# On the restricted divisor function in arithmetic progressions

### Igor E. Shparlinski

University of New South Wales, Sydney, Australia

## Abstract

We obtain several asymptotic estimates for the sums of the restricted divisor function

$\tau_{M,N}(k) = \# \{1 \le m \le M, \ 1\le n \le N : mn = k\}$

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 $\alpha k^2$, $k=1,2,\dots$ with a real $\alpha$.

## Cite this article

Igor E. Shparlinski, On the restricted divisor function in arithmetic progressions. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 231–238

DOI 10.4171/RMI/675