Measure-theoretic aspects of oscillations of error terms
Tom 187 / 2019
Acta Arithmetica 187 (2019), 201-217
MSC: 11N37, 11M06.
DOI: 10.4064/aa170126-23-4
Opublikowany online: 21 December 2018
Streszczenie
We consider fluctuations of error terms 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.
