The distribution of $\gcd(n,φ(n))$

TL;DR

This work analyzes the distribution of $\gcd(n,\varphi(n))$ and its divisors by introducing the generalized sum $S_g(x)=\sum_{d\le x} g(d)\mathcal{A}_d(x)$ with bounded multiplicative $g$, where $\mathcal{A}_d(x)=\sum_{n\le x,\; d\mid(n,\varphi(n))}1$. The author proves a sharp asymptotic for $S_g(x)$ of the form $S_g(x)=x\Big(\prod_{p\le\log_2 x}\sum_{j\ge0}\frac{g(p^j)}{p^j}\Big)\exp(\mathcal{Q}_g(x))+O_{G,K}\big( x/(\log_3 x)^K\big)$, with $\mathcal{Q}_g(x)$ a computable prime-sum and the product/primes regime governed by primes near $\log_2 x$; this provides a tractable, product-structured description of the main term. A substantial decomposition and cleaning procedure reduces the problem to primes in two ranges around $\log_2 x$, and a sieve-based tool (Corollary NoDivide) yields the precise main-term evaluation, with controlled error. The paper then specializes the framework to several natural sets via their multiplicative indicators, obtaining asymptotics for cases like $r$th powers, $r$-free integers, and sums of two squares, and also derives the average order of $\tau((n,\varphi(n)))$, paralleling and extending results of Erdős and Pollack. Overall, the work quantifies how $\gcd(n,\varphi(n))$ distributes across arithmetic subsets and provides a versatile method to extract asymptotics with explicit prime-sum corrections, yielding a suite of refined results in the spirit of Pollack–Poincaré expansions.

Abstract

Let $φ(n)$denote Euler's phi function. We study the distribution of the numbers $gcd(n,φ(n))$ and their divisors. Our results generalize previous results of Erdős and Pollack.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof