Erratum to "Corrections to: Weyl Sums for Quadratic Roots"
Bibliographic record
Abstract
In the paper 2, to prove Theorem 7.1, we apply our old result about sums of Salié sums (Theorem 4 of 1) and this is then applied to prove Lemma 9.1. Although these applications go through without any slips, unfortunately 1, Theorem 4 turns out to be incorrect; specifically, the main term in (1.11) of 1 should be slightly different from the one claimed. We owe these observations to Lilu Zhao and are grateful to him for pointing them out. Fortunately, the exact form of the main term (1.11) of 1 does not play a role in the application to this paper 2, so all the conclusions remain unchanged. Indeed, our Theorem 7.1, which comes by recalling 1, Theorem 4, does not contain a main term, however, the condition “aq not a square” must be replaced by “ not a square”. Moreover, in the comments following Theorem 7.1, aq must be replaced by twice. Consequently, when we move on to the application to Lemma 9.1, we must change the sentence after (9.3) to “Note that can be a square only if n|r”. Then, six lines later, we need to change the ending of the paragraph into “say, where T′(x) and T′′′(x) are the partial sums over r≤R such that is, or is not, a square, respectively, and T′′(x) is the partial sum over R<r≤x”. Lastly, we need to replace the estimations for T′′′(x) leading to (9.6) by the following: ... ...
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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 | 0.009 | 0.031 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.005 |
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".