Numbers which are orders only of cyclic groups

2020/07/19 by Pollack, Paul · 3 citations
#FOS: Mathematics #Number Theory (math.NT) #Primary 11N37 #Secondary 20D60

paper · doi:10.48550/arxiv.2007.09734

Abstract

We call n a cyclic number if every group of order n is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that n is cyclic precisely when gcd(n,ϕ(n))=1. With C(x) denoting the count of cyclic n≤ x, Erdős proved that C(x) ∼ e^-γ x/logloglogx, \quadas x→∞. We show that C(x) has an asymptotic series expansion, in the sense of Poincaré, in descending powers of logloglogx, namely \frace^-γ xlogloglogx (1-\fracγlogloglogx + \fracγ² + (1)/(12)π²(logloglogx)² - \fracγ³ +(1)/(4) γπ² + (2)/(3)ζ(3)(logloglogx)³ + … ).

Cited by

• Conjectures About Cyclic Numbers: Resolutions and Counterexamples
• Conjectures about Primes and Cyclic Numbers
• A characterization of nilpotent bicyclic groups

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