On the zeros of odd weight Eisenstein series

TL;DR

This paper investigates the distribution of zeros of odd weight Eisenstein series across all fundamental domains related to SL_2(Z), providing insights into their complex structure and zero locations.

Contribution

It offers a comprehensive count of zeros of odd weight Eisenstein series in all SL_2(Z)-translates of the fundamental domain, expanding understanding of their zero distribution.

Findings

Zeros are distributed in specific patterns across fundamental domains.

The number of zeros is explicitly counted for all SL_2(Z)-translates.

Results enhance knowledge of Eisenstein series' zero structures.

Abstract

We count the number of zeros of the holomorphic odd weight Eisenstein series in all $\mathrm{SL}_2(\mathbb{Z})$-translates of the standard fundamental domain.