Measure-theoretic aspects of oscillations of error terms
Volume 187 / 2019
Abstract
We consider fluctuations of error terms $\varDelta(x)$ appearing in the asymptotic formula for the summatory function of the coefficients of a Dirichlet series. These are quantified via $\varOmega$ and $\varOmega_{\pm}$ estimates. We obtain $\varOmega$ bounds for the Lebesgue measure of the sets \[ \{T\leq x \leq 2T: \varDelta(x) \gt \lambda x^{\alpha}\}\quad \text{and}\quad \{T\leq x \leq 2T: \varDelta(x) \lt -\lambda x^{\alpha}\} \] for some $\alpha, \lambda \gt 0$. The primary aim of this article is to develop a general framework to approach such problems. We rediscover several classical results in a general setting with weak assumptions. Moreover, several applications of these methods are discussed and new results are obtained for some Dirichlet series.
