Sign changes in sums of the Liouville function
Bibliographic record
Abstract
The Liouville function λ ( n ) \lambda (n) is the completely multiplicative function whose value is − 1 -1 at each prime. We develop some algorithms for computing the sum T ( n ) = ∑ k = 1 n λ ( k ) / k T(n)=\sum _{k=1}^n \lambda (k)/k , and use these methods to determine the smallest positive integer n n where T ( n ) > 0 T(n)>0 . This answers a question originating in some work of Turán, who linked the behavior of T ( n ) T(n) to questions about the Riemann zeta function. We also study the problem of evaluating Pólya’s sum L ( n ) = ∑ k = 1 n λ ( k ) L(n)=\sum _{k=1}^n\lambda (k) , and we determine some new local extrema for this function, including some new positive values.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.006 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.028 | 0.006 |
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 source (direct Gemma or distilled Codex), 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".