On the Euler function of the Catalan numbers (Q415249)

In the paper under review, the authors are motivated by a Carmichael's conjecture asserting that for every positive integer \(n\) there exists positive integer \(m\) with \(m\neq n\) such that \(\varphi(m)=\varphi(n)\), where as usual, \(\varphi\) refers to the Euler function. The authors restrict the argument to the Catalan numbers \(C_n\), defined by \(C_n=\frac{1}{n+1}{2n\choose n}\). They start by proving that the equality \(\varphi(C_{n+1})=4\varphi(C_n)\) holds, if \(n=2p-2\), where \(p\geq 5\) is a prime such that \(4p-3\) is also a prime, and if \(n=3p-2\), where \(p\geq 7\) is a prime such that \(2p-1\) is a prime, too. The authors explain their numerical experiences, verifying that the case with coefficient \(4\) is special among the others. NEWLINENEWLINEMore precisely, for a fixed positive rational number \(r\), they let \(\mathcal{N}_r\) to be the set of all positive integers \(n\) such that the equality \(\frac{\varphi(C_m)}{\varphi(C_n)}=r\) holds for some positive integer \(m\) with \(m\neq n\), they set \(\mathcal{N}_r(x)=\mathcal{N}_r\cap [1,x]\), and they prove validity of the estimates NEWLINE\[NEWLINE\#\mathcal{N}_r(x)\leq \frac{x}{(\log x)^{3+o(1)}}NEWLINE\]NEWLINE when \(r\neq 4,\frac{1}{4}\) as \(x\to\infty\), and NEWLINE\[NEWLINE\#\mathcal{N}_r(x)\ll\frac{x}{(\log x)^2}NEWLINE\]NEWLINE when \(r=4,\frac{1}{4}\) for \(x>10\).