The number of preimages of iterates of \(\phi\) and \(\sigma\) (Q6150380)

Let \(a:\mathbb{N}\rightarrow\mathbb{R}\) be an arithmetic function. Denote \[ N_a^k(n)=\#\big\{m:a_k(m)=n\big\}, \] where \(a_k\) is the \(k\)-th iterate of the arithmetic function \(a\). For fixed integer \(k\) and real \(\beta<k-1\), author of the paper proves that \[ \max\Big\{N_\phi^k(n),N_\sigma^k(n)\Big\}\leqslant\frac{n}{\Big(L_{k,\beta+1}(n)\Big)^{1+o(1)}} \] if \(n\) is sufficiently large. Here \(\phi\) is the Euler's totient function, \(\sigma(m)\) denotes the sum of divisors of \(m\), and \[ L(n)=\exp\bigg\{\frac{(\log\log\log n)^{\beta+1}}{(\log\log n)^k}\,\log n\bigg\}. \]