TY - JOUR
AU - Hadjicostas, Petros
AB - Using Ramanujan sums, which generalize the Möbius function and the Euler totient function, we enumerate q-ary (fixed) necklaces over the color set {a1,…,aq}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{a_1,\ldots ,a_q\}$$\end{document} with ni\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n_i$$\end{document} beads of color ai\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_i$$\end{document} for i=1,…,q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,\ldots ,q$$\end{document} and co-periods dividing a fixed non-negative integer v. (The co-period of a necklace is its length divided by its period.) We also provide Witt-type infinite products and Lambert-type multiple infinite series for these quantities. Furthermore, we provide regions in Cq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb C}^q$$\end{document} where our infinite products and series converge absolutely.
TI - On the number of necklaces whose co-periods divide a given integer
JF - The Ramanujan Journal
DO - 10.1007/s11139-023-00723-3
DA - 2023-08-01
UR - https://www.deepdyve.com/lp/springer-journals/on-the-number-of-necklaces-whose-co-periods-divide-a-given-integer-0ENvWkYIRG
SP - 1021
EP - 1035
VL - 61
IS - 4
DP - DeepDyve
ER -