Measure Theoretic Aspects of Oscillations of Error Terms

Abstract

We consider fluctuations of error terms $\Delta(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $\Omega$ and $\Omega_{\pm}$ estimates. We obtain $\Omega$ bounds for Lebesgue measure of the sets $$\{T\leq x \leq 2T: \Delta(x)>\lambda x^{\alpha}\} \text{ and } \{T\leq x \leq 2T: \Delta(x)< -\lambda x^{\alpha}\}$$ for some $\alpha, \lambda>0$. Primary aim of this article is to develop a general framework to approach these problems. We rediscover several classical results in general setting with weak assumptions. Moreover, several applications of these methods have been discussed and new results have been obtained for some Dirichlet series.