Estimates for Character Sums and Dirichlet<i>L</i>-Functions to Smooth Moduli
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Bibliographic record
Abstract
We use the <f>$q$</f>-analogue of van der Corput's method to estimate short character sums to smooth moduli. If <f>$\\chi $</f> is a primitive Dirichlet character modulo a squarefree, <f>$q^\\delta $</f>-smooth integer <f>$q$</f> we show that <fd id="rnv285UM1">\\[ L\\left(\\tfrac12,\\chi\\right)\\ll_\\epsilon q^{\\frac{27}{164}+O\\left(\\delta\\right)+\\epsilon}. \\]</fd>
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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 | 0.004 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it