About $\pi(x)Today I found that in 1914, Littlewood proved that
(1) there are arbitrarily large values of $x$ for which
$$\pi(x)<li(x)-\frac{1}{3}\frac{\sqrt{x}}{\log(x)}\log(\log(\log(x)))$$

*

*First: Is this true?


*Second: Is (1) equivalent to:
(2) There are infinitely many $x$ such that
$$\pi(x)<li(x)-\frac{1}{3}\frac{\sqrt{x}}{\log(x)}\log(\log(\log(x)))$$?
Thanks!
 A: Concerning lower and upper bounds of $\pi(x)-li(x)$ Saouter and Demichel have shown in $2010$ that
$$
\frac{-0.2x}{\log^3 x}-\frac{12x}{\log^4 x}-C_1-C_2\le \pi(x)-li(x)\le \frac{0.51x}{\log^3 x}-C_1 \quad \forall \; x\ge 355991,
$$
with
$$
C_1=li(2)-\frac{2}{\log 2}\left(1+\frac{1}{\log 2}+\frac{2}{\log^2 2}\right),
$$
$$
C_2=\int_2^{e^8}\frac{48}{\log^5 t}dt-\frac{24}{\log^4 2}.
$$
The best result for the difference is obtained if we believe in RH:
Theorem (Schoenfeld 1976): If the Riemann hypothesis holds, then for all $x\ge 2657$ we have
$$
|\pi(x)-li(x)|\le \frac{1}{8\pi}\sqrt{x}\log x.
$$
A: The result in the question is first proved by J.E. Littlewood in Comptes Rendus de l'Academie des Sciences, June 1914. 
The result is expressed as: 
$$\pi(x)-Li(x)<-K\frac{\sqrt{x}\log\log\log x}{\log x} $$
$$\pi(x)-Li(x)>K\frac{\sqrt{x}\log\log\log x}{\log x}. $$
He concludes that the inequality $\pi(x)<Li(x),$ "presumed by many authors for empirical reasons, cannot obtain for any value of $x$ however large." 
He does not specify K.
His conclusion may be interpreted to mean that for any value of $x,$ however large, for which one sense of the inequality holds, one can find a larger $x$ for which the other sense holds, and so on. 
Ingham's later proof is somewhat easier (The Distribution of Prime Numbers, Ch. V). 
A: The result you quote from Littlewood is stated and proved in that paper of Hardy  and Littlewood (see theorem 5.8 and below, page 194). 
