A short note on the Bruiner–Kohnen sign equidistribution conjecture and Halász’ theorem

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 Halász' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.Mathematics Subject Classification (2010): 11F37 (Forms of half-integer weight; nonholomorphic modular forms); 11F30 (Fourier coefficients of automorphic forms).

A note on the extended Bruinier-Kohnen conjecture

“…Moreover, they conjectured that the signs of the sequence of Fourier coefficients of a cusp form which lies in the Kohnen's plus space are equidistributed. Through this problem in [8,9,4] Arias, Inam and Wiese proved the Bruinier-Kohnen conjecture in the special case when the Fourier coefficients of a Hecke eigenforms of half-integral weight are indexed by tn 2 with t a fixed square-free number and n ∈ N. In [1], the results of [8,4] were generalized to Hecke eigenforms with not necessarily real Fourier coefficients, based on this and empirical evidence the first author generalized the Bruinier-Kohnen conjecture to cusp forms in Kohnen's plus space with not necessarily real Fourier coefficients a(n). More precisely, he conjectured that…”

On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture

“…This gives us confidence in our belief that prime indexed normalised coefficients are not distributed differently than those with squarefree indices. Combining the Shimura lift with the (proved) celebrated Sato-Tate Conjecture for integral weight Hecke eigenforms, it is not very difficult to prove equidistribution of signs for the coefficients indexed by squares, see [IW13], [AdRIW15], [IW16]. Note that this is only a partial result and the full proof of the conjecture is still an open problem and, for the moment, it is likely that there is no theoretical tool to attack this problem.…”

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

“…They also showed with Arias-de-Reyna in [11], that (a(tn 2 )) n∈N are equidistributed when F t has CM and the equidistribution was reformulated in both CM and not CM cases using Dedekind-Dirichlet and natural densities. Later, those results were obtained in [6] by removing the error term assumption.…”