A note on the sup norm of Eisenstein series

TL;DR

This paper establishes a new upper bound on the maximum size of real-analytic Eisenstein series in the upper half-plane, improving understanding of their growth behavior as the spectral parameter T increases.

Contribution

It provides a uniform sup norm bound for Eisenstein series of the form $E(z, 1/2 + iT) \

Findings

Sup norm bound: $E(z, 1/2 + iT) \\ll T^{3/8 + \\\varepsilon}$

Bound holds uniformly for $z$ in fixed compact subsets of $\\\mathbb{H}$

Advances understanding of Eisenstein series growth

Abstract

We prove a sup norm bound on the real-analytic Eisenstein series, of the form $E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$, uniformly for $z$ in a fixed compact subset of $\mathbb{H}$.