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

Let $\Gamma$ be a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$ such that $\Gamma \backslash \mathbb{H}$ has at least one cusp of width one. We give a bound on the Fourier coefficients of the weight-zero Eisenstein series $E_0^{(j)}(z, r+it)$ for $1/2<r \leq 3/2$, which extends the known bound for $r=1/2$. The proof relies on our main result, namely a suitable estimate for the Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded. In particular, we compute the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y \rightarrow \infty$.


Introduction
Let G := PSL 2 (R), Γ ⊂ G be a cofinite Fuchsian group, and H be the upper-half plane model of the hyperbolic plane (i.e. with the Poincare metric). The group G acts transitively on the left of H via Möbius transformations and, moreover, these actions are orientationpreserving isometries. We assume that there exists at least one cusp of width one, which we can take to be the standard cusp κ 1 := ∞. Let Γ 1 := Γ ∞ be the stabilizer of κ 1 . Moreover, in general, there are a finite number of inequivalent cusps {κ j } q j=1 ⊂ R ∪ {∞} with stabilizer Γ j . For each cusp, we choose σ j ∈ G such that σ j (κ j ) = ∞, namely taking the cusp κ j into the standard cusp. (We always choose σ 1 to be the identity.) Note that σ j is not in Γ for any j ∈ {2, · · · , q}. By modifying σ j for j ∈ {2, · · · , q}, we can ensure that holds for all j ∈ {1, · · · , q} and for all B ≥ B 0 > 1 (see [10, (2.2)] or [7,Page 268]). Here B 0 is a fixed constant depending only on Γ. Let us denote the j-th cuspidal region in F by C j,B : And define the bounded region of F by We will study Eisenstein series. Recall that these are important automorphic functions because they are eigenfunctions of the non-Euclidean Laplacian (i.e. the operator D := y 2 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 ). For background on Eisenstein series, see [8] for example. Let z := x + iy, s ∈ C. There is an Eisenstein series E (j) (z, s) of weight 0 for each inequivalent cusp: The Fourier expansion is the following (see [9,Lemma 2.6] or [7, Page 280] for example): ψ n,j (r + it) √ yK r−1/2+it (2π|n|y)e 2πinx .
Remark 1.4. Since y sinh µ = t 2 − y 2 and µ = cosh −1 t y hold, our result, in the special case of purely imaginary order, reduces to the standard result for purely imaginary order (see [6,Page 88 (19)] for example), namely: as y → ∞.
For completeness, we also give a result for small y: where the implied constant depends only t 0 and is uniformly bounded for all large enough t 0 .
Remark 1.6. Here, t 0 ≥ 1 is chosen to be a fixed large constant (large enough to use the first term in the Stirling asymptotic series for the gamma function for the approximation in the proof of the proposition below).
As an application of our main result, we give a bound on the Fourier coefficients of the Eisenstein series for large enough |t|: where the implied constant depends only on the lattice subgroup Γ and t 0 .
Remark 1.8. Note that [10,Proposition 4.1] gives the result for the case of r = 1 2 . Here, ω(t) denotes the spectral majorant function whose properties are ω(−R) = ω(R) ≥ 1 and 1.2. Outline of paper. Section 2 is devoted to the proof of the main results, Theorems 1.1 and 1.3. Section 3 gives a proof of Proposition 1.5. Finally, Section 4 gives a proof Theorem 1.7.
2. Bounds for K ν (y) where ℑ(ν) large, ℜ(ν) bounded, and y is real and positive There are two related methods, sometimes called the same, that could allow us to obtain the bounds we require. We will use what E. T. Copson [4,Chapter 8] calls the saddle point method, which is the simpler of these methods, to obtain dominant behavior and thus the required bounds. 2 For background on asymptotic expansions, see [4] (especially Chapter 7) for example.
The saddle points and paths of steepest descent for the function K it (y) (i.e. purely imaginary order) have been obtained by N. M. Temme [11]. The saddle points and paths of steepest descent for our function K ν (y) are the same as we now show. In addition, we give a proof of the dominant behavior.
In this section (Section 2), let us set where r, t ∈ R. An integral representation for K ν (z) (see [12, Page 182 (7)] for example) is There are two cases: y ≥ t ≥ 0 and 0 < y ≤ t. Remark 2.1. Note that if t < 0, then applying (2.1) allows us to be in one of these two cases.
2 Going further, one may be able to obtain uniform asymptotic expansions or fast and accurate computation using what Copson [4,Chapter 7] calls the method of steepest descents. For ν purely imaginary, these exist ( [2], [3]).
Using the Cauchy-Goursat theorem, we can replace the integral along the real axis from (2.1) with an integral along the path of steepest descent though the saddle point R 0 . Doing so, we obtain the following integral representation: whenever y > t > 0 holds. (We will show below that this integral representation for K ν (y) is also valid for the exceptional cases t = 0 and t = y.) Consider the neighborhood of the saddle point R 0 along the path of steepest descent determined by |u| < min( cot θ 2 , cot 4 θ 16 , θ 4 ). (Note that 0 < cot θ < ∞ as 0 < θ < π/2.) Using the Taylor series for sinh(x), 1/(1 + x) and the geometric series, we have that for u in our small neigbhorhood. Here κ := 1 6 and the implied constant is less than 1/2. Thus, for our small neighborhood. Here the implied constant less than 10 9 sin 2 θ. Using the Taylor series for √ 1 + x, we have that for u in our small neighborhood. Here the implied constant is less than 1/2. Note that, since that 0 < w ≤ θ, cos w ≥ 0.
By the mean value theorem, we have that there exists w ∈ (w, θ) such that for our small neighborhood. Note that cos θ ≤ cos w < 1. Here the implied constant is less than sin θ 2 cos w ≤ 1 2 tan θ. Pick 1/3 < δ < 1/2 and let We now estimate the integral in (2.3) around a small neighborhood of the saddle point R 0 along the path of steepest descent determined by |u| < y −δ where y ≥ y 0 . As we are interested in the behavior of K ν (y) as y → ∞, we may assume that y is large. Using the proceeding estimates and the Taylor series for cosh(x), we have e −y(cosh u cos w+w sin θ) = e −y(cos θ+θ sin θ) e −yAu 2 e −yBu 3 = e −y(cos θ+θ sin θ) e −yAu 2 (1 + O(y 1−3δ )) and over our small neighborhood. Here A = 1 2 cos θ + κ sin θ tan θ − κ sin 2 θ cos w > 0 and |B| < 11/12. Note that since we are on a small neighborhood around 0, the approximation e x = 1 + O(x) holds. Thus, we have (Note that 3δ − 1 < δ for our choice of δ.) Pick an ε > 0 and set A := A 0 1+ε . Following the method in Copson [4, Chapter 8, (36.6) ff], namely changing variables u 2 → y Au 2 and estimating, we obtain Similarly, we have (Note that it is important for δ < 1/2 here.) Then, for every ε > 0, there exists y 1 > 0 such that, whenever y ≥ y 1 , we have that Taking the limit as y → ∞ and noting that ε is arbitrary, we have that  [3, (14)] in the special case of purely imaginary order. We now consider the rest of the integral, which we now show is negligible. We will first integrate along the contour ℓ + := {u + iw(y −δ ) : u ≥ y −δ } and then use the Cauchy-Goursat theorem. The details are as follows. Let The integral along this contour is given by Note that the integrand comes from (2.2). Now we have that Here we have used the fact that cosh u ≥ 1 + u 2 /2 and applied (2.6, 2.7). Since κ sin θ tan θ − κ sin 2 θ/ cos w ≥ 0 and cos w ≥ cos θ, we have that where the second inequality follows from (2.9), the equality from changing variables yu 2 cos θ/4 → u 2 , and the final inequality from a standard bound for erfc(x) (see [1, 7.1.13] for example). This shows that, for large y, |C| is negligible compared to the dominant behavior that we computed in (2.8). Now let us integrate over the contour ℓ − (U ) : When U is large enough, we have that is a simple closed contour and the Cauchy-Goursat theorem implies that the integral over c(U ) is negligible compared to (2.8). Letting U → ∞ shows that the integral over the piece of the path of steepest descent for which u ≥ y −δ is also negligible compared to (2.8). For the integral over the remaining piece of the path of steepest descent, we note that w(u), cosh u are even functions and that the analogous proof also shows that it is negligible compared to (2.8). This proves the desired result in the case 0 < θ < π/2. We now prove the case θ = 0. The proof is a simplification of the previous case. The details are as follows. The integral representation for this case is because t = 0. Pick 1/3 < δ < 1/2 and let y 0 := 2 1/δ . As in the previous case, the dominant behavior of the integral comes from a small neighborhood around the origin, namely |u| < y −δ where y ≥ y 0 . On this neighborhood, cosh u = 1 + u 2 /2 + O(u 4 ). Let us consider this contribution first: The final equality is obtained, as in the case 0 < θ < π/2, by changing variables u 2 → y 2 u 2 and estimating erf(x) = 1 − erfc(x) with the standard bound for erfc(x).
Away from this neighborhood, the integral is negligible. Let We have that For the rest of the integral over −∞ to −y −δ , we obtain a similar bound. This gives the desired result for the case θ = 0. We now prove the final case of θ = π/2. Unlike in the previous two cases, here the saddle points R k are no longer of order 1 but are, instead, of order 2 (see [5,Page 40] for the definition of order). 3 It still suffices to consider the saddle point R 0 = iπ/2. We now have that We remark that w ′ (u) is a bounded, odd function. It has a jump discontinuity at u = 0.
Pick 1/4 < δ < 1/3. Consider the neighborhood of the saddle point R 0 along the path of steepest descent determined by |u| < 1/4. We have for u in our small neighborhood. Here κ = 1 6 , λ := 7 320 , and the implied constant is less than 1/2. Using this and the Taylor series for √ 1 + x, we obtain where λ := 21 200 and the implied constant is less than 2. (Note that the precise value of λ does not come into the computation of the dominant behavior.) Now using the Taylor series for arccos(x), we have and taking the derivative with respect to u yields Using the Taylor series for cosh(x), we have where the third equality follows from changing variables u → −u.
Consequently, we have where in the second-to-last equality we have changed variables u → u 1/3 and in last equality we have used a standard estimate for the incomplete gamma function Γ(a, x) (see [1, 6.5.32] for example). Consequently we have that , which agrees with [12,Pages 78,247] and [3, (14)] in the special case of purely imaginary order.
We now show that the rest of the integral is negligible. As in the case 0 < θ < π/2, we will use the Cauchy-Goursat theorem and the contours given by C and D. Let 3 y −δ because the implied constant has norm less than 2. Applying (2.13), we have that 72 √ 3 and 1−δ 2 > 1 3 , we have that |C| is negligible compared to the dominant behavior that we computed.
For D, we have the following estimate: which, when U is large enough, is negligible compared to the dominant behavior that we computed. Note that cos W ≥ cos(w(y −δ )). The integral over the remaining piece of the path of steepest descent is handled in a manner analogous to the 0 < θ < π/2 case. This proves the desired result in all cases.
We start by finding the saddle points and suitable integral representations. Using (2.1), we have The saddle points (values of R for which φ ′ (R) = 0) are as follows [11] (see also [3, Section 2.1] ): Let us now write R = u + iw and thus we have The paths of steepest descent/ascent through the saddle points R ± k is given by ℑ(−φ(R)) = ℑ(−φ(R ± k )) and is the following family of curves [11]: We use only the parts of these curves as shown in [11, Figure 3.3], which we will refer to as the path of steepest descent. Notice that this path is the union of two branches L − ∪ L + , separated by the imaginary axis, where -L − runs from − ∞ to 0 and from 0 to + i∞, -L + runs from + i∞ to 0 and from 0 to + ∞.
What is important about this path is that, on both of the branches, the function yφ(R) has constant imaginary part, namely for L + and L − , respectively.
Using the Cauchy-Goursat theorem, we can replace the integral along the real axis from (2.1) with an integral along the path of steepest descent: Let us first show that we can obtain the integral representation: Remark 2.3. In the special case of purely imaginary order, (2.16) reduces to [11, (3.5)] and the proof of (2.16) is similar to that in [11].
Proof. To begin the proof of (2.16), first note that Also, u(w) is two-valued, one for each branch. Let us, for clarity, temporarily use u + (w) to denote the value on L + and u − (w) to denote the value on L − , then we have that Now let us change variables u → −u on the branch L − so that all integrals will be over only the branch L + : We now have two integrals with respect to dw, and these are both integrated over a piece of L + . Moreover, these integrals can both be reduced to integrals over a finite interval. Given P (w) := e −y cosh u cos w+ru du dw + i Q(w) := e −y cosh u cos w−ru − du dw + i , we have that P (w) = P (w + 2π) and Q(w) = Q(w + 2π) because u(w) = u(w + 2π) holds over the bounds of integration of the integrals with respect to dw. Consequently, using the geometric series (which is valid because t > 0), we have that Combining all of this gives (2.16), as desired. Now take the piece of L + from w = 3 2 π to w = 5 2 π (inclusive of the endpoints) and shift it down the vertical axis by 2π. Call this shifted piece L 0 . Note that L 0 does not meet L − and meets L + only at one point (namely, the saddle point R + 0 ). Although w(u) is multiple-valued on L + ∪ L 0 , we note that w(u) is single-valued on L 0 and on the piece of L + corresponding to 0 ≤ w ≤ 3π/2.
Let µ − := u(−π/2) = u(3π/2). For the task of computing the dominant behavior, we need to modify the integral representation (2.16) in two ways. The first is to shift the integrals from 3 2 π to 5 2 π by −2π (and this piece of the integrals will be over L 0 ) and the second is to convert some of the integrals with respect to w to integrals with respect to u. Proposition 2.4. We have the following integral representation: e −yψ(u) e ru e irw du dw + i dw.
Remark 2.5. Note that only one saddle point (namely R + 0 ) appears in this integral representation.
Proof. Changing variables w → w − 2π, we have that Now note that w(u) is a bijection on L 0 and on L + , and, thus, on each of these pieces, we may change our integrals with respect to w to integrals with respect to u.
By calculus, we have that µ − is the minimum value of u on L + ∪ L 0 . In particular, we have that 0 < µ − < µ. Hence, using the substitution theorem, we have that The desired result now follows.
Proof of Theorem 1.3. The integral representation in Proposition 2.4 is particularly suited to the saddle point method. In each of the integrals from the proposition, we now change variables, letting ρ := u − µ, and estimate the integrals near ρ = 0. First note that we have sin w = sin(w(ρ + µ)) = ρ cosh µ + sinh µ sinh ρ cosh µ + cosh ρ sinh µ .
Choose ρ 0 > 0 small enough so we may apply the geometric series in the second equality of (2.18). Let |ρ| ≤ ρ 0 . Using the Taylor series for sinh x and cosh x, we have Note that sinh µ = 0 because µ > 0. Now choose ρ 1 > 0 small enough so we may apply the Taylor series for the function √ 1 + x in the third equality of (2.19). Let us further restrict ρ by requiring that |ρ| ≤ min(ρ 0 , ρ 1 ) holds. We have Now choose ρ 2 > 0 small enough so that Here the implied constant is the same as the implied constant in (2.19). Let again restrict ρ by requiring that |ρ| ≤ min(ρ 0 , ρ 1 , ρ 2 ) holds.
For clarity, let us define the following: and let ψ 0 (u) := cosh u cos w 0 + w 0 cosh µ ψ + (u) := cosh u cos w + + w + cosh µ denote ψ(u) on on L 0 and on L + corresponding to 0 ≤ w ≤ 3π/2, respectively. Note that on L + , for small enough δ > 0, we have cos (w + (δ + µ)) < 0 if δ < 0 cos (w + (δ + µ)) > 0 if δ > 0 , and, thus, we have cos w + = cos(w + (ρ + µ)) = ρ + 1 2 On L 0 , for small enough δ > 0, we have cos w 0 > 0 if δ < 0, and, thus, we have Note that the implied constants in the expressions for cos w + and cos w 0 are the same. Note that ρ ≤ 0 on L 0 . Now choose ρ 3 > 0 small enough so we may apply the Taylor series for the function arccos(x) in the expressions for cos w + and cos w 0 . Let us again restrict ρ by requiring that |ρ| ≤ min(ρ 0 , ρ 1 , ρ 2 , ρ 3 ) holds. We have Consequently, using the above and the Taylor series for cosh(x) and sinh(x), we have that ψ + (ρ + µ) = cosh ρ cosh µ cos w + + sinh ρ sinh µ cos w + + w + cosh µ = cosh µ ρ + 1 2 and, analogously, Let us now consider the integrals in (2.17). Applying the change of variables ρ = u − µ, we have that Pick 1/3 < δ < 1/2 and let y −δ 0 = min(ρ 0 , ρ 1 , ρ 2 , ρ 3 , µ − µ − , 1/2). Pick y 1 > 0 such that holds for all y ≥ y 1 . Here, the implied constant is the same as that for cos w + above. We now estimate the integrals in a small neighborhood of the saddle point R + 0 on L + determined by |ρ| < y −δ where y ≥ max(y 0 , y 1 ). Using the estimates above, we simplify in manner similar that in Section 2.1 to obtain where in the last equality we have changed variables ρ 2 → y(sinh µ)ρ 2 and applied the standard bounds for erfc(x). Likewise, we have We now compute the dominant term, namely the following: π y sinh µ e −y π 2 cosh µ+ir π 2 (cosh(iχ − rµ) + i sinh(iχ − rµ)) = π y sinh µ e −y π 2 cosh µ+ir π 2 [cosh(rµ) (cos χ − sin χ) − i sinh(rµ) (cos χ + sin χ)] = 2π y sinh µ e −y π 2 cosh µ+ir π 2 cosh(rµ) sin Here, the last equality follows from the elementary observation To prove that (2.24) is the dominant term, we now show that the rest is negligible. As in Section 2.1, we shall pick a more suitable contour and use the Cauchy-Goursat theorem. Let us consider the integral and replace the piece of the path of steepest descent that the integral is over with the contours ℓ 1 ∪ ℓ 2 where The integral over ℓ 1 is where the integrand comes from (2.14). Changing variables, ρ = u − µ, we have Thus, as −y −δ ≥ µ − − µ, we have Here the first inequality follows because cos w + (µ − y −δ ) < 0 and the second from the approximations we computed above and the fact that (2.20) holds. By the mean value theorem, there exists µ − y −δ < µ < µ such that cosh(µ − y −δ ) = cosh µ − y −δ sinh µ. Since 0 < µ − ≤ µ − y −δ holds, we have that sinh µ > 0. Consequently, we have the following upper bound: As 1 − 2δ > 0, this shows that A 1 is negligible compared to the dominant term, as desired. The integral over ℓ 2 is Thus, as 3π/2 > w + (µ − y −δ ) > π/2, we have where the second inequality follows from the fact that π/2 − θ ≤ cos θ for all θ ≥ π/2. This shows that A 2 and, thus, A are negligible compared to the dominant term, as desired. Let us consider the integral for any U ≥ U 0 . The integral over ℓ 4 is where the integrand comes from (2.14). Changing variables, ρ = u − µ, we have, for all U ≥ U 0 , where the third inequality follows because 0 < w + (µ + y −δ ) < π/2, cosh ρ ≥ 1, and sinh ρ ≥ ρ for ρ > 0, where the fourth inequality follows from our approximations above, and where the fifth and sixth inequalities follow from (2.23). This shows the integral over C 1 (U ) is negligible compared to the dominant term for all U ≥ U 0 , as desired. The integral over ℓ 5 is where the integrand comes from (2.14). We have where the second inequality follows from the observation that 0 < cos w + (µ + y −δ ) ≤ cos U over the bounds of integration. This shows that the integral over C 2 (U ) is negligible compared to the dominant term for all large enough U , as desired. Consequently, the integral over C is negligible compared to the dominant term, as desired.
For the remaining integrals, we note that we have changed variables u → −u the integrals over L − to arrive at the integral representation in Proposition 2.2. We have already estimated above the integral around (−µ, π/2) along L − . Using the Cauchy-Goursat theorem, we now replace the remaining with integrals along the following contours The integrals over these contours are, respectively where each of the integrands comes from (2.14). Changing variables u → −u, using the observations that w(x) and cosh x are both even functions, and applying the analogous proofs we used to show that A 1 , A 2 , B, C 1 , C 2 , respectively, are negligible compared to the dominant term, we have that respectively, are negligible compared to the dominant term, as desired. This proves the theorem.
3. Bounds for K r−1/2+it (y) for 0 < y < 1 and 1/2 ≤ r ≤ 3/2 Finally, for completeness, we give an estimate of K r−1/2+it (y) for small positive real argument. Recall that we pick t 0 ≥ 1 to be a fixed large constant (large enough to use the first term in the Stirling asymptotic series for the gamma function for the approximation below).
4. The proof of Theorem 1.7 Using our result on the asymptotics of the K-Bessel function, we now bound the sum of the c n .