**1.** It is known unconditionally that, as $x$ tends to infinity,
$$\psi(x)-x=\Omega_{\pm}(x^{1/2}).\tag{$1$}$$
This is Corollary 15.4 in Montgomery-Vaughan: Multiplicative number theory I.

**2.** In fact Hardy and Littlewood proved the stronger result
$$\psi(x)-x=\Omega_{\pm}(x^{1/2}\log\log\log x).\tag{$2$}$$
This is Theorem 15.11 in Montgomery-Vaughan: Multiplicative number theory I.

**3.** Schmidt (1903) proved the analogue of $(1)$ for the function
$$f(x):=\sum_{k=1}^\infty\frac{1}{k}\pi(x^\frac{1}{k}).$$
His proof is essentially the same as of the above quoted Corollary: if the Riemann Hypothesis is false, then one has a better result, while if the Riemann Hypothesis is true, then one has a precise form of the stated result with an implied constant given in terms of the lowest lying zero of $\zeta(s)$. So Schmidt's result is unconditional as well, but it differs slightly from the statement attributed to him in the Wikipedia article.

**4.** Hardy and Littlewood (1916) attribute $(1)$ to Schmidt, and they quote it as Theorem 2.241. Precisely, they say that "This is substantially the well-known result of Schmidt". The stronger statement $(2)$ is Theorem 5.8 in their paper.

**P.S.** As Greg Martin kindly pointed out, $(2)$ is really due to Littlewood (1914).