Large gaps in the image of the Euler's function

TL;DR

This paper establishes an upper bound on the count of integers up to x that are Euler's totient values, showing that these values have arbitrarily large gaps, with the estimate roughly proportional to x divided by the fourth root of ln x.

Contribution

It provides a new upper bound on the number of integers representable as Euler's totient function, highlighting the existence of arbitrarily large gaps in its image.

Findings

Number of integers ≤ x that are Euler's totient values is at most proportional to x/ln(x)^{1/4}

The set of Euler's totient values contains arbitrarily large gaps

The estimate improves understanding of the distribution of Euler's totient values

Abstract

The aim of this note is to provide an upper bound of the number of positive integers $\le x$ which can be written as $\varphi(n)$ for some positive integer $n$, where $\varphi$ stands for the Euler's function. The order of magnitude of this estimate, which is roughly $x/\sqrt[4]{\ln x}$, implies that the set of Euler's values contains arbitrarily large gaps.