Omega theorems for the twisted divisor function

Abstract

For a fixed $\theta\neq 0$, we define the twisted divisor function \[ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ . \] In this article, we consider the error term $\Delta(x)$ in the following asymptotic formula \[ \sum_{n\leq x}|\tau(n, \theta)|^2=\omega_1(\theta)x\log x + \omega_2(\theta)x\cos(\theta\log x) +\omega_3(\theta)x + \Delta(x),\] where $\omega_i(\theta)$ for $i=1, 2, 3$ are constants depending only on $\theta$. We obtain \[\Delta(T)=\Omega\left(T^{\alpha(T)}\right) \text{ where } \alpha(T) =\frac{3}{8}-\frac{c}{(\log T)^{1/8}} \text{ and } c>0,\] along with an $\Omega$-bound for the Lebesgue measure of the set of points where the above estimate holds.