Exponential sums over Möbius convolutions with applications to partitions

We consider partitions $p_w(n)$ of a positive integer $n$ arising from the generating functions $$\sum _{n=1}^{\mathrm{\infty }}{p}_w(n)z^n=\prod _{m\in \mathbb{N}}(1-z^m{)}^{-w(m)},$$ where the weights $w(m)$ are Möbius convolutions. We establish an upper bound for $p_w(n)$ and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.