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.
References
[1]
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
[2]
Haukkanen, P. (2002) On an Inequality Related to the Legendre Totient Function. Journal of Inequalities in Pure and Applied Mathematics, 3, Article 37.
[3]
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
[4]
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).
[5]
Dirichlet, G.L. and Dedekind, R. (1879) Vorlesungen über Zahlentheorie Braunschweig. Cambridge University Press, Cambridge.
[6]
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
[7]
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).
[8]
Kendall, R. and Osborn, R. (1965) Two Simple Lower Bounds for the Euler ϕ-Function. Texas Journal of Science, 17, 324-326.
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