From quickly glanced through the paper, it doesn't seem to me that the authors have actually proved any theorems - at least not in the mathematical sense of the word "proof." They *have* provided ample numerical evidence that indicates Benford's Law applies to primes, but their explanation for why this is so is because the primes have a 1/log(x) density - nothing deeper than that. It's a cool paper to be sure, but there doesn't seem to be much significance for mathematics.

Extremely highly-regarded mathematician Terence Tao has said, in response to a comment left on his blog, that the proof is probably incorrect.