In Mazur and Stein's book, "Prime Numbers and the Riemann Hypothesis" there is a discussion about the square root error in a random walk (page 38). It says there that this quantity is $\sqrt{2/\pi}\sqrt{x}$. How is the coefficient $\sqrt{2/\pi}$ obtained?
Edit The comment answers the original question. A more interesting scenario, and the one I am interested in, is as follows. You have three possible moves: left, right, or no move at all (but time increments by one). We can assume the simplest probability scenario. Each possibility has probability 1/3. What is the square root error now?