Omega Theorems for The Twisted Divisor Function

TL;DR

This paper investigates the oscillatory behavior of the error term in the asymptotic formula for the twisted divisor function's second moment, establishing Omega bounds that describe its magnitude and distribution.

Contribution

It provides new Omega bounds for the error term in the twisted divisor function's mean square, revealing its growth rate and measure-theoretic properties.

Findings

The error term elta(T) exhibits growth t least T^{rac{3}{8}-rac{c}{(\u00a0log T)^{1/8}}}.

The paper establishes measure-theoretic bounds for the set where the Omega estimate holds.

It advances understanding of the oscillatory nature of divisor-related error terms in analytic number theory.

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.