The generalized divisor problem in arithmetic progressions (Q1803574)

The author announces sharp asymptotic formulas for the sums \[ \begin{aligned} D_ z(x,q,\ell) &= \sum_{n\leq x, n\equiv\ell\pmod q} d_ z(n)\\ \text{and} \pi_ k(x,q,\ell) &= \sum_{n\leq x, n\equiv\ell\pmod q, n=p_ 1\dots p_ k (p_ i\neq p_ j)} 1,\end{aligned} \] which are uniform in \(k\), \(\ell\), \(q\) (\((q,\ell)=1\)) and \(z\in\mathbb{C}\), and \(p_ j\) denotes primes. Here \(d_ z(n)\) for any \(z\) denotes the generalized divisor function generated by \(\zeta^ z(s)\). The method of proof, based on complex integration, is similar to the method of the author's work [Nagoya Math. J. 122, 149--159 (1991; Zbl 0731.11053)] in which he evaluated \(D_ z(x,1,1)\). The detailed proofs appeared in another paper [Sci. Rep. Kanazawa Univ. 37, 23--47 (1992; Zbl 0774.11056)].