Effective Erd\H os--Wintner theorems

TL;DR

This paper provides effective estimates for the remainder term in Erdős–Wintner theorems, enhancing understanding of limiting distributions of additive arithmetical functions with conditional cases considered.

Contribution

It introduces explicit bounds for the remainder term in Erdős–Wintner theorems under certain conditions, including cases with conditional distributions.

Findings

Derived effective estimates for the remainder term.

Analyzed the case of conditional limiting distributions.

Enhanced the applicability of Erdős–Wintner theorems.

Abstract

The classical theorem of Erd\H os \& Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.