I really wonder if you have studied basic analytic number theory, because that bound is actually well-known and is equivalent to the PNT. Anyway, Montgomery-Vaughan explicitly and clearly states in equation 6.12 that there exists some constant c>0 such that

psi(x) = x + O(x exp(-c√log x))

uniformly for x \geq 2. Since psi(x)=0 for x in [1, 2), this uniformity trivially extends to x \geq 1.

Also, arXiv moderators don't review papers, they are too busy for that. Submissions claiming to solve famous open problems are classified basing more on author reputation/submission history. I personally know a few moderators in math and physics.