The expectation values of the [1D simple random walk](https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk) $S\_n$ can be [shown](https://mathworld.wolfram.com/RandomWalk1-Dimensional.html) to have the asymptotic behavior of
$$ \lim\_{n\to\infty} \frac{a\_n}{n^{1/2}} = \sqrt{\frac{2}{\pi}}, \tag{1}\label{1}$$


with $a\_n = S\_n$.


On the other hand, the Riemann Hypothesis is famously equivalent to the statement that


$$ \lim\_{n\to\infty} \frac{a\_n}{n^{1/2 +\varepsilon}} = 0, \tag{2}\label{2}$$
for any fixed $\varepsilon>0$ and setting $a\_n=L(n)$, the [cumulative (or summatory) Liouville function](https://mathworld.wolfram.com/LiouvilleFunction.html) $L(n)=\sum\_n \lambda(n)$.


I am interested in about what we know on the difference between the two asymptotic behaviours \eqref{1} and \eqref{2}. Clearly the random walk asymptotics \eqref{1} implies \eqref{2} but not vice versa.


How can it be shown that $a\_n=L(n)$ violates \eqref{1}, would its fulfillment have interesting implications? Are there other/simple examples of $a\_n$ that fulfill \eqref{2}?



Asymptotics of cumulative Liouville function under RH versus simple random walk
I figured the remarks I gave in the comments deserve to be gathered up into a more coherent form as an answer.


One thing I will start with is that comparing $L(n)$ (or $M(n)$, the Mertens function) to the values $S\_n$ is arguably not the right heuristic. Indeed, $S\_n$ averages over all random walks, which can have their peaks and troughs at different places, and so they cancel out. Analysing a single random walk gives a different asymptotic: this is governed by [the law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm), which asserts that the random walk will oscillate with extremal values on the order of $\pm\sqrt{2n\log\log n}$.


However it turns out that for subtle arithmetic reasons we do not expect the arithmetic functions of interest to reach the same bounds. Most literature (see e.g. [this paper](https://arxiv.org/abs/math/0310381v1)) concerns the Mertens function $M(n)$ so I will discuss this one, but all the relevant heuristics should also hold for $L(n)$. An unpublished conjecture of Gonek is that the maximal order of $|M(n)|$ is not $\sqrt{n\log\log n}$, but rather $\sqrt{n}(\log\log\log n)^{5/4}$, which in particular implies that the ratio $|M(n)|/\sqrt{n}$ is unbounded.


While Gonek's conjecture is still open, as is unboundedness of $|M(n)|/\sqrt{n}$, we do have unconditional results which imply that this sequence does not have a limit. Specifically, we know that $M(n)/\sqrt{n}$ has strictly negative liminf and strictly positive limsup (indeed, they are at least $\pm1.8$ respectively, see [Wikipedia](https://en.wikipedia.org/wiki/Mertens_conjecture#Disproof_of_the_conjecture) for more details and references.) This shows that $|M(n)|/\sqrt{n}$ crosses zero infintiely often, and infinitely often it takes values greater than $1$, so it cannot have a limit. (Same is known for Liouville though with different bounds on liminf and limsup, see [Wikipedia](https://en.wikipedia.org/wiki/Liouville_function#Conjectures_on_weighted_summatory_functions).)


