An equivalent inequality for the Riemann hypothesis

We present a purely analytical inequality which is equivalent to the Riemann hypothesis (RH). The proof of the equivalence is based on a representation of the modulus of the Riemann $\xi$ function. As the first step to analyze the inequality, we consider polynomial approximations. We also show that the RH is equivalent to the statement that some wave functions constructed using the Brownian motion never evolve into perfectly distinguishable states.