Arithmetic functions and learning theory

TL;DR

This paper links number theory and learning theory by showing the M"obius function is statistically hard to learn, requiring many samples, and establishes lower bounds on its learnability based on Fourier analysis.

Contribution

It introduces a novel connection between analytic number theory and computational learning theory, providing lower bounds on the learnability of the M"obius function.

Findings

The M"obius function has a Fourier Ratio lower bound of R^{-1/4-ε}.

The class of functions with high Fourier Ratio has VC dimension at least proportional to R.

Any distribution-independent learning algorithm needs at least linear samples to learn the M"obius function class.

Abstract

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the restriction of the M\"obius function to the squarefree integers in $\{1,\dots,R\}$. Using a recent lower bound of Pandey and Radziwi{\l}{\l} for the $L^1$ norm of exponential sums with M\"obius coefficients, we prove that \[ \FR(\mu_R) \gg R^{-1/4-\epsilon} \] for every $\epsilon>0$. We then show that, for a suitable absolute constant $c_0>0$, the class of $\{-1,1\}$-valued functions on the squarefree integers with Fourier Ratio at least $c_0$ has Vapnik--Chervonenkis dimension at least $cR$. It follows that any distribution-independent learning algorithm that succeeds uniformly on the class $\mathcal{H}_R(\eta_R)$ containing $\mu_R$, where $\eta_R \to 0$, requires at least $\Omega(R)$ samples. We also discuss a conditional improvement under a strong uniform bound for additive twists of the M\"obius function, and we note that the same method applies to the Liouville function.