A Generalization of the Goldfeld-Sarnak Estimate on Selberg’s Kloosterman Zeta-function

Wladimir de Azevedo Pribitkin

May 21, 1998

1 Introduction

Let Γ denote a discrete subgroup of SL(2, ℝ) which has a fundamental region with finite hyperbolic area but which is not compact. For convenience we assume

that ∞ is cuspidal and also that Γ contains −I. Let S_{λ} = ( \begin{matrix}1 & λ \\ 0 & 1 \end{matrix} ), λ > 0,

be the minimal translation in Γ and let Γ_{∞} denote the subgroup generated by S_{λ} and −I. It is well-known that Γ supports automorphic forms and that the Fourier coefficients of these forms (and certain automorphic integrals), when expanded at ∞, involve the generalized Kloosterman sum

A_{c}(m, n) = A(c; m, n, k, v, Γ) = ∑_{M∈Γ_{∞}\Γ/Γ_{∞}} \overline{v}(M)e(\frac{m̃a + ñd}{c}),

where M = ( \begin{matrix}a & ∗ \\ c & d \end{matrix} ), c > 0, m̃ = \frac{m+κ}{λ}, ñ = \frac{n+κ}{λ}, and m, n are arbitrary

integers. For brevity we use the usual notation e(z) = e^{2πiz}. Here v is a multipier system (MS) for Γ in the weight k. This means that v is a function from Γ into the unit circle which, for z in the complex upper half-plane, must satisfy the “consistency condition”

v(M_{3})(c_{3}z + d_{3})^{k} = v(M_{1})v(M_{2})(c_{1}M_{2}z + d_{1})^{k}(c_{2}z + d_{2})^{k},

where M_{3} = M_{1}M_{2}, and M_{j} = ( \begin{matrix}∗ & ∗ \\ c_{j} & d_{j} \end{matrix} ), j = 1, 2, 3. We also suppose that

v(−I)(−1)^{k} = 1 (otherwise nontrivial automorphic forms would not exist). The parameter κ is determined from v by v(S_{λ}) = e(κ), 0 ≤ κ < 1. Finally, note that although k can be any real number, we could suppose that 0 ≤ k < 2, because a MS in weight k′ is also a MS in weight k ≡ k′(mod 2).

1991 Mathematics Subject Classification: Primary 11F30; Secondary 11F72, 11L05.

Atle Selberg, who wrote a seminal paper [9] which discusses the problem of estimating the order of magnitude of Fourier coefficients of cusp forms, introduced the Dirichlet series

Z_{m,n}(s) = Z(s; m, n, k, v, Γ) = ∑_{c>0} \frac{A_{c}(m, n)}{c^{2s}},

known today as Selberg’s Kloosterman zeta-function. Observe that Z_{m,n}(s) converges absolutely for σ > 1, s = σ + it, and is analytic in this half-plane. (This is obvious when Γ is a subgroup of SL(2, ℤ) of finite index but is nontrivial for general Γ.) Selberg proved that Z_{m,n}(s) has an analytic continuation to a function which is meromorphic in the whole s-plane.

Good estimates on the growth of Z_{m,n}(s), however, were not obtained until the early 80’s, when Goldfeld and Sarnak [1] jointly established that, for m̃ñ ≠ 0,

Z_{m,n}(s) = O(|(m + 1)(n + 1)| · \frac{|s|^{1/2}}{σ − \frac{1}{2}} ),

for σ > \frac{1}{2} as |t| → ∞. The implied constant depends at most upon k, v and Γ. Actually, both Selberg and Goldfeld-Sarnak assumed that Γ is a subgroup of SL(2, ℤ) of finite index, but (and this was mentioned in [1]) there is no need for this restriction and so we are working in a more general setting.

In this note we provide a minor improvement of the above bound (but not in the s-aspect) and present estimates on |Z_{m,n}(s)| which are valid for certain pole-free lacunary strips throughout the whole s-plane. We achieve the latter at the cost of placing a restriction on v which guarantees that the spectrum of the hyperbolic Laplacian in weight k is discrete.

Theorem Let k be any real number and m, n be any integers.

(a) Assume that m̃ñ ≠ 0. If 0 < ε ≤ \frac{1}{2} and \frac{1}{2} < σ ≤ 1 + ε, |t| ≥ 1, then

Z_{m,n}(s) = O(|(m + κ)(n + κ)|^{1−σ+ε} · \frac{|t|^{1−σ+ε}}{σ − \frac{1}{2}} ),

where the implied constant depends at most upon ε, v and Γ.

(b) Assume that κ ≠ 0 (and also make the analogous assumption on v in case there are other parabolic cusps inequivalent to ∞). If 0 < ε_{j} ≤ 1 and −j + \frac{1}{2} < σ < −j + \frac{3}{2}, |t| ≥ 1, then

Z_{m,n}(s) = O(|(m + κ)(n + κ)|^{2j+ε_{j}} · \frac{|t|^{1/2}}{(σ + j − \frac{1}{2})(−j + \frac{3}{2} − σ)}),

where j = 1, 2, 3, … and the implied constant depends at most upon j, ε_{j}, v and Γ.

For comments about the implied constants and also some refinements see the remarks concluding this paper.

2 Proof of Theorem

As usual (see [9] and [1]) we shall exploit the fact that Z_{m,n}(s) occurs in the Fourier expansion of the nonanalytic Poincaré series

P_{m}(z|s) = P_{m}(z|s; k, v, Γ) = ∑_{M∈Γ_{∞}\Γ} \overline{v}(M)(\frac{cz + d}{|cz + d|})^{−k}e(m̃Mz)[Im(Mz)]^{s}.

Here k is real, m is an integer, z = x + iy ∈ H (the complex upper half-plane)

and M = ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix} ). Observe that P_{m}(z|s) is holomorphic (in s) for σ > 1 and

that

P_{m}(V z|s) = v(V)(\frac{γz + δ}{|γz + δ|})^{k}P_{m}(z|s)

∀ V = ( \begin{matrix}∗ & ∗ \\ γ & δ \end{matrix} ) ∈ Γ. By employing spectral theory Selberg [9] proved that

P_{m}(z|s) has an analytic continuation to a function which is meromorphic in the whole s-plane. In fact, he then used this to extract the aforementioned analytic behavior of Z_{m,n}(s). In particular he showed that the pole sets of both these functions consist of points on the real ray −∞ < σ ≤ 1, points on the vertical lines σ = −j + \frac{1}{2}, j = 0, 1, 2, … and (in case the MS v permits Eisenstein series) points in σ < \frac{1}{2}.

We first consider the case where m̃ > 0 and ñ > 0. Our lemmas are generalizations of those found in [1]. The first lemma, which provides a fundamental identity, shows how the inner product of two nonanalytic Poincaré series that are “out of phase” can give us direct access to Selberg’s Kloosterman zeta-function.

Lemma 1 For m̃ > 0, ñ > 0 and Re(w) − L > σ > 1, L = 0, 1, 2, … we have that

⟨P_{m}(z|s), P_{n}(z|w)⟩ = δ_{m,n} · λ(4πñ)^{1−s−\overline{w}}Γ(s + \overline{w} − 1)

+ (−i)^{k}4^{−\overline{w}+1}π^{s−\overline{w}+1}ñ^{s−\overline{w}}Γ(s + \overline{w} − 1)

× ∑_{p=0}^{L} \frac{(−4π^{2}m̃ñ)^{p}Γ(\overline{w} − s − p)}{p!Γ(s + k/2 + p)Γ(\overline{w} − k/2 − p)} · Z_{m,n}(s + p)

+ R_{m,n}(s|w; L),

where δ_{m,n} is the Kronecker symbol and

R_{m,n}(s|w; L) = ∑_{c>0} \frac{A_{c}(m, n)}{c^{2s}} ∫_{0}^{∞} y^{\overline{w}−s−1}e^{−2πñy}

× ∫_{−∞}^{∞} \frac{(x + i)^{−k}}{(x^{2} + 1)^{s−k/2}} e(−ñyx) ∑_{p=L+1}^{∞} \frac{(\frac{−2πim̃}{c^{2}y(x+i)})p}{p!} dxdy.

Proof: A standard computation which uses unfolding (note that R_{Γ} (resp. R_{Γ_{∞}}) is a fundamental region for Γ (resp. Γ_{∞})) shows that

⟨P_{m}(z|s), P_{n}(z|w)⟩ := ∫ ∫_{R_{Γ}} P_{m}(z|s)\overline{P_{n}(z|w)} \frac{dxdy}{y^{2}}

= ∫ ∫_{R_{Γ∞}} P_{m}(z|s)e(−ñ\overline{z})y\overline{w} \frac{dxdy}{y^{2}}

= δ_{m,n} · λ(4πñ)^{1−s−\overline{w}}Γ(s + \overline{w} − 1)

+ ∑_{c>0} \frac{A_{c}(m, n)}{c^{2s}} ∫_{0}^{∞} ∫_{−∞}^{∞} \frac{y^{\overline{w}−s−1}}{(x^{2} + 1)^{s}} [\frac{x + i}{(x^{2} + 1)^{1/2}} ]^{−k}

× e(\frac{−m̃}{yc^{2}(x + i)} − ñ(y(x − i))) dxdy

= δ_{m,n} · λ(4πñ)^{1−s−\overline{w}}Γ(s + \overline{w} − 1)

+ ∑_{p=0}^{L} \frac{(−2πim̃)^{p}}{p!} ∑_{c>0} \frac{A_{c}(m, n)}{c^{2s+2p}} ∫_{0}^{∞} y^{\overline{w}−s−1−p}e^{−2πñy}

× ∫_{−∞}^{∞} \frac{(x + i)^{−k−p}}{(x^{2} + 1)^{s−k/2}} e(−ñyx) dxdy + R_{m,n}(s|w; L),

where R_{m,n} is defined in the lemma. Now, we know that [5, pp. 208–209]

∫_{−∞}^{∞} \frac{e(−Y x)}{(x + i)^{α}(x − i)^{β}} dx = \frac{π(−i)^{α−β}(πY)^{\frac{α+β}{2} −1}}{Γ(α)} W_{\frac{α−β}{2} , \frac{α+β−1}{2}} (4πY),

for Y > 0, Re(α + β) > 1, and where W_{·,·}(Y) is the Whittaker function that decays exponentially in Y as Y → ∞. So the middle term above equals

4(−i)^{k}ñ^{s−1}π^{s} ∑_{p=0}^{L} \frac{(−2π^{3/2}m̃ñ^{1/2})^{p}}{p!Γ(s + k/2 + p)} · Z_{m,n}(s + p)

× ∫_{0}^{∞} y^{\overline{w}−p/2−2}e^{−2πñy}W_{\frac{k+p}{2} , \frac{2s+p−1}{2}} (4πñy) dy.

The lemma now follows from the formula

∫_{0}^{∞} W_{β,μ}(y)e^{−y/2}y^{s−1} dy = \frac{Γ(s + 1/2 + μ)Γ(s + 1/2 − μ)}{Γ(s − β + 1)},

valid for Re(s + 1/2 ± μ) > 0, which can be found by using the integral representations for W_{β,μ} and the Γ-function (see, e.g., [7, p. 32]). □

The case L = 0 in Lemma 1 was considered by Goldfeld and Sarnak [1]. Note that the factor of λ which immediately follows δ_{m,n} is missing in [1, p. 248]. The next two lemmas establish bounds on R_{m,n} and ||P_{m}||.

Lemma 2 Assume that m̃ > 0 and ñ > 0.

(a) R_{m,n}(s|\overline{s} + 2; 0) is analytic for σ > 0. For ε > 0 and ε ≤ σ ≤ 1 + ε we have that

R_{m,n}(s|\overline{s} + 2; 0) = O(\frac{(m̃ñ)^{1−σ+ε}}{ñ^{2}} ),

where the implied constant depends upon ε and Γ.

(b) R_{m,n}(s|\overline{s}+2j+2; 2j) is analytic for σ > −j, j = 1, 2, 3, … For σ > −j+ \frac{1}{2} and 0 < ε_{j} ≤ 1 we have that

R_{m,n}(s|\overline{s} + 2j + 2; 2j) = O(\begin{matrix} & m̃^{2j+ε_{j}} \\ & ñ^{2−ε_{j}} \end{matrix} ),

where the implied constant depends at most upon j, ε_{j} and Γ.

Proof: The definition of R_{m,n} gives us that

|R_{m,n}(s|w; L)| ≤ ∑_{c>0} \frac{|A_{c}(m, n)|}{c^{2σ}} ∫_{0}^{∞} y^{Re(\overline{w})−σ−1}e^{−2πñy}

× ∫_{−∞}^{∞} \frac{1}{(x^{2} + 1)^{σ}} | ∑_{p=L+1}^{∞} \frac{(\frac{−2πim̃}{c^{2}y(x+i)})p}{p!} | dxdy.

But

|e^{z} − ∑_{p=0}^{L} \frac{z^{p}}{p!} | < B_{r}|z|^{r}, L ≤ r ≤ L + 1,

for Re(z) ≤ 0, z ≠ 0, and where

B_{r} = \{ \begin{matrix}\frac{1}{r!} & if r = L + 1, \\ e + 1 & if L ≤ r < L + 1. \end{matrix}

As an aside we remark that this inequality is a generalized version of the fact that e^{z} is “enveloped” by its Maclaurin series for Re(z) ≤ 0, z ≠ 0. Since m̃ > 0 we can apply the above basic inequality to produce

|R_{m,n}(s|w; L)| ≤ B_{r}(2πm̃)^{r} ∑_{c>0} \frac{|A_{c}(m, n)|}{c^{2σ+2r}} · \frac{Γ(Re(\overline{w}) − σ − r)}{(2πñ)^{Re(\overline{w})−σ−r}}

× \sqrt{π} \frac{Γ(σ + r/2 − 1/2)}{Γ(σ + r/2)}.

To show part (a) we set w = \overline{s} + 2, L = 0, r = 1 − σ + ε, where ε > 0, and use the fact that the series of absolute values associated with Z_{m,n}(1+ε) is O_{ε,Γ}(1). To prove part (b) set w = \overline{s} + 2j + 2, L = 2j, r = 2j + ε_{j}, where 0 < ε_{j} ≤ 1 and again appeal to the absolute convergence properties of Z_{m,n}. The case ε_{j} = 1 establishes analyticity for σ > −j. In this special case the implied constant does not depend upon j. □

Lemma 3 Assume that m̃ > 0.

(a) For 0 < ε ≤ \frac{1}{2} and \frac{3}{2} ≤ σ ≤ B, B ≥ \frac{3}{2}, we have that

||P_{m}(z|s)|| = O(m̃^{1−σ+ε}),

where the implied constant depends at most upon ε, B, v and Γ.

(b) For 0 < ε ≤ \frac{1}{2} and \frac{1}{2} < σ ≤ \frac{3}{2}, |t| ≥ 1, we have that

||P_{m}(z|s)|| = O(\frac{m̃^{1−σ+ε}}{σ − \frac{1}{2}} ),

where the implied constant depends at most upon ε, k, v and Γ.

(c) Assume that κ ≠ 0 (and also make the analogous assumption on v in case there are other parabolic cusps inequivalent to ∞). For 0 < ε_{j} ≤ \frac{1}{2} and −j + \frac{1}{2} < σ < −j + \frac{3}{2}, |t| ≥ 1, we have that

||P_{m}(z|s)|| = O(\frac{m̃^{1−σ+ε_{j}}}{(σ + j − \frac{1}{2})(−j + \frac{3}{2} − σ)}),

for j = 1, 2, 3, … and where the implied constant depends at most upon j, ε_{j}, k, v and Γ.

Proof: To prove part (a) we first observe that

||P_{m}(z|s)||^{2} ≤ ∫_{0}^{∞} ∫_{0}^{λ} |P_{m}(z|s)|e^{−2πm̃y}y^{σ−2} dxdy.

But we also have that

|P_{m}(z|s)| ≤ ∑_{M∈Γ_{∞}\Γ} e^{−2πm̃Im(Mz)}[Im(Mz)]^{σ}

≤ \frac{⌈r⌉!}{(πm̃)^{r}} ∑_{M∈Γ_{∞}\Γ} e^{−πm̃Im(Mz)}[Im(Mz)]^{σ−r},

where r ≥ 0. Here we used our assumption that m̃ > 0 and the simple inequality e^{−u} < \frac{⌈r⌉!}{u^{r}}, u > 0, r ≥ 0. (Note that ⌈·⌉ denotes the “ceiling” function.) The proof of part (a) follows from letting r = σ − 1 − ε, ε > 0, and recalling the fact that the series of absolute values connected with P_{m}(z|1 + ε) is O_{ε,v,Γ}(1) throughout H.

Part (b) is a consequence of part (a) and the recursion relation (first observed by Selberg [9])

P_{m}(z|s) = −4πm̃(s − k/2)R_{s(1−s)}(P_{m}(z|s + 1)), σ > \frac{1}{2}, s ∉ D_{0},

where D_{0} = D_{0}(k, v, Γ) is a finite subset of the real segment \frac{1}{2} < σ ≤ 1. (Actually, this relation is first computed for σ > 1 and then extended by analytic continuation.) Here R_{μ} = (Δ_{k} + μ)^{−1} is the resolvent at μ of

Δ_{k} = y^{2}(\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − iky \frac{∂}{∂x},

the hyperbolic Laplacian in weight k. It is known that Δ_{k} admits a self-adjoint extension to L^{2}(Γ\H, k, v), the Hilbert space of functions f : H → ℂ which satisfy

f(V z) = v(V)(\frac{γz + δ}{|γz + δ|})^{k}f(z)

∀ V = ( \begin{matrix}∗ & ∗ \\ γ & δ \end{matrix} ) ∈ Γ and

||f|| < ∞.

Because Δ_{k} is self-adjoint spectral theory tells us that spectrum(Δ_{k}) ⊂ ℝ and that

||R_{μ}|| ≤ \frac{1}{distance(μ, spectrum(Δ_{k}))},

for μ ∉ spectrum(Δ_{k}). Therefore we get that

||R_{s(1−s)}|| ≤ \frac{1}{|t|(2σ − 1)}

for σ > \frac{1}{2}, t ≠ 0. Combining this with the recursion relation and part (a) (the case B = \frac{5}{2}) we obtain

||P_{m}(z|s)|| = O(\frac{m̃^{1−σ+ε}|s − k/2|}{|t|(σ − \frac{1}{2})}),

for \frac{1}{2} < σ ≤ \frac{3}{2}, t ≠ 0, where the implied constant depends at most upon v, Γ and ε, 0 < ε ≤ 1/2. In the lacunary strip \frac{1}{2} < σ ≤ \frac{3}{2}, |t| ≥ 1 we clearly have that \frac{|s−k/2|}{|t|} = O_{k}(1). This proves part (b).

To show part (c) we need to extend the domain of validity for Selberg’s recursion relation. In fact, because of our assumption on v (and this is crucial) we know that the spectrum of Δ_{k} is discrete in L^{2}(Γ\H, k, v). The absence of continuous spectra means that R_{s(1−s)} has no “bad” singularities (i.e., it has only poles) and analytic continuation gives us that, for j = 1, 2, 3….,

P_{m}(z|s) = −4πm̃(s − k/2)R_{s(1−s)}(P_{m}(z|s + 1)), σ > −j + \frac{1}{2}, s ∉ D_{j},

where D_{j} = D_{j}(k, v, Γ) is the union of a finite set of points on the real segment −j + \frac{1}{2} < σ ≤ 1 and a discrete set of points on the vertical lines σ = −j + \frac{1}{2} + p, p = 1, …, j. From this formula we infer

||P_{m}(z|s)|| ≤ (2π)^{j+1}m̃^{j+1}( ∏_{p=1}^{j+1} \frac{|s + p − 1 − k/2|}{|t||σ + p − \frac{3}{2}|})||P_{m}(z|s + j + 1)||,

for −j+ \frac{1}{2} < σ < −j+ \frac{3}{2}, |t| ≠ 0, where j = 1, 2, 3, … The proof now follows from obvious considerations of (euclidean) distance coupled with part (a). □

We can now finish the proof of the Theorem in the case m̃ > 0 and ñ > 0. Let w = \overline{s} + 2j + 2 and L = 2j, j = 0, 1, 2, … in the identity of Lemma 1 and use analytic continuation to obtain

Z_{m,n}(s) = (−i)^{−k}4^{s+2j+1}π^{2j+1}ñ^{2j+2} \frac{Γ(s + k/2)Γ(s + 2j + 2 − k/2)}{Γ(2j + 2)Γ(2s + 2j + 1)}

× \{⟨P_{m}(z|s), P_{n}(z|\overline{s} + 2j + 2)⟩ − R_{m,n}(s|\overline{s} + 2j + 2; 2j)\}

− \frac{Γ(s + k/2)Γ(s + 2j + 2 − k/2)}{Γ(2j + 2)}\{δ_{m,n} · λ(−i)^{−k}4^{−s}π^{−2s}ñ^{−2s+1}

+ ∑_{p=1}^{2j} \frac{(−4π^{2}m̃ñ)^{p}Γ(2j + 2 − p)}{p!Γ(s + k/2 + p)Γ(s + 2j + 2 − k/2 − p)} · Z_{m,n}(s + p)\},

where σ > −j + \frac{1}{2}, s ∉ D_{j}. The Theorem now follows by induction from the estimates in Lemma 2, Lemma 3 (use the Cauchy-Bunyakovsky-Schwarz inequality), Stirling’s formula and the boundedness of Z_{m,n}(u) for Re(u) ≥ \frac{3}{2}. For part (a) we also invoked the convexity principle of Phragmén-Lindelöf. Note that the estimates in the Theorem do not depend on k because, as was mentioned in our introduction, we can assume that 0 ≤ k < 2.

Because the proof for the case m̃ > 0 and ñ < 0 parallels the one just completed, we shall omit it. We only mention that whereas before we considered ⟨P_{m}(z|s; k, v, Γ), P_{n}(z|w; k, v, Γ)⟩, here it is preferable to examine

⟨P_{m}(z|s; k, v, Γ), \overline{P_{n^{∗}}(z|w; −k, \overline{v}, Γ)}⟩,

where

n^{∗} = \{ \begin{matrix}−n − 1 & if 0 < κ < 1, \\ −n & if κ = 0. \end{matrix}

Recall that κ is the parameter associated with v and note that if v is a MS in weight k, then \overline{v} is a MS in weight −k.

In order to conclude the proof altogether, we must still analyse the case when m̃ < 0. But this is an immediate implication of the case m̃ > 0 coupled with the nice observation that

\overline{Z(\overline{w}; m, n, k, v, Γ)} = \{ \begin{matrix}Z(w; −m − 1, −n − 1, −k, \overline{v}, Γ) & if 0 < κ < 1, \\ Z(w; −m, −n, −k, \overline{v}, Γ) & if κ = 0. \end{matrix}

Remark 1 Although the implied constants in our estimates do depend upon v and Γ, this dependence is at most upon κ, λ and c_{min}, the minimal positive element which occurs in the left column and lower row of a matrix in Γ. In particular, and this is in contrast with the Goldfeld-Sarnak estimate, our implied constants never depend upon Area(R_{Γ}), the hyperbolic area of R_{Γ}. This is clear from our proof of Lemma 3, part (a), and is not surprising because the inner product of two classical holomorphic Poincaré series (which are cusp forms) also does not depend upon Area(R_{Γ}), but does depend upon λ. We are in a generalized situation. Also note that for certain families of groups the area of the fundamental region can grow without bound while λ remains constant. This is the case for the Hecke congruence groups of level N, Γ_{0}(N), N = 1, 2, 3, …, where Area(R_{Γ_{0}(N)}) → ∞ as N → ∞ and yet λ = 1 ∀ N.

Remark 2 We do not claim that our estimates are the best possible in the maspect or the n-aspect. For certain groups and MS the bound for \frac{1}{2} < σ ≤ 1 + ε, |t| ≥ 1 could be improved by making use of stronger estimates on the generalized Kloosterman sum. Moreover, for general Γ one ought to be able to improve the estimate for −j + \frac{1}{2} < σ < −j + \frac{3}{2}, |t| ≥ 1 by replacing the exponent 2j + ε_{j}, 0 < ε_{j} ≤ 1, with j + \frac{1}{2} + ε_{j}, 0 < ε_{j} ≤ \frac{1}{2}, where j = 1, 2, 3, . . .. Although we cannot prove this, we can show (with the same assumption on the MS v stated in part (b) of the Theorem) that for 0 < ε_{j} ≤ \frac{1}{2} and any compact subset S_{j} of σ > −j + \frac{1}{2}, s ∉ D_{j}, we have that

Z_{m,n}(s) = O(|(m + κ)(n + κ)|^{j+1/2+ε_{j}}),

where j = 1, 2, 3, … and the implied constant depends at most upon j, S_{j}, ε_{j}, v and Γ. Our proof of this relies on the fact (stated here for m̃ > 0 and ñ > 0) that R_{m,n}(s|\overline{s} + j + 2; j) is actually analytic for σ > −j, j = 1, 2, 3, … and for 0 < ε_{j} ≤ \frac{1}{2} we have that

R_{m,n}(s|\overline{s} + j + 2; j) = O(\begin{matrix} & m̃^{j+1/2+ε_{j}} \\ & ñ^{3/2−ε_{j}} \end{matrix} ),

in any compact subset T_{j} of σ > −j + \frac{1}{2}, where the implied constant depends at most upon j, T_{j}, ε_{j} and Γ.

Remark 3 If we lift the restriction on the MS v in part (b) of the Theorem, then the situation becomes more delicate. Certainly estimates of the kind stated in part (b) are in general impossible because of the existence of poles which occur to the left of σ = \frac{1}{2} but are not in D_{j}. That is, they are in sets of the form −j + \frac{1}{2} < σ < −j + \frac{3}{2}, t ≠ 0, where j = 1, 2, 3, . . .. Nonetheless, with suitable restrictions on s (and perhaps on Γ), there ought to be estimates. We hope to return to this problem at a later time.

Remark 4 The estimate in part (b) can be used to find explicit formulas for the Fourier coefficients of modular forms (and certain modular integrals) on SL(2, ℤ) of weight k, 0 < k ≤ 1, and arbitrary MS v. We shall present this in a forthcoming article.

Remark 5 For a nice discussion of the Goldfeld-Sarnak estimate (in the case where Γ is the full modular group and A_{c}(m, n) is the classical Kloosterman sum) as well as references to applications see Sarnak’s book [8, pp. 33–41]. For a rather comprehensive proof of this estimate for general Γ see Hejhal’s volume [3, pp. 665–709]. For more information on the spectral theory of automorphic forms refer to the two tomes by Hejhal [2, 3] and the more recent books by Venkov [10] and Iwaniec [4]. Lastly, for the pretty notion of enveloping series consult the classic by Pólya and Szegö [6, pp. 32–36].

Acknowledgement: The author wishes to thank the Max Planck Institute for Mathematics in Bonn, Germany, for providing support as well as a wonderful working environment.

References

[1] D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent. Math. 71 (1983), pp. 243–250.

[2] D. Hejhal, The Selberg Trace Formula for P SL(2, ℝ) Volume 1, Springer Lecture Notes in Math. 548, Springer-Verlag, Berlin Heidelberg New York 1976.

[3] —–, The Selberg Trace Formula for P SL(2, ℝ) Volume 2, Springer Lecture Notes in Math. 1001, Springer-Verlag, Berlin Heidelberg New York Tokyo 1983.

[4] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Revista Matemática Iberoamericana 1995.

[5] H. Maass, Lectures on Modular Functions of One Complex Variable, Springer-Verlag (Distributed for the Tata Institute of Fundamental Research), Berlin Heidelberg New York Tokyo, revised ed. 1983.

[6] G. Pólya and G. Szegö, Problems and Theorems in Analysis I, Die Grundlehren der Mathematischen Wissenschaften 193, Springer-Verlag, Berlin Heidelberg New York 1978.

[7] W. Pribitkin, The Fourier Coefficients of Modular Forms and Modular Integrals Having Small Positive Weight, Doctoral Dissertation, Temple University, Philadelphia, 1995.

[8] P. Sarnak, Some Applications of Modular Forms, Cambridge Tracts in Math. 99, Cambridge University Press, Cambridge 1990.

[9] A. Selberg, On the estimation of Fourier coefficients of modular forms, Theory of Numbers: Proc. Sympos. Pure Math. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 1–15.

[10] A. B. Venkov, Spectral Theory of Automorphic Forms and its Applications, Soviet Series 51, Kluwer Academic Pub., Dordrecht 1990.

Max-Planck-Institut für Mathematik Gottfried-Claren-Str. 26 53225 Bonn Germany

E-mail: pribitki@mpim-bonn.mpg.de; w pribitkin@msn.com