I was looking into Littlewood's 1914 result that pi(x) and Li(x) cross infinitely many times, and I came across this Wikipedia page: https://en.wikipedia.org/wiki/Skewes%27s_number#Riemann's_formula. This paper references Montgomery-Vaughan's Multiplicative Number Theory I, in particular Theorem 15.11. As the Wikipedia article linked above says, Dirichlet's approximation theorem is used; but also a gnarly lemma 15.9:
My questions are: is there an easier proof of Littlewood's theorem, following the formula and outline on Wikipedia:
The largest error term in the approximation ${\displaystyle \pi (x)\approx \operatorname {li} (x)}$ (if the Riemann hypothesis is true) is negative ${\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}$, showing that ${\displaystyle \operatorname {li} (x)}$ is usually larger than $\pi (x)$. The other terms above are somewhat smaller, and moreover tend to have different, seemingly random complex arguments, so mostly cancel out. Occasionally however, several of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term ${\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}$.
The reason why the Skewes number is so large is that these smaller terms are quite a lot smaller than the leading error term, mainly because the first complex zero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of $N$ random complex numbers having roughly the same argument is about 1 in $2^N$. This explains why $\pi (x)$ is sometimes larger than ${\displaystyle \operatorname {li} (x),}$ and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function.
The argument above is not a proof, as it assumes the zeros of the Riemann zeta function are random, which is not true. Roughly speaking, Littlewood's proof consists of Dirichlet's approximation theorem to show that sometimes many terms have about the same argument. In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms $\displaystyle \operatorname {li} (x^{\rho })$ for zeros violating the Riemann hypothesis (with real part greater than 1/2) are eventually larger than ${\displaystyle \operatorname {li} (x^{1/2})}$.
and if so, is there a quick/easy way derive this particular "explicit formula" for $\pi(x)$ from the Riemann-van Mangoldt "explicit formula" $\psi(x):=\sum_{n\leq x} \Lambda(n) = x - \lim_{T\to\infty} \sum_{\rho: |\text{Im}(\rho)|\leq T} \frac{x^\rho}\rho - \ln(2\pi)- \frac 12 \ln(1-x^{-2})$ found in Terry Tao's notes (Theorem 8 of 246B Notes 4)? Summation by parts allows us to relate $\sum_{n\leq x} \frac{\Lambda(n)}{\log n}$ and $\psi(x):=\sum_{n\leq x} \Lambda(n)$, and $\pi(x)$ is close to $\sum_{n\leq x} \frac{\Lambda(n)}{\log n}$, but I don't see how the logarithmic integral pops up.
The reason I ask about a proof coming from this particular explicit formula for $\pi(x)$ is because the proof from Montgomery-Vaughan (again see their Thm. 15.11) actually proves some facts about how $\psi(x)-x$ deviates, and then uses Thm. 13.2 to translate over those results to $\pi(x)-\text{li}(x)$. If the Wikipedia sketch can be made complete and rigorous, then there will be no need to first work with $\psi(x)-x$.
As an aside: I did look at Littlewood's 1914 proof relating to Skewes' number which does helpfully provide a link to the original Littlewood paper from 1914 (written by Hadamard?), and I found an English translation here: https://foundationalperspectives.wordpress.com/2018/10/14/littlewoods-1914-theorem-on-pix-li-x/. Montgomery-Vaughan (at the end of Chapter 15) note that the 1914 paper is really just a sketch, and a more fleshed-out version can be found in Section 5 of Hardy-Littlewood's joint 1918 paper.