English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let    $${{\mathbb {Z}}}_{n}$$    Z  nbe the additive group of residue classes modulo  n . Let    $$c(n_1,n_2,n_3)$$    c  (   n  1   ,   n  2   ,   n  3   )denote the number of cyclic subgroups of the group    $${{\mathbb {Z}}}_{n_1}\times {{\mathbb {Z}}}_{n_2}\times {{\mathbb {Z}}}_{n_3}$$Z   n  1    ×   Z   n  2    ×   Z   n  3, where    $$n_1, n_2$$n  1   ,   n  2and    $$n_3$$    n  3are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum    $$\sum _{n_1,n_2,n_3\le _x} c(n_1,n_2,n_3).$$∑    n  1   ,   n  2   ,   n  3    ≤  xc   (   n  1   ,   n  2   ,   n  3   )   .

research in number theory

On the average number of cyclic subgroups of the groups $${\mathbb {Z}}_{n_1} \times {\mathbb {Z}}_{n_2}\times {\mathbb {Z}}_{n_3}$$ with $$n_1,n_2,n_3\le x$$