The divisor problem for arithmetic progressions (Q5917670)

Let the divisor function \(d_r (n)\) denote the number of solutions of \(n_1 n_2 \dots n_r = n\) in natural numbers \(n_1,n_2, \dots, n_r\) and let \[ D_r (x,q,a) = \sum_{{n \leq x \atop n \equiv a\pmod q}} d_r(n) \] with \((a,q) = 1\). Now find numbers \(\theta_r\) as large as possible with the following property: For each \(\varepsilon > 0\) there exists \(\delta > 0\) such that \[ D_r (x,q,a) - {x \over \varphi (q)} P_r (\log x) \ll_\varepsilon {x^{1 - \delta} \over \varphi (q)} \] provided that \(q < x^{\theta_r- \varepsilon}\). \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Acta Arith. 45, 273-277 (1985; Zbl 0572.10033)] proved that the inequality holds with \(\theta_r = \min (8/3r, 5/12)\) for \(r\geq 6\). This is now improved by \(\theta_r = \min (3/r, 5/12)\) for \(r \geq 7\). The proof uses a theorem of \textit{D. A. Burgess} [J. Lond. Math. Soc., II. Ser. 33, 219-226 (1986; Zbl 0593.10033)].