Bibliographic record
Abstract
Let $d(n)$ denote the number of divisors of $n$. Erd{\H{o}}s, in 1968, proved the remarkable result whereby there exists a positive constant $ d_{2} < \infty$ such that $$ \lim_{x \to \infty} \frac{\sum_{n \leq x} d(d(n))}{x \log \log x} = d_{2}, $$ and this was improved by Rieger in 1972. If we compare the above result to Ramanujan's formula for $ \prod_{n=1}^{x} d(n)$ and its relation to the Dirichlet formula for $\sum_{n = 1}^{x} d(n)$, this raises questions as to the behaviour of products of the form $\prod_{n=1}^{x} d(d(n))$. By the AM-GM inequality, Erd{\H{o}}s' result gives us that $$ \frac{\left( \prod_{n=1}^{x} d(d(n)) \right)^{\frac{1}{x}}}{\log \log x} $$ is bounded above by a positive, finite constant, but this does not give that the limit of the above expression exists as a positive constant, and known bounds for expressions such as $d(n)$ and $\log d(d(n))$ cannot be used in any direct way. Moreover, as we discuss, it seems that Erd{\H{o}}s' and Rieger's summation techniques cannot be altered so as to be applicable to expressions such as $\sum_{n \leq x} \log d(d(n))$ or $\prod_{n \leq x} d(d(n))$. We prove that: For all $\epsilon > 0$, the bounds $$ \log 2 - \epsilon \leq \frac{\sum_{n \leq x} \log d(d(n)) }{x \log \log \log x} \leq 1 + \epsilon $$ hold for all for sufficiently large $x$, yielding bounds for the product obtained by replacing $d(n)$ with $d(d(n))$ in Ramanujan's product.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.003 |
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".