$N^{1/2}$ and randomness

Question

I apologize if this question is overly vague, but part of the reason I am asking is because I don't know a more precise way of discussing these ideas.

To state a general question:

What, if any, connection is there between $O(N^{1/2})$ (or related ideas if big $O$ is not appropriate) and randomness?

Here are a few things which lead me to this vague (sorry!) question:

I was recently talking with an analytic number theorist and he stated something along the lines of

"Such and such a sequence $\{a_n\}$ has partial sums $\displaystyle\sum_{n<N}a_n$ has asymptotic behavior of $O(N^{1/2})$, so the sequence is essentially random in its distribution."

I would like to understand more what he is referring to.

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EDIT: I understand that $O(\sqrt N)$ can occur for very deterministic sequences. That was never on question, so I apologize if my quote was misleading. However, I would like to understand what the reason the he would (admittedly imprecisely and incorrect in general) such a comment would be.

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In other contexts, I have seen $N^{1/2}$ related to random walks and the term "root-mean-squared", but these are unknown waters for me.

There is also the appearance of $O(x^{1/2}\log(x))$ in the Riemann hypothesis as the best error bound for the prime number theorem.

Is this somehow related to the "randomness" $N^{1/2}$ implies in other contexts, with the $\log(x)$ term handling the non-random fact that primes become more sparse for $x>>0$?

Any information about these specific things would be greatly appreciated.

Answer 1

Suppose $X_1, X_2, \dots X_n$ is a sequence of independent and identically distributed random variables with mean $0$ and standard deviation $\sigma$. Then the sum $S_n = X_1 + \dots + X_n$ has mean $0$ and standard deviation $\sqrt{n} \sigma$.

There are a large class of results in probability, the simplest being Chebyshev's inequality and the most famous being the central limit theorem, guaranteeing that random variables are not too far from their expected values, where "too far" is measured in terms of the standard deviation. So what the above tells you is that while you might have expected the sum above to vary as much as $O(n)$ from its mean (which occurs if the $X_i$ are not independent but are instead identical, say), in fact independence implies that it only varies about $O(\sqrt{n})$ from its mean. This is a fundamental observation in probability and statistics.

Edit, 7/22/24: I didn't give any examples from number theory! Here are two. First, consider the quadratic Gauss sums

$$\sum_{n=0}^{p-1} \zeta_p^{n^2}$$

where $p$ is an odd prime and $\zeta_p = \exp \frac{2 \pi i}{p}$ is a primitive $p^{th}$ root of unity. This is a sum of $p$ terms each of which is a root of unity, so if these roots of unity were distributed "sufficiently randomly" around the unit circle we'd expect the sum to have size $O(\sqrt{p})$. In fact it has absolute value exactly $\sqrt{p}$.

Second, consider the Möbius function $\mu(n)$. This is equal to $1$ on squarefree numbers with an even number of prime factors, $-1$ on squarefree numbers with an odd number of prime factors, and $0$ otherwise. You might heuristically expect this to behave randomly in some sense (this goes under the name "the Möbius pseudorandomness principle"), so you might expect cancellations between these signs to make the partial sums

$$M(n) = \sum_{k=1}^n \mu(k)$$

small (this is the Mertens function). In fact the claim that $M(n) = o(n)$ is known to be equivalent to the prime number theorem, and the claim that $M(n) = O(n^{\frac{1}{2} + \varepsilon})$ for all $\varepsilon > 0$ is known to be equivalent to the Riemann hypothesis. However, the Mertens conjecture that $|M(n)| < \sqrt{n}$ is known to be false, although apparently no explicit counterexamples are known. Quoting from Wikipedia:

In other words, the Riemann hypothesis is in some sense equivalent to saying that $\mu(x)$ behaves like a random sequence of coin tosses.

Relevant here for thinking about the difference between $O(\sqrt{n})$ and $O(n^{\frac{1}{2} + \varepsilon})$ is the law of the iterated logarithm.

Answer 2

if you the average of $N$ independent random variables you get something that has standard deviation of size roughly $N^{-1/2}$. This is the central limit theorem. So the sum has error $O(N^{1/2}).$

The law of iterated logarithms might also be relevant in terms of the rate of convergence of a sequence to the average.

Answer 3

I'm not sure how that remark about an $O(N^{1/2})$ sequence being random came about, because you can easily come up with a deterministic (non-random) sequence such that the sum of the first $N$ terms is exactly $N^{1/2}$.

However square root does have some notable importance in randomness. If you have a random sample of $n$ i.i.d. samples from a distribution with standard deviation $\sigma$, then the standard deviation for the mean of these samples (trying to estimate the mean of the distribution) is $\sigma / \sqrt{n}$. For related reasons, if you take a random walk of $N$ steps either $+1$ or $-1$ on the real line, then the standard deviation of your final position relative to your initial position is $\Theta(\sqrt{N})$