Last week, I was at the Mathematics and Machine Learning program at Harvard's Center of Mathematical Sciences and Applications.^{1} ^{1}Later Update: I'll be talking about this and related experiments on October 29th at Harvard. The talk should be made available on youtube. The underlying topic was on number theory and I've been studying various number theoretic problems from a machine learning perspective.
I've been computing several experiments related to estimating the Mobius function $\mu(n)$. I don't expect $\mu(n)$ to be easily approximable; all earlier attempts to study $\mu$ using machine learning have resisted much success. This is perhaps related to Mobius Randomness.^{2} ^{2}See for example Peter Sarnak's Three Lectures on the Mobius Function Randomness and Dynamics.
Previous machine learning experiments on studying $\mu(n)$ have used neural networks or classifiers. Francois Charton made an integer sequence to integer sequence transformerbased translator, Int2Int, and I thought it would be fun to see if this works any different.
Initially, I sought to get Int2Int to work. Then I set it on various examples. I describe some of them here.
I'm splitting my description into two parts: a general report and a technical report. This is the general report.^{3} ^{3}This is also available as a pdf. The technical^{4} ^{4} By "technical" here, I mean pertaining to technology (i.e. to programming). Both notes are nonelementary. But I acknowledge that there are very few people who are experts in both number theory and machine learning. report includes technical details for running or rerunning Int2Int experiments and other programmingrelated aspects.
Mobius Function
Recall that the mobius function $\mu(n)$ is $0$ if the square of any prime divides $n$, and otherwise is $(1)^\omega(n)$, where $\omega(n)$ is the number of prime divisors of $n$. For example, $\mu(1) = 1, \mu(2) = \mu(3) = 1, \mu(4) = 0, \mu(5) = 1, \mu(6) = 1,$ and so on.
Int2Int takes as input a sequence of integers, and the output is a sequence of integers. I struggled to make sense of studying many outputs, but this is really my own problem.
Inputs and Outputs for Möbius
Int2Int takes sequences of integers as input and produces sequences of integers as output. I tried several variations to estimate $\mu(n)$, including

Input just $n$ and output $\mu(n)$. (Or rather, make sure I can get Int2Int to process anything at all with the simplest possible example).

Input $n \bmod p$ and $p$ for the first $100$ primes.

Input $n \bmod p$ and $p$ for the second $100$ primes.

Input the Legendre symbol $(n/p)$ for the first $100$ primes.

Input $n$, $n \bmod p$, and $(n/p)$ for the first $100$ primes.
For each of these, I estimated $\mu(n)$, $\mu^2(n)$, and $\mu(n+1)$. The input $n$ were sampled uniformly randomly from $n$ between $2$ and $10^{13}$ (with a few larger experiments here and there), using training sets between $2\cdot10^6$ for initial runs and $5\cdot10^{7}$ to investigate further. I also trained over $n$ restricted to be squarefree.
Better than Random: squarefree detection
I quickly saw that Int2Int could guess $\mu(n)$ better than random guesses. But the reason why was because it was determining if $n$ was squarefree or not with reasonable accuracy.^{5} ^{5}This is similar to a pattern observed by Jordan Ellenberg when attempting to train neural networks to estimate $\mu(n)$. The network seemed to figure out eventually that $4 \mid n \implies \mu(n) = 0$, and then sometime later that $9 \mid n$ also implies $\mu(n) = 0$. Presumably it would figure out other squares later, eventually.
The Int2Int models were determining whether $n$ was squarefree or not with very high accuracy, and then guessing $\mu(n) = \pm 1$ randomly when it thought $n$ was squarefree. Some of these models were guessing $\mu(n)$ correctly around $60$ percent of the time: far better than chance.
Looking closer, the bestperforming model (which also had the most data: $n, n \bmod p,$ and $(n/p)$ for the first $100$ primes $p$) correctly recognized almost $92$ percent of squareful numbers,^{6} ^{6}Be careful with what is the condition here. In particular it doesn't say that the model computes $\mu(n)$ correctly $92$ percent of the time. but only correctly recognized whether $\mu(n) = \pm 1$ about $40$ percent of the time. Using that the density of squareful numbers is about $0.39$, this gave the overall correctness at \begin{equation*} 0.39 \cdot 0.92 + 0.61 \cdot 0.4 \approx 0.6, \end{equation*} recovering the approximately $60$ percent overall correctness. The model tended to overestimate the number of squareful numbers and guessed that several squarefree numbers were squareful.
This occurred quickly when trained using quadratic residue symbols. I wasn't initially surprised by this because of course Legendre symbols include information about squares. Thus it should be possible to quickly train a network to recognize most squares given $(n/p)$ for the first $100$ primes (most numbers are divisible mostly by small primes, and hence checking small prime behavior usually suffices).
But here we're looking at numbers that are or are not squarefree: multiplying a square by a squarefree number mixes up all the quadratic residues and nonresidues.
With a bit more training, having only $n \bmod p$ for the first $100$ primes produced very similar behavior. How was it doing this?
This is an interesting purely mathematical question: how would you guess whether $n$ is squarefree or not given $n \bmod p$ for lots of primes $p?$
One way would be to perform the Chinese remainder theorem, reconstruct $n$, and then actually check. Is the model recognizing something like this?
To test, I ran several experiments along the following lines:
 Given $(n \bmod p)$ for the first $100$ primes, output if $n$ is in the interval $[10^6, 2 \cdot 10^6]$.
 Given $(n \bmod p)$ for the first $100$ primes but excluding $7$, output $n \bmod 7$.
These probe CRTtype knowledge. I sample input $n$ uniformly at random from large intervals. The frequencies of each residue class should be approximately uniformly randomly distributed.
But the model never did better than random guessing on either of this type of experiment. I guess the model isn't recovering CRTlike information.
I'm also looking to determine behavior mod $p^2$ or $p^3$ using this type of transformer model. This is similar to CRTlike information, but slightly different. I'll talk about this later.
How to guess if $\mu(n) = 0$ given $n \bmod p$
After talking with Noam Elkies and Andrew Sutherland, I think I know how the model is guessing when $\mu(n) = 0$ with such high accuracy. The point is that numbers that are not squarefree are probably divisible by a small square and thus likely to be $0$ mod a small prime. Numbers that are squarefree might be $0$ mod a small prime, but not as often.
Let's look at this in greater detail.
The zeta function associated to squarefree numbers is
\begin{equation} \zeta_{\mathrm{SF}}(s) = \prod_p\Big( 1 + \frac{1}{p^s} \Big) = \zeta(s) / \zeta(2s). \end{equation} Thus the ratio of numbers up to $X$ that are squarefree is about^{7} ^{7}This is tangentially related to my note from yesterday
\begin{equation} \mathrm{Res}_{s = 1} \zeta(s)/\zeta(2s) = 1/\zeta(2) = \frac{6}{\pi^2} \approx 0.6079 \end{equation}
The default algorithm to use would be to guess that every integer is squarefree: this is right just over $60$ percent of the time. We need to do better than that.
The zeta function associated to even squarefree numbers is \begin{equation} \frac{1}{2^s} \prod_{\substack{p \\ p \neq 2}} \Big( 1 + \frac{1}{p^s} \Big) = \frac{1}{2^s} \frac{\zeta^{(2)}(s)}{\zeta^{(2)}(2s)} = \frac{1}{2^s} \frac{(1  1/2^s)}{(1  1/4^s)} \frac{\zeta(s)}{\zeta(2s)}. \end{equation} It follows that the ratio of numbers up to $X$ that are even and squarefree is about \begin{equation} \frac{1}{2} \frac{1/2}{3/4} \frac{6}{\pi^2} = \frac{1}{3} \frac{6}{\pi^2}. \end{equation} This implies that the remaining $\frac{2}{3} \frac{6}{\pi^2} X$ squarefree integers up to $X$ are odd. We could see this directly by noting that the corresponding zeta function is \begin{equation} \prod_{\substack{p \\ p \neq 2}} \Big( 1 + \frac{1}{p^s} \Big) = \frac{(1  1/2^s)}{(1  1/4^s)} \frac{\zeta(s)}{\zeta(2s)}, \end{equation} and computing the residue as $(1/2)/(3/4) \cdot \frac{6}{\pi^2} = \frac{2}{3} \frac{6}{\pi^2}$.
A squarefree integer is twice as likely to be odd as even.
For this classification problem, we're interested in the converse conditional: what is the probability that $n$ is squarefree given that it is even (or odd)? Basic probability shows that \begin{equation} P(\mathrm{sqfree}  \mathrm{even}) = \frac{P(\mathrm{even \; and \; sqfree})}{P(\mathrm{even})} = \frac{\frac{1}{3} \frac{6}{\pi^2}}{\frac{1}{2}} \approx 0.4052 \end{equation} and \begin{equation} P(\mathrm{sqfree}  \mathrm{odd}) = \frac{P(\mathrm{odd \; and \; sqfree})}{P(\mathrm{odd})} = \frac{\frac{2}{3} \frac{6}{\pi^2}}{\frac{1}{2}} \approx 0.8105. \end{equation}
This already gives a betterthannaive strategy: if $n$ is even, guess that it's not squarefree (correct about $1  0.4052 \approx 0.6$ of the time); if $n$ is odd, then guess squarefree (correct about $0.8105$ of the time). This should be correct about $0.5 \cdot (1  0.4052) + 0.5 \cdot (0.8105) \approx 0.7$ (or actually $0.7026423\ldots$) of the time.
As $0.7 > 6/\pi^2$, this type of thinking is an improvement.
This readily generalizes to other primes. The Dirichlet series for squarefree numbers that are divisible by a fixed prime $q$ is \begin{equation}\label{eq:euler} \frac{1}{q^s} \prod_{\substack{p \\ p \neq q}} \Big( 1 + \frac{1}{p^s} \Big) = \frac{1}{q^s} \frac{(1  1/q^s)}{(1  1/q^{2s})} \frac{\zeta(s)}{\zeta(2s)}, \end{equation} and the series for squarefree numbers that aren't divisible by a fixed prime $q$ is the same, but without $q^{s}$. Thus the percentage of integers that are squarefree and divisible by $q$ or not divisible by $q$ are, respectively, \begin{equation}\label{eq:local_densities} \frac{1}{q+1} \frac{6}{\pi^2} \quad \text{and} \quad \frac{q}{q+1} \frac{6}{\pi^2}. \end{equation} Playing conditional probabilities as above shows that \begin{align*} P(\text{sqfree}  \text{qeven}) &= \frac{P(\text{sqfree and qeven})}{P(\text{qeven})} = \frac{q}{q+1} \frac{6}{\pi^2} \\ P(\text{sqfree}  \text{qodd}) &= \frac{P(\text{sqfree and qodd})}{P(\text{qodd})} = \frac{q^2}{q^2  1} \frac{6}{\pi^2}. \end{align*} I use the adhoc shorthand $q$even to mean divisible by $q$, and $q$odd to mean not divisible by $q$.
The differences are the largest when the prime $q$ is small. A good strategy would then be to look at a couple of small primes $q$ and then predict whether $n$ is squarefree based on divisibility rules for the primes $q$.
I've ignored all the joint probabilities. These are explicitly computable by computing the local densities at the appropriate primes, as above. But the point is that divisibility by primes $q$ correlates nontrivially with being squarefree, and this sort of table correlation is something that we should expect machine learning to figure out.
Explicit computation shows that using the first $20$ primes and guessing
squarefree
or not squarefree
based on which divisibility pattern of those
primes is more common yields an overall correct rate of $70.3$ percent, only
$0.1$ percent higher than using $2$ alone.
We can hope that machine learning algorithms could do better. Computing the table of cross correlations given sufficient data isn't hard. But ML models should also determine weights to use to better predict outcomes. Predicting what ML can or can't do is much harder.
Pure squarefree detection
With the same inputs, I looked at guessing $\mu(n)^2$. That is, I tried to look just at the squarefree detection powers.
Overall, the models were correct about $70$ percent of the time. This is consistent with the above behavior and with the heuristic that it could only use mod $2$ information.
Restricting to squarefree $n$
In the other direction, I also restricted all inputs to squarefree $n$. This balances the expected outputs: about $50$ percent each should correspond to $1$ and about $50$ percent should correspond to $1$. Any prediction with accuracy greater than $50$ percent would be a major achievement.
But ultimately none of the models I checked did any better than $50$ percent consistently.
Removing $2$
Still input $n \bmod p$ for $100$ primes, but use the $100$ primes after $2$. As we saw above, $2$ has the most explanatory power using pure Bayesian probability. This asks: is the machine learning doing anything else other than the $2$based cross correlations described above?
In short, the performance plummeted to less than $50$ percent accuracy for guessing $\mu(n)$. The performance was consistent with determining whether $n$ was squarefree correctly about $60$ percent of the time, and then guessing randomly between $+1$ and $1$ when $n$ was determined to be squarefree.
And this is consistent with using the pure Bayesian probabilistic approach on exactly the prime $3$. Indeed, the probability that $n$ is squarefree given that $3$ divides $n$ is $(3/4) \cdot 6/\pi^2 \approx 0.4559$, and the probability that $n$ is squarefree given that $3$ doesn't divide $n$ is $(9/8) \cdot 6/\pi^2 \approx 0.6839$. Thus $1/3$ of the time, we would guess "not squarefree" with accuracy $1  0.4559$ and the rest of the time we would guess "squarefree" with accuracy $0.6839$, giving a total accuracy around \begin{equation*} (1/3) \cdot (1  \tfrac{3}{4} 6/\pi^2) + (2/3) \cdot \tfrac{9}{8} 6/\pi^2 \approx 0.6372. \end{equation*}
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