The frequency and the structure of large character sums
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Bibliographic record
Abstract
Let M(\chi) denote the maximum of |\sum_{n\le N}\chi(n)| for a given non-principal Dirichlet character \chi modulo q , and let N_\chi denote a point at which the maximum is attained. In this article we study the distribution of M(\chi)/\sqrt{q} as one varies over characters modulo q , where q is prime, and investigate the location of N_\chi . We show that the distribution of M(\chi)/\sqrt{q} converges weakly to a universal distribution \Phi , uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \Phi 's tail. Almost all \chi for which M(\chi) is large are odd characters that are 1-pretentious. Now, M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)| , and one knows how often the latter expression is large, which has been how earlier lower bounds on \Phi were mostly proved. We show, though, that for most \chi with M(\chi) large, N_\chi is bounded away from q/2 , and the value of M(\chi) is little bit larger than \frac{\sqrt{q}}{\pi} |L(1,\chi)| .
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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