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Omega theorems for the twisted divisor function. (English) Zbl 1457.11133

Summary: For a fixed \(\theta\neq 0\), we define the twisted divisor function \[ \tau(n, \theta):=\sum\limits_{d\mid n}d^{i\theta} . \] In this article, we consider the error term \(\Delta(x)\) in the following asymptotic formula \[ \sum\limits_{n\leqslant x}|\tau(n, \theta)|^2=\omega_1(\theta)x\log x + \omega_2(\theta)x\cos(\theta\log x) +\omega_3(\theta)x + \Delta(x), \] 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.

MSC:

11N37 Asymptotic results on arithmetic functions
11M41 Other Dirichlet series and zeta functions
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References:

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