Counting zeros of Dedekind zeta functions
Bibliographic record
Abstract
Given a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Subscript upper K"> <mml:semantics> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">n_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with absolute discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript upper K"> <mml:semantics> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">d_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we obtain an explicit bound for the number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript upper K Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">N_K(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of non-trivial zeros (counted with multiplicity), with height at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , of the Dedekind zeta function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta Subscript upper K Baseline left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ζ </mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta _K(s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . More precisely, we show that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">T \geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper N Subscript upper K Baseline left-parenthesis upper T right-parenthesis minus StartFraction upper T Over pi EndFraction log left-parenthesis d Subscript upper K Baseline left-parenthesis StartFraction upper T Over 2 pi e EndFraction right-parenthesis Superscript n Super Subscript upper K Superscript Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to 0.228 left-parenthesis log d Subscript upper K Baseline plus n Subscript upper K Baseline log upper T right-parenthesis plus 23.108 n Subscript upper K Baseline plus 4.520 comma"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mi>T</mml:mi> <mml:mi> π </mml:mi> </mml:mfrac> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>e</mml:mi> </mml:mrow> </mml:mfrac> <mml:msup> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD">
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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 | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".