A note on the extended Bruinier-Kohnen conjecture

TL;DR

This paper investigates the sign distribution of Fourier coefficients of half-weight cusp forms, extending a conjecture on their equi-distribution, with partial verification for specific prime-related sequences.

Contribution

It provides a partial verification of an extended Bruinier-Kohnen conjecture on sign equi-distribution for Fourier coefficients of half-weight cusp forms.

Findings

Partial verification of the conjecture for prime-related Fourier coefficients.

Extension of the conjecture to sequences involving prime powers.

Insights into sign distribution patterns of Fourier coefficients.

Abstract

Let $f$ be a cuspform of integral half-weight $k+1/2$, whose Fourier coefficients $a(n)$ not necessarily real. We verify partially an extension of a conjecture of Bruinier and Kohnen on the equi-distribution of the signs of $a(n)$ (when are real), conjectured by the first author in \cite{Amri2} for the sequence $\{a(tp^{2\nu})\}_{p,\text{prime}}$, where $\nu$ an odd positive integer and $t$ a square-free integer.