Zhong Zhuang

Coprime pairs with equal sum of divisors, assuming smooth shifted primes

Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erd\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\de...