Revista Matemática Iberoamericana

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Volume 28, Issue 1, 2012, pp. 231–238
DOI: 10.4171/RMI/675

Published online: 2012-01-20

On the restricted divisor function in arithmetic progressions

Igor E. Shparlinski[1]

(1) University of New South Wales, Sydney, Australia

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 $\alpha$.

Keywords: Divisor function, congruences, character sums

Shparlinski Igor: On the restricted divisor function in arithmetic progressions. Rev. Mat. Iberoam. 28 (2012), 231-238. doi: 10.4171/RMI/675