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

Shisheng Li, Zhong Zhuang

Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd (a,b)=1$ and $\sigma (a)=\sigma (b)$. Erdős Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this…