Bibliographic record
Abstract
The coefficients that appear in the Laurent series of Dedekind zeta functions and their logarithmic derivatives are mysterious and seem to contain a lot of arithmetic information. Although the residue and the constant term have been widely studied, not much is known about the higher coefficients. In this thesis, we study these coefficients $\gamma_{K,n}$ that appear in the Laurent series expansion of $\frac{\zeta_K'(s)}{\zeta_K (s)}$ about $s=1$, where $K$ is a global field. For example, when $K$ is a number field, we unconditionally prove certain arithmetic formulas satisfied by these coefficients and we give bounds for them under GRH. Analogous bounds for function fields of curves defined over a finite field are also shown. \vspace{2mm}\\ We also study the distribution of values of higher derivatives of $\mathcal{L}(s,\chi)= L'(s, \chi)/L(s, \chi)$ at $s=1$ where $\chi$ ranges over all non-trivial Dirichlet characters with a given large prime conductor $m$. In particular, we compute moments, i.e. the average of $P^{(a,b)}(\mathcal{L}^{(n)}(1, \chi))$, where $P^{(a,b)}(z) = z^a \overline{z}^b$ and study their asymptotic behaviour as $m \rightarrow \infty$. We then construct a density function $M_{\sigma}(z)$, for $\sigma= $ Re$(s)$ and show that for Re$(s) > 1$ $$\text{Avg}_{\chi} \Phi(\mathcal{L}'(s, \chi)) = \int_{\mathbb{C}} M_{\sigma}(z) \Phi(z) |dz| $$ holds for any continuous function $\Phi$ on $\mathbb{C}$.
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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 | 0.001 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 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".