On the local behavior of the Carmichael \(\lambda\)-function (Q874429)

Let \(\varphi\) be Euler's function, and let \(\lambda\) be Carmichael's function. Theorem 1. Let \(t\) be a positive integer and let \(a\) be any permutation of the set \(\{1,2,\ldots,t\}.\) There exist infinitely many positive integers \(n\) such that the inequality \(\lambda(n+a(i))>\lambda(n+a(i+1))\) holds for all \(i=1,\dots,t-1.\) Furthermore, if \(n:=n(t)\) denotes the minimal value of \(n\) such that the preceding inequality holds, then the estimate \(t>>\log_2n(t)\) holds as \(t\) tends to infinity. Theorem 2. Let \(t\) be a positive integer and let \(a\) and \(b\) be any permutations of the set \(\{1,2,\dots,t\}.\) There exist infinitely many positive integers \(n\) such that the inequalities \(\lambda(n+a(i))>\lambda(n+a(i+1))\) and \(\varphi(n+b(i))>\phi(n+b(i+1))\) hold for all \(i=1,\dots,t-1.\) Furthermore, if \(n:=n(t)\) denotes the minimal value of \(n\) such that the preceding inequalities hold, then the estimate \(t>>\left(\frac{\log_3n(t)}{\log_5n(t)}\right)^{\frac{1}{2}}\) holds as \(t\) tends to infinity.