An effective universality theorem for the Riemann zeta function
Bibliographic record
Abstract
Let 0 < r < 1/4 , and f be a non-vanishing continuous function in |z|\leq r , that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function \zeta(3/4 + z + it) can approximate f uniformly in |z| < r to any given precision \varepsilon , and moreover that the set of such t \in [0, T] has measure at least c(\varepsilon) T for some c(\varepsilon) > 0 , once T is large enough. This was refined by Bagchi who showed that the measure of such t \in [0,T] is (c(\varepsilon) + o(1)) T , for all but at most countably many \varepsilon > 0 . Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T . Our method is flexible, and can be generalized to other L -functions in the t -aspect, as well as to families of L -functions in the conductor aspect.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
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".