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Omega Theorems for The Twisted Divisor Function (1807.10047v1)
Published 26 Jul 2018 in math.NT
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.