On the variance of squarefree integers in short intervals and arithmetic progressions
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Bibliographic record
Abstract
Abstract We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>6</mml:mn><mml:mo>/</mml:mo><mml:mn>11</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>></mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>5</mml:mn><mml:mo>/</mml:mo><mml:mn>11</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> and $$q > x^{1/3 + \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>></mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mi>ε</mml:mi></mml:msup></mml:math> in the full range $$H < x^{1 - \varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ε</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.013 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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