# Eisenstein series and an asymptotic for the $K$-Bessel function

## TL;DR

This paper derives asymptotic estimates for the $K$-Bessel function with large complex order and applies these results to bound Eisenstein series in the context of automorphic forms.

## Contribution

It provides new asymptotic formulas for the $K$-Bessel function of large complex order and applies these to obtain bounds on Eisenstein series for certain ranges of parameters.

## Key findings

- Asymptotic expansion of $K_{r + i t}(y)$ as $y o
abla$
- Uniform estimates when $t$ and $y$ are close
- Bounds on Eisenstein series for $1/2 \,\leq r \leq 3/2$

## Abstract

We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y \cosh \mu$ for a fixed parameter $\mu>0$. In particular, we compute the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y \rightarrow \infty$. When $t$ and $y$ are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $E_0^{(j)}(z, r+it)$ for each inequivalent cusp $\kappa_j$ when $1/2 \leq r \leq 3/2$.