Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

Given a positive integer n, let τ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (n)$$\end{document} count the number of divisors of n. We obtain several asymptotic formulas for the sum of τ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (n)$$\end{document} with n≤x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le x$$\end{document} in an arithmetic progressions n≡a(modq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \equiv a \pmod q$$\end{document} on average over a from a set of several consecutive elements of reduced residues modulo q and on average over arbitrary sets. The main goal is to obtain nontrivial results for q≥x2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \ge x^{2/3}$$\end{document} with a small amount of averaging over a. We recall that for individual values of a the limit of current methods is q≤x2/3-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \le x^{2/3-\varepsilon }$$\end{document} for an arbitrary fixed ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}. Our method builds on an approach due to Blomer (Q J Math 139:707–768, 2008) based on the Voronoi summation formula which we combine with some recent results on bilinear sums of Kloosterman sums due Kowalski et al. (Ann Math 186:413–500, 2017) and Shparlinski (Trans Am Math Soc 371:8679–8697, 2019). We also make use of extra applications of the Voronoi summation formula after expanding into Kloosterman sums and this reduces the problem to estimating the number of solutions to multiplicative congruences.