Well lacunary series and modular forms of weight one

Abstract

A series f=∑n=0∞a(n)qn is lacunary if the set of n for which a(n)=0 has density 1. We say f is well lacunary if f is lacunary and a(n) assumes every integer value infinitely often. A well-known theorem of Deligne and Serre states that each modular form of weight one is lacunary. In this paper, we show that each modular form of weight one is well lacunary provided that certain special values can be attained. We also construct a family of well lacunary series via binary quadratic forms.