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

Abstract

We produce an estimate for the $K$-Bessel function $K_{r+it}(y)$ 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 $0\le \theta \le \pi /2$ or $t=y\cosh \mu$ for a fixed parameter $\mu >0$. In particular, we compute the dominant term of the asymptotic expansion of $K_{r+it}(y)$ as $y\to \infty$. 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 $1/2\le r\le 3/2$.

20 pages. The bounds for the Eisenstein series have been extended to all of $y>0$. Error terms for all the estimates have been added. Accepted version. To appear in the Ramanujan Journal