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 $$ψ(x)>x+K√x$$


Since the original paper is in German. I read this paper: <https://projecteuclid.org/download/pdf_1/euclid.acta/1485887467>


My **question** is: Is the result of Schmidt conditional to Riemann Hypothesis.



Is the result of Schmidt conditional to RH
**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)](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002259192) 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)](https://projecteuclid.org/euclid.acta/1485887467) 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)](https://gallica.bnf.fr/ark:/12148/bpt6k3111d/f1873.image).


