On the Signs of Fourier Coefficients of Cusp Forms

Abstract

Let Γ be a discrete subgroup of SL(2, ℝ) with a fundamental region of finite hyperbolic volume. (Then, Γ is a finitely generated Fuchsian group of the first kind.) Let

$$ f\left( z \right) = \sum\limits_{n + \kappa > 0} {a{{\left( n \right)}^{e2\pi i\left( {n + \kappa } \right)z/\lambda }}} ,z \in H. $$

be a nontrivial cusp form, with multiplier system, with respect to Γ. Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon Γ.

Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign.

Theorem. Either the sequence (Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence [Im a(n)}.

In memory of Robert A. Rankin