Eisenstein series and an asymptotic for the K-Bessel function

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 0le 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 + 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 le r le 3/2.