How often is \(d(n)\) a power of a given integer? (Q2116755)

For a given integer \(k\) denote by \(A_k(x)\) the number of integers \(n\le x\) with \(d(n)\) being a power of \(k\). The author observes that if \(k\) is a prime power, then asymptotics for \(A_k(x)\) can be deduced from a result of [\textit{E. Wirsing}, Acta Math. Acad. Sci. Hung. 18, 411--467 (1967; Zbl 0165.05901)] and shows on examples \(k=6, 30\) that for other \(k\) this can be reduced to an evaluation of certain complex integrals, for example \[ A_6(x) = (1+o(1))\frac x{\log x}\cdot\frac1{2\pi i}\int_C\frac{c(z)\log^zx}zdz, \] where \(C\) encircles anticlockwise the origin and \[ c(z) = \frac1{\Gamma(z)}\prod_p\left(1-\frac1p\right)^z\sum_{m,n\ge0}\frac{z^{m-n}}{p^{2^m3^n-1}}. \] In the section ``Examples'' at the end of the paper the author shows that under the Riemann Hypothesis one can prove the equality \[ \sum_{n\le x, d(n)\equiv bmod\ 6}1 = ax^{1/4} + (b+\eta c)x^{1/6} + O(x^{1/8+\varepsilon}), \] for \(\eta\in\{1,-1\}\) \[ a=\frac{\zeta(3/2)}{2\zeta(2)},\ b=\frac{\zeta(2/3)}{2\zeta(4/3)},\ c=\frac1{2\zeta(2/3)}. \]