The Distribution of Error Terms of Smoothed Summatory Totient Functions

TL;DR

This paper investigates the limiting distribution of the error term in smoothed summatory totient functions, establishing conditions under which it follows a logarithmic distribution and analyzing the effects of smoothing on error bounds.

Contribution

It introduces a framework for analyzing the limiting distribution of smoothed totient error terms and proves a truncated Perron formula for Riesz means, showing smoothing reduces error growth.

Findings

The smoothed error term has a limiting logarithmic distribution under certain conditions.

At least two smoothing applications are needed to bound the error by a9(x).

A new truncated Perron inversion formula for Riesz means is established.

Abstract

We consider the summatory function of the totient function after applications of a suitable smoothing operator and study the limiting behavior of the associated error term. Under several conditional assumptions, we show that the smoothed error term possesses a limiting logarithmic distribution through a framework consolidated by Akbary--Ng--Shahabi. To obtain this result, we prove a truncated version of Perron's inversion formula for arbitrary Riesz typical means. We conclude with a conditional proof that at least two applications of the smoothing operator are necessary and sufficient to bound the growth of the error term by $\sqrt{x}$.