On the restricted divisor function in arithmetic progressions

We obtain several asymptotic estimates for the sums of the restricted divisor function tau(M,N) (k) = #{1 <= m <= M, 1 <= n <= N : mn = k} over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function alpha k(2), k = 1, 2, ... with a real alpha.