Propagation of Zero-Free Regions for the Riemann Hypothesis
Bibliographic record
Abstract
We prove that the Riemann zeta function has no zeros in the half-strip0.5 < Re(s) < 1. Combined with the functional equation of \zeta(s)and the symmetry of its nontrivial zeros, this establishes that allnontrivial zeros lie on the critical line Re(s)=0.5. The proof is based on an explicit analytic sieve applied to the Dirichleteta function, leading to a family of filtered alternating Dirichlet series.We show that these series converge uniformly on compact subsets ofRe(s)>0.5 to a simple nonvanishing holomorphic function. The exactmultiplicative relation between the filtered series and \eta(s) thenexcludes the existence of zeros off the critical line.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.027 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.005 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.008 | 0.011 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".