An improved upper bound for the argument of the Riemann zeta-function on the critical line
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Bibliographic record
Abstract
This paper concerns the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</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">S(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the argument of the Riemann zeta-function along the critical line. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is proved that for sufficiently large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</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 S left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to 0.1013 log t period"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mn>0.1013</mml:mn> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} |S(t)| \leq 0.1013 \log t. \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> Theorem 2 makes the above result explicit, viz. it enables one to select values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than t 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">t>t_{0}</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 S left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to a plus b log t period"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} |S(t)| \leq a + b\log t. \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula>
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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 | 0.002 | 0.002 |
| 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 it