Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support
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Bibliographic record
Abstract
We study the $1$ -level density of low-lying zeros of Dirichlet $L$ -functions attached to real primitive characters of conductor at most $X$ . Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$ , which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$ . We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$ , both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$ , and is analogous to a lower order term which was isolated by Rudnick in the function field case.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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