Sign changes of the Eisenstein series on the critical line

TL;DR

This paper establishes a quantitative lower bound on the number of nodal domains of Eisenstein series on the critical line, using a novel restricted QUE theorem with shrinking test function support.

Contribution

It introduces a new quantitative restricted QUE theorem that enables analysis of nodal domains of Eisenstein series with variable support.

Findings

Proves a lower bound on nodal domains of Eisenstein series.

Develops a restricted QUE theorem with shrinking support.

Provides new insights into the structure of Eisenstein series on the critical line.

Abstract

We prove a quantitative lower bound on the number of nodal domains of the real-analytic Eisenstein series. The main tool in the proof is a quantitative restricted QUE theorem where the support of the test function is allowed to shrink with the Laplace eigenvalue.