On (not) learning the M\"obius function

TL;DR

This paper establishes fundamental lower bounds on the ability of various standard learning algorithms to learn the M"obius and Liouville functions, linking these bounds to digital character correlations and prime number theorems.

Contribution

It provides the first quantitative bounds on learning these functions using kernel, gradient, and statistical query methods, connecting number theory and learning theory.

Findings

Lower bounds on learning M"obius with kernel and gradient methods

Correlation bounds relate to digital characters of finite abelian groups

Results connect to digital prime number theorems

Abstract

We prove lower bounds on learning the M\"obius or Liouville function with a variety of standard learning techniques, including kernel methods, noisy gradient methods, and correlational statistical query algorithms. These results follow from quantitative bounds on the correlation of M\"obius with digital characters of various finite abelian groups, where the group is dictated by the type of input data the algorithm is given. Using residues mod $p$ for many different primes corresponds to a cyclic group, and using the base $p$ expansion for a fixed prime corresponds to an elementary abelian $p$-group. We also note that lower bounds of this form are closely related to certain types of digital prime number theorems.