods, Exponential Sums, and their Applications in Number Theory , G. R. H. Greaves, G. Harman, and M. N. Huxley (eds.), Cambridge University Press, 1997, 325--344. See Section 1 for Hooley's weaker zero-free hypothesis. doi:10.1017/CBO9780511526091.022 (https://doi.org/10.1017/CBO9780511526091.022).DOI

Version history

  1. v1Submitted by Manjul BhargavaInitial depositCurrentOct 14, 2026