From this page: https://en.wikipedia.org/wiki/Chebyshev_function#Asymptotics_and_bounds
A theorem due to Erhard Schmidt states that, for some explicit positive constant $K'$, there are infinitely many natural numbers $x$ such that $$ψ(x)>x+K'√x$$
My question is: Is it possible to take the constant $K$ arbitrary large in the sense that the above inequality holds true for any $K≥K'$
and what is known about the smallest positive integer that verifying the above inequality.