On the concentration of certain additive functions

TL;DR

This paper investigates how additive functions concentrate in distribution, especially when prime values decay rapidly and are well spaced, confirming a conjecture about a specific additive function's concentration.

Contribution

It proves a conjecture by Erdos and Katai regarding the concentration behavior of a particular additive function under certain decay and spacing conditions.

Findings

Confirmed the Erdos-Katai conjecture for the additive function with decay parameter c>1.

Established conditions under which the distribution of the additive function concentrates.

Provided insights into the behavior of additive functions with rapidly decaying prime contributions.

Abstract

We study the concentration of the distribution of an additive function, when the sequence of prime values of $f$ decays fast and has good spacing properties. In particular, we prove a conjecture by Erdos and Katai on the concentration of $f(n)=\sum_{p|n}(\log p)^{-c}$ when $c>1$.