A note on the extended Bruinier-Kohnen conjecture
Mohammed Amin Amri, M'hammed Ziane

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 be a cuspform of integral half-weight , whose Fourier coefficients not necessarily real. We verify partially an extension of a conjecture of Bruinier and Kohnen on the equi-distribution of the signs of (when are real), conjectured by the first author in \cite{Amri2} for the sequence , where an odd positive integer and a square-free integer.
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