On integers n with \(J_ t(n)<J_ t(m)\) for \(m>n\) (Q1114728)

Let \(\phi\) denote Euler's totient function and define F to be the set of all sparsely totient numbers, i.e. the set of all integers \(n>1\) such that \(m>n\) implies \(\phi (m)>\phi (n)\). \textit{D. W. Masser} and \textit{P. Shiu} [Pac. J. Math. 121, 407-426 (1986; Zbl 0538.10006)] obtained a number of results about the set F, some of which are listed in the sequel: (1) Any given prime number divides all sufficiently large elements of F, (2) any positive integer divides infinitely many elements of F, (3) if n and \(n'\) denote consecutive elements of F, then \(n'/n\to 1\) as \(n\to \infty\) in F, and \((4)\quad \log F(x)\ll \sqrt{\log x},\) where \(F(x)=\#\{n\in F\); \(n\leq x\}\) denotes the counting function of F. By replacing Euler's function in the definition of F with the more general Jordan totient function \(J_ t\) of order t, defined by \(J_ t(n)=n^ t\cdot \prod_{p| n}(1-p^{-t}),\) the present authors introduce the corresponding sets \(F_ t\). They use essentially the methods of Masser and Shiu to extend the above results to the sets \(F_ t\), \(t\in {\mathbb{N}}\).