Arithmetic functions and learning theory

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.