Research ArticleOpen AccessGoogle Scholar indexed
Average Probability of an Element Being a Generator in the Cyclic Group
Faculty of Economics, Hokkaido University, Sapporo, Japan
- 1 Faculty of Economics, Hokkaido University, Sapporo, Japan
American Journal of Computational Mathematics·Volume 13 (2023)·Pages 230–235·Published 5 May 2023·DOI10.4236/ajcm.2023.132012
Copy link · social · email
Abstract
All elements in the cyclic group are generated by a generator g . The number of generators of of , namely is known to be Euler’s totient function ; however, the average probability of an element being a generator has not been discussed before. Several analytic properties of have been investigated for a long time. However, it seems that some issues still remain unresolved. In this study, we derive the average probability of an element being a generator using previous classical studies.
KeywordsGeneratorCyclic GroupEuler Product
- Wang, S.D. (2022) A Study of the Use of Euler Totient Function Is RSA Cryptosystem and the Future of RSA Cryptosystem. Journal of Physics: Conference Series, 2386, Article ID: 012030. https://doi.org/10.1088/1742-6596/2386/1/012030
- Haukkanen, P. (2002) On an Inequality Related to the Legendre Totient Function. Journal of Inequalities in Pure and Applied Mathematics, 3, Article 37.
- Zhai, W.G. (2020) On a Sum Involving the Euler Function. Journal of Number Theory, 211, 199-219. https://doi.org/10.1016/j.jnt.2019.10.003
- Dirichlet, G.L. (1849) über die Bestimmung der mittleren Werthe in der Zahlentheorie. Verlag nicht ermittelbar, Berlin. (Reprinted in G. Lejeune Dirichlet’s Werke, Chelsea Publishing Company, New York, 1969).
- Dirichlet, G.L. and Dedekind, R. (1879) Vorlesungen über Zahlentheorie Braunschweig. Cambridge University Press, Cambridge.
- Chen, Y.-P. (2012) A Probabilistic Look at Series Involving Euler’s Totient Function. Integers, 12, 649-657. https://doi.org/10.1515/integers-2011-0125
- Dirichlet, G.L. (1837) Beweis des Satzes, dass jede unbegrentze arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlichen Preussischen Akademie der Wissenschaften zu Berlin, 48, 45-71. arXiv:0808.1408v2 (2014).
- Kendall, R. and Osborn, R. (1965) Two Simple Lower Bounds for the Euler ϕ-Function. Texas Journal of Science, 17, 324-326.