$\sigma _{0}(n)=\sigma _{0}(n+1)$ will occur infinitely often. In 1984, Roger Heath-Brown proved that will occur $\sigma _{0}(n)=\sigma _{0}(n+1)$ infinitely often. How did he prove that? I couldn't find the paper on the internet.

The paper is probably

The divisor function at consecutive integers, Mathematika, 31 (1984), 141–149

This is not freely available, unfortunately.
See also this paper by Hildebrand which extends Heath-Brown's and is freely available:

The divisor function at consecutive integers, Pacific Journal of Mathematics,   129 (1987) 307–319

