On the values of the divisor function (Q926304)

As usual, let \(\tau(n)\) denote the number of positive integer divisors of the positive integer \(n\). In the paper under review, the authors show that as \(x\to\infty\) \[ \frac{1}{x}\sum_{n\leq x}\frac{\tau(n+1)}{\tau(n)}\asymp(\log x)^\frac12, \] \[ \log\big(\#\{\tau(p-1) : p\leq x \text{ and \(p\) prime}\}\big)\geq \left(\frac{\pi\sqrt{10}}{3}+o(1)\right)\frac{(\log x)^\frac12}{(\log\log x)^\frac32}, \] and \[ \#\{\tau(2^n-1)\leq x : n\leq x\}\leq\frac{x}{(\log x)^{\alpha+o(1)}}, \] where \(\alpha\approxeq 0.2587966321\) is a solution of the equation \(\alpha=1-\frac{\alpha}{\log 2}\log(\frac{\mathrm{e}\log 2}{\alpha})\). Here we note that while authors (see Theorem 3) claim that this solution is unique, but it has another solution \(\alpha=-\frac{\log 2}{W(-\frac{1}{2\mathrm{e}})}\approxeq 2.988206298\), where \(W\) refers to the Lambert \(W\) function. The authors conclude their paper with some remarks on possible extensions and generalizations of their results and also discuss some conjectures.