On the \(k\)-fold iterates of the Euler totient function
at shifted primes

Abstract. Let \(\gamma(n)\) stand for the product of the prime factors of \(n\). The index of composition \(\lambda(n)\) of an integer \(n\ge 2\) is defined as \(\lambda(n)=\log n / \log \gamma(n)\) with \(\lambda(1)=1\). Given an arbitrary integer \(k\ge 0\) and letting \(\phi_k(n)\) stand for the \(k\)-fold iterate of the Euler totient function, we show that, given any real number \(\varepsilon>0\), \(\lambda(\phi_k(p-1)) <1+\varepsilon\) for almost all prime numbers \(p\).