Omega Theorems for The Twisted Divisor Function

Abstract

For a fixed $\theta \ne 0$, 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 $$\sum _{n\le x}^{\ast }|\tau (n,\theta ){|}^2={\omega }_1(\theta )x\log x+{\omega }_2(\theta )x\cos (\theta \log x)+{\omega }_3(\theta )x+\Delta (x),$$ 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.

13 pages, This is Chapter 4 of arXiv:1512.03144 (v3). New version of 1512.03144 will not have this chapter