Harper's beyond square-root conjecture

TL;DR

This paper explores how the Ratios Conjecture could extend Harper's randomization approach to demonstrate nontrivial cancellation of the Liouville function in arithmetic progressions beyond the square-root barrier, linking random matrices and multiplicative functions.

Contribution

It proposes a conjectural extension of Harper's method using the Ratios Conjecture to achieve cancellation results beyond the square-root barrier for the Liouville function.

Findings

Suggests the Ratios Conjecture implies nontrivial cancellation in progressions exceeding the square-root modulus.

Connects random matrix theory with multiplicative number theory.

Provides a framework for extending cancellation results to larger moduli.

Abstract

We explain how the (shifted) Ratios Conjecture for $L(s,\chi)$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of ``beyond square-root cancellation'' for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.