A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Hal\'asz' Theorem

TL;DR

This paper advances the understanding of the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular forms by leveraging Halász's Theorem and Serre's results to remove unproved assumptions.

Contribution

It improves previous results on the conjecture's natural density and eliminates unverified assumptions using advanced number theory tools.

Findings

Established improved bounds on sign equidistribution

Removed all unproved assumptions from previous results

Applied Halász' Theorem and Serre's results effectively

Abstract

In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Hal\'asz' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.