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Mathematics  2015 

Measure Theoretic Aspects of Oscillations of Error Terms

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Abstract:

In this paper we obtain $\Omega$ and $\Omega_\pm$ estimates for a wide class of error terms $\Delta(x)$ appearing in Perron summation formula. We revisit some classical $\Omega$ and $\Omega_{\pm}$ bounds on $\Delta(x)$, and obtain $\Omega$ bounds for Lebesgue measure of the following types of sets: \begin{align*} \A_+&:=\{T\leq x \leq 2T: \Delta(x)> \lambda x^{\alpha}\},\\ \A_-&:=\{T\leq x \leq 2T: \Delta(x)< -\lambda x^{\alpha}\},\\ \A~&:=\{T\leq x \leq 2T: |\Delta(x)|>\lambda x^{\alpha}\}, \end{align*} where $\alpha, \lambda>0$. We also prove that if Lebesgue measure of $\A$ is $\Omega(T^{1-\delta})$ then \[\Delta(x)=\Omega_\pm(x^{\alpha-\delta})\] for any $0<\delta<\alpha$.

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