Methodology
How the rankings are built
This page documents how the ranked list is constructed, what is in the data, and what is deliberately excluded. The n2+1 ranking uses a three-source composite design: arXiv preprint output, OpenAlex topical citations, and zbMATH MSC classifications. Because the landmark results in this area predate arXiv, the zbMATH layer carries more weight here than it does for younger fields: it is the primary signal for senior researchers such as Iwaniec and Friedlander.
Data sources
| Source | What it gives | Limitations |
|---|---|---|
| arXiv (math.NT) | Preprint-level: titles, abstracts, authors, dates, co-author graph | Biased toward researchers who post preprints. Senior figures and pre-1991 results are not covered. The n2+1 community is small and specialized; many on-topic papers are published in journals without a preprint. |
| OpenAlex | Author-level: paper count, citations, affiliations, country | Concept tagging is noisy in mathematics; surname-only matching can misidentify researchers. Broad adjacent terms pull in authors with only tangential connections to n2+1. |
| zbMATH Open | Curated math review database; editor-assigned MSC classification (11N32, 11N35, 11N36) | Coverage is excellent for older and non-Western mathematicians. The REST API requires a one-time Terms-of-Use acceptance. |
Search terms
The arXiv and OpenAlex pulls use 12 terms restricted to math.NT: prime values of polynomials; Friedlander-Iwaniec; Bunyakovsky conjecture; Schinzel hypothesis H; almost-prime values of polynomials; Bateman-Horn conjecture; primes represented by polynomials; Merikoski; primes of the form n^2+1; prime polynomial values quadratic; n^2+1 prime; quadratic polynomial prime values. The zbMATH pull uses three MSC classes: 11N32 (primes represented by polynomials), 11N35 (Goldbach-type theorems and other additive questions involving primes), and 11N36 (applications of sieves). Together these collected over 4,600 zbMATH documents.
Pipeline
Title-weighting. A paper can mention a term without being about it. To separate genuine work from passing mentions, the arXiv and OpenAlex pipelines weight a keyword match by where it appears: a match in the paper title counts at full weight, and a match only in the abstract counts at half (a factor of 0.5). zbMATH is not title-weighted because its documents are classified by human editors; the subject class is the relevance signal.
- arXiv pull: 12 search terms restricted to math.NT.
Each paper contribution to an author is title-weighted as above.
A co-authorship graph is built and eigenvector centrality is the second factor in an arXiv composite of
0.60 * pr(weighted papers) + 0.40 * pr(eigen). Authors with at least 3 topical papers qualify. - OpenAlex pull: the same 12 phrase queries, with an author cap of 10 per work to remove large collaborative papers
only tangentially related to the topic. Works and their citations are title-weighted as above.
Composite:
0.60 * pr(weighted works) + 0.40 * pr(weighted citations). - zbMATH pull: documents tagged with any of the three MSC classes. The editor-assigned MSC classes correct a systematic gap in the other sources: pre-1991 number theorists and specialists who publish in journals with sparse arXiv presence.
- Merge and scoring: the rankings are surname-deduplicated and joined.
The available ranks are combined with a weighted order statistic:
each researcher ranks are sorted and weighted
0.70on the best,0.20on the middle, and0.10on the worst. Sorting before weighting means the method rewards excellence in any one pipeline, while a researcher strong across all of them still finishes ahead. Lower combined score ranks higher. - Estimating a missing rank (interpolation): a researcher ranked by only one of the pipelines is not given a flat penalty. To estimate a missing rank, we order the whole pool by a pipeline the researcher does appear in, then walk outward to the two nearest researchers above and below who carry a real rank in the missing pipeline, and average those (up to four) values. Estimated ranks show in [square brackets] on the ranked table; measured ranks show plain.
- Hand-curated edits: an exclusions file removes researchers the automated pipeline surfaced in error. The merge does not hand-place any researcher; everyone earns their rank from the pipeline scores.
What is not in this list
- Researchers without a strong digital footprint. The pipeline indexes arXiv well and OpenAlex moderately; the zbMATH layer corrects for journal-only publishing, but very old work may still be under-represented.
- Single-breakthrough authors. A researcher whose n2+1 contribution is one landmark paper will rank lower than a productive researcher with many related papers. This is a known limitation of any output-based metric.
- Adjacent topics. Some researchers work mainly on related questions (Goldbach-type results, primes in arithmetic progressions, general sieve theory) rather than polynomial prime values directly. Title-weighting reduces, but does not eliminate, this adjacency.