Omega Theorems for The Twisted Divisor Function

For a fixed $\theta \mathrm{\ne }0$, we define the twisted divisor function $$\tau (n,\theta ):=\sum _{d\mid n}{d}^{i\theta }\mathrm{.}$$ In this article we consider the error term $\mathrm{\Delta }(x)$ in the following asymptotic formula $$\sum _{n\le x}^{\ast }\mathrm{\mid }\tau (n,\theta ){\mathrm{\mid }}^2={\omega }_1(\theta )x\log x+{\omega }_2(\theta )x\cos (\theta \log x)+{\omega }_3(\theta )x+\mathrm{\Delta }(x),$$ where ${\omega }_i(\theta )$ for $i=1,2,3$ are constants depending only on $\theta$. We obtain $$\mathrm{\Delta }(T)=\mathrm{\Omega }\left(T^{\alpha (T)}\right)\text{ where }\alpha (T)=\frac{3}{8}-\frac{c}{(\log T{)}^{1\mathrm{/}8}}\text{ and }c>0,$$ along with an $\mathrm{\Omega }$-bound for the Lebesgue measure of the set of points where the above estimate holds.