A note on the extended Bruinier-Kohnen conjecture

Abstract

Let f be a cusp form of half-integral weight k + 1/2, whose Fourier coefficients a(n) not necessarily real. We prove an extension of the Bruinier-Kohnen conjecture on the equidistribution of the signs of a(n) for the families {a(tp 2ν )} p,prime , where ν and t be fixed odd positive integer and square-free integer respectively.

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

“…The study of sign changes of Fourier coefficients of a single cusp form has been the focus of much recent study (cf. [1,3,4,8,7,13,10]) due to their various number theoretic applications (see for instance [5]) and his long history which goes back at least to Siegel [17]. The question of simultaneous sign change of Fourier coefficients of modular forms was first considered by Kohnen and Sengupta in [9].…”