A resolution of Erdős Problem 1061 on the sum-of-divisors function

Abstract

We resolve Erdős Problem 1061, the question whether the number \[ S(x)=\#\{(a,b)\in\mathbb{N}^2:a+b\le x, \ σ(a)+σ(b)=σ(a+b)\} \] of ordered solutions has a linear asymptotic $S(x)\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every \(R>0\), \[ \lim_{x\to\infty}\frac{S(x)}{x(\log x)^R}=+\infty. \] The construction begins with three integers having the same abundancy index and reduces the divisor-sum identity to two equations in six primes. After a linear change of variables, these equations lie on a split quadric. A three-parameter rational ruling of the quadric supplies many affine systems of six linear forms. An exact lattice-index calculation, an elementary codimension-two parameter sieve, and Bienvenu's higher-dimensional Siegel--Walfisz theorem give prime points uniformly on these planes. Coprime multiplier amplification then yields the stated resolution.