On the ratio of surjective group homomorphisms to ring homomorphisms between cyclic rings

Abstract

# Zenodo record: what to edit The archive already exists. It was created automatically from GitHub release v1.1.2. - Version DOI (cited in the paper): https://doi.org/10.5281/zenodo.23106785 - Concept DOI (always points to the newest version): https://doi.org/10.5281/zenodo.23106784 Do not make a new upload on zenodo.org. That would publish a second, separate record with its own DOI. To correct the record, open https://doi.org/10.5281/zenodo.23106785 while logged in, click **Edit**, change the fields below, and click **Publish**. Editing keeps the same DOI. | Zenodo field | Change | |---|---| | Creator 3 | Replace "[Third Author Surname], [Given names]" and "[Department, Institution, City PIN, Country]" with the real name, ORCID and institution when they are known | | Description | Replace the sentence ending "checked against exact counts up to 2*10^10." with the text below | | Subjects (optional) | MSC 11A15, 11N25, 11N37, 11R18, 13M05 | ## Description (paste as is) For n | m there are φ(n) surjective group homomorphisms Z_m → Z_n and 2^ω(n) ring homomorphisms. Their ratio ρ(n) is 2^ε(n) times the number of squares in (Z/nZ)^×, where ε(n) ∈ {−1, 0, 1} is fixed by the power of 2 dividing n, and for odd n the ring homomorphisms correspond to the subfields of degree at most 2 of Q(ζ_n). The paper gives two-term asymptotic formulas, with explicit constants, for the number of n ≤ x with ρ(n) not an integer and for the partial sums of ρ, checked against exact counts up to 2·10^10 and 10^10, respectively. The archive contains the manuscript (PDF and LaTeX source with figures), the C programs for the exact counts, the Python programs for the constants, tables, figures and algebraic checks, and the computed data. ## New GitHub releases Zenodo is now linked to the GitHub repository, so every new GitHub release creates a new Zenodo version with a new version DOI under the same concept DOI. Zenodo versions cannot be deleted once published. After a new release, the DOI in the paper's Data availability section should be switched to the new version DOI, or to the concept DOI.