Improved estimates for the argument and zero-counting function of the Riemann zeta-function
Bibliographic record
Abstract
In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper N left-parenthesis upper T right-parenthesis minus StartFraction upper T Over 2 pi EndFraction log left-parenthesis StartFraction upper T Over 2 pi e EndFraction right-parenthesis EndAbsoluteValue less-than-or-equal-to 0.10076 log upper T plus 0.24460 log log upper T plus 8.08344 comma"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>e</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mn>0.10076</mml:mn> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mn>0.24460</mml:mn> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8.08344</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left | N (T) - \frac {T}{ 2 \pi } \log \left ( \frac {T}{2\pi e}\right ) \right |\le 0.10076\log T+0.24460\log \log T+8.08344,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> for every <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 e"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mi>e</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T\ge e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <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(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the number of non-trivial zeros <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho equals beta plus i gamma"> <mml:semantics> <mml:mrow> <mml:mi> ρ </mml:mi> <mml:mo>=</mml:mo> <mml:mi> β </mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi> γ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho =\beta +i\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than gamma less-than-or-equal-to upper T"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi> γ </mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0>\gamma \le T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , of the Riemann zeta-function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <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 (s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main source of improvement comes from implementing new subconvexity bounds for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis sigma plus i t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> σ </mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta (\sigma +it)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k">
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 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".