Zeros of the Eisenstein series 𝐸₂

Abstract

In this paper we investigate the zeros of the Eisenstein series E 2 E_2 . In particular, we prove that E 2 E_2 has infinitely many SL 2 ⁡ ( Z ) \operatorname {SL}_2(\mathbb {Z}) -inequivalent zeros in the upper half-plane H \mathfrak {H} , yet none in the standard fundamental F \mathfrak {F} . Furthermore, we go on to investigate other fundamental regions in the upper half-plane H \mathfrak {H} for which there do or do not exist zeros of E 2 E_2 . We establish infinitely many such regions containing a zero of E 2 E_2 and infinitely many which do not.