Measure Theoretic Aspects of Oscillations of Error Terms

TL;DR

This paper develops a general measure-theoretic framework to analyze oscillations of error terms in asymptotic formulas, providing bounds on the measure of sets where these errors exceed certain thresholds, with applications to Dirichlet series.

Contribution

It introduces a unified approach to quantify error term oscillations using measure estimates, extending classical results and deriving new bounds for Dirichlet series.

Findings

Established $ ext{Omega}$ bounds for measure of oscillation sets.

Reproduced classical results under weak assumptions.

Derived new results for specific Dirichlet series.

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.