Eisenstein series and an asymptotic for the K-Bessel function

Abstract

We produce an estimate for the K-Bessel function Kr+it(y) with positive, real argument y and of large complex order r + it where r is bounded and t = y sin θ for a fixed parameter 0 ≤ θ ≤ π/2 or t = y cosh µ for a fixed parameter µ > 0. In particular, we compute the dominant term of the asymptotic expansion of Kr+it(y) as y → ∞. 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 (j) 0 (z, r + it) for each inequivalent cusp κj when 1/2 ≤ r ≤ 3/2.