The divisor function d(n), which counts the divisors of the integer n, over arithmetic progressions has been widely studied. We consider the divisor sum related to the Piatetski-Shapiro sequence \( \mathcal {N}^{(c)} :=(\lfloor n^{c} \rfloor )_{n=1}^\infty \) in arithmetic progressions, namely \( \sum _{\begin{array}{c} n \leqslant x, n \in \mathcal {N}^{(c)} \\ n \equiv a \pmod q \end{array}} d(n). \) We obtain an asymptotic formula with a main term and an error term for \(1< c < \frac{6}{5}\) and \(q \ll x^{\frac{3}{5c}-\frac{1}{2}-\varepsilon }\) . The range of c is the same as the previous result related to the divisor sum over Piatetski-Shapiro sequences. A key idea is that we have a new bound on exponential sums over arithmetic progressions. As an application, we also extend the admissible range of c for Piatetski-Shapiro primes in arithmetic progressions.
