Euler’s function on products of primes in a fixed arithmetic progression
Bibliographic record
Abstract
We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler’s function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\varphi (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Riemann Hypothesis. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis q comma a right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(q, a)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the constant appearing in the asymptotic formula <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript StartLayout 1st Row p less-than-or-equal-to x 2nd Row p identical-to a left-parenthesis mod reverse-solidus left-brace q right-brace right-parenthesis EndLayout Endscripts left-parenthesis 1 minus StartFraction 1 Over p EndFraction right-parenthesis tilde StartFraction upper C left-parenthesis q comma a right-parenthesis Over left-parenthesis log x right-parenthesis Superscript StartFraction 1 Over phi left-parenthesis q right-parenthesis EndFraction Baseline EndFraction comma"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∏ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle scriptlevel="1"> <mml:mtable rowspacing="0.1em" columnspacing="0em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>x</mml:mi> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>p</mml:mi> <mml:mo> ≡ </mml:mo> <mml:mi>a</mml:mi> <mml:mtext> </mml:mtext> <mml:mtext>(mod\ {q})</mml:mtext> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mrow> </mml:munder> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo> ∼ </mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\prod _{\substack {p \leq x \\ p \equiv a\ \text {(mod\ {q})}}} \left (1 - \frac {1}{p}\right ) \sim \frac {C(q, a)}{(\log {x})^\frac {1}{\varphi (q)}},</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\rightarrow \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Among other things, we prove that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to q less-than-or-equal-to 10"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>q</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1\leq q\leq 10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals 12 comma 14"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>12</mml:mn> <mml:mo>,</mml:mo> <mml:mn>14</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q=12, 14</mml:annotation>
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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".