On the Restricted Divisor Function in Arithmetic Progressions

We obtain several asymptotic estimates for the sums of the restricted divisor function $$ τ_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ 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 $αk^2$, $k=1,2,...$ with a real $α$.

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