Eisenstein series and an asymptotic for the K-Bessel function

Abstract

We produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ 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 $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

An asymptotic for the K-Bessel function using the saddle-point method

“…The original proof uses Laplace's method, which is a powerful tool but does not yield much insight. Our proof, unlike that in [19], explains, using elementary tools, how these asymptotics come about. We will find a suitable integral representation of the K-Bessel function and the relevant saddle point and will show that the integral is dominated by its restriction to a small (suitable) neighborhood of the saddle point and is negligible outside of this neighborhood.…”

“…The K-Bessel function (see (2.1) for the definition) is one of these variants and it, in particular, has important applications in analytic number theory and ergodic theory, especially as it appears in the Fourier expansion of eigenforms of the non-Euclidean Laplacian, such as Hecke-Maass forms and Eisenstein series, (see [10,Chapter 3] and [21] for example). Understanding the K-Bessel function and, in particular, its asymptotics will have applications in science and mathematics such as, for example, computing bounds on the Eisenstein series [19,Theorem 1.12] (see also [17]).…”

“…In this paper, we give a proof of these asymptotics, first proved in [19,Theorems 1.1 and 1.3]. The original proof uses Laplace's method, which is a powerful tool but does not yield much insight.…”

“…Let ν := r+it. The asymptotics of the K-Bessel function that we compute using our illustrative method are stated for the two cases y ≥ t ≥ 0 and 0 < y < t in Theorems 1.1 and 1.2, respectively, and were first proved in [19] ([19, Theorems 1.1 and 1.3], respectively). When t < 0, see Remark 3.1.…”

“…For a comparison of Theorems 1.1 and 1.2 with related results in the literature, see [19,Remarks 1.2 and 1.4]. In [19,Remark 1.4], note that = should replace ∼ and that the righthand side of the displayed expression should be multiplied by (1 + o(1)). Also, there are some related uniform results [2,19].…”