ON THE ESTIMATION OF EIGENVALUES OF HECKE OPERATORS

M. RAM MURTY

Dedicated to the Memory of E.G. Straus and R.A. Smith

1. Introduction

Let A denote the adele ring of the rational numbers ℚ and suppose that we have a cuspidal automorphic representation π of GL_{2}(A). (For the terminology and the details, we refer the reader to Gelbart [?] and Jacquet and Langlands [5]). Langlands [9] has described how one can attach an L-function to π. To describe this construction briefly, one can associate to π a family of local representations π_{p} for each prime p of ℚ. This family is uniquely determined by π such that

(i) π_{p} is irreducible for every p, (ii) for all but finitely many primes p, π_{p} is unramified (that is, the restriction of

π_{p} to GL_{2}(ℤ_{p}) contains the identity representation exactly once), (iii) π can be factored as the restricted infinite tensor product π = ⊗_{p}π_{p}.

Let S denote the set of primes p for which π_{p} is ramified. For p ∉ S, it is known that π_{p} corresponds canonically to a semisimple conjugacy class σ_{p} in GL_{2}(ℂ), where σ_{p} contains a matrix of the form

|\begin{matrix}α_{p} & 0 \\ 0 & β_{p} \end{matrix}| .

If r denotes any finite dimensional complex representation of GL_{2}(ℂ), one can attach an L-series L(s, π, r) as follows.

L(s, π, r) = ∏_{p} L(s, π_{p}, r),

where

L(s, π_{p}, r) = det(1 − r(σ_{p})p^{−s})^{−1}

whenever π_{p} is unramified. If π_{p} is ramified, and r is standard, we refer to J-L [5] for the definition of L(s, π_{p}, r). It is known [9] that each L(s, π, r) converges in some half-plane and we conjecture that it has an analytic continuation and satisfies a suitable functional equation. We also have the Ramanujan-Petersson conjecture

|α_{p}| = |β_{p}| = 1.

If r is ρ_{2}, the standard representation, we shall simply write L(s, π) for L(s, π, ρ_{2}). In fact, L(s, π) has classical origins. L(s, π) is the Mellin transform of either a cusp form or a Maass wave form on Γ_{0}(N), for some natural number N.

More precisely, let

f(z) = ∑_{n=1}^{∞} a(n)e^{2πinz}

be a cusp form on Γ_{0}(N) of integral weight k and Nebentypus χ (mod N). Suppose further that f is a normalised eigenfunction for all the Hecke operators. Then there exists a cuspidal automorphic representation π_{f} of GL_{2}(A) such that

L(s, π_{f}) = (2π)^{−(s+k−1/2)}Γ (s + \frac{k − 1}{2}) ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} ,

where a_{n} = n^{−(k−1)/2}a(n). Moreover, for this π_{f}, we have

α_{p}β_{p} = χ(p) and α_{p} + β_{p} = a_{p}.

The Ramanujan-Petersson conjecture in this case of holomorphic forms was settled by Deligne [2].

Now, let f be a real analytic modular form on Γ_{0}(N) with character χ (mod N). That is, f satisfies f((az + b)(cz + d)) = χ(d)f(z). A Maass form (of weight 0) is an eigenfunction of the non-Euclidean Laplacian,

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) ,

which also satisfies polynomial growth conditions at each cusp of Γ_{0}(N)\𝔥, where 𝔥 denotes the upper half plane. Let f, satisfying Δf = λf, be an eigenfunction of the Hecke operators. Such an f necessarily has a Fourier expansion of the form

f(z) = ∑_{n≠0} a_{n}\sqrt{y}K_{ir}(4π|n|y)e^{2πinx},

where K_{s}(u) = ∫_{0}^{∞} e^{−y/2(u+1/u)}u^{s−1} du, and λ = 1/4 + r^{2} is the eigenvalue. The Ramanujan-Petersson conjecture for Maass forms then reads

|a_{n}| ≤ |a_{1}| · d(n),

where d(n) is the number of divisors of n. This conjecture is still open. As a consequence of recent developments, Serre has observed that one can show

a_{n} = O(n^{1/5}d(n)).

In connection with the eigenvalue λ, one knows that r is purely real, or purely imaginary. We have the conjecture (Selberg)

λ ≥ \frac{1}{4}.

Selberg showed [13] that λ ≥ 3/15. In [9], Langlands viewed the Selberg conjecture as the Ramanujan conjecture at the infinite prime and showed how one can deduce both of these conjectures if we had an analytic continuation of L(s, π, r_{m}), where r_{m} denotes the m-th symmetric power of the standard embedding ρ_{2} of GL_{2}(ℂ) into GL_{2}(ℂ). We describe Langlands conjectural approach below.

2. Upper Bounds for Eigenvalues

If m = 2, 3 or 4, it is known that L(s, π, r_{m}) has a meromorphic continuation to the entire complex plane and satisfies a suitable functional equation. The case m = 2 (for which L(s, π, r_{2}) is in fact proved to be entire) is due to Shimura [15], when π is associated to a holomorphic modular form, and the general case for m = 2 is due to Gelbart and Jacquet [4]. Shahidi [14] recently settled the cases of m = 3 and 4. These results are valid over any field and not just over ℚ.

The Euler factors of L(s, π, r_{m}) are easy to describe. We have, for unramified π_{p},

L(s, π_{p}, r_{m}) = ∏_{j=0}^{m} (1 − \frac{α_{p}^{m−j}β_{p}^{j}}{p^{s}} )^{−1} .

From the analytic properties of L(s, π, r_{m}), we would like to infer the size of the α_{p}.

The philosophy behind this idea can be revealed by the following simple observation. Suppose a_{n} ≥ 0 and f(s) = ∑_{n=1}^{∞} a_{n}/n^{s} represents an analytic function for Re s > 1. Then, it is almost immediate that a_{n} = O(n^{1+ε}). To see this, we first note that

e^{−1/x} = \frac{1}{2πi} ∫_{2−i∞}^{2+i∞} Γ (s)x^{s} ds.

Therefore

∑_{n=1}^{∞} a_{n}e^{−n/x} = \frac{1}{2πi} ∫_{2−i∞}^{2+i∞} Γ (s)f(s)x^{s} ds.

Moving the line of integration to σ = 1 + ε, we obtain ∑_{n=1}^{∞} a_{n}e^{−n/x} = O(x^{1+ε}), as the Γ -function decays exponentially as t → ∞ in any half-plane and f(s) is bounded for σ ≥ 1 + ε. It follows that a_{n} = O(n^{1+ε}).

We remark that this result cannot be substantially improved. Indeed, consider

ζ(ks − k + 1) = ∑_{n=1}^{∞} \frac{n^{k−1}}{n^{ks}} .

This series converges for Re s > 1 and a_{n} = Ω(n^{1−1/k}). As k can be arbitrarily large, we find that there exist series with a_{n} ≥ 0 and a_{n} = Ω(n^{1−ε}).

Now, suppose L(s, π, r_{m} ⊗ \overline{r}_{m}) is analytic for Re s > 1, where \overline{r}_{m} denotes the contragredient representation of r_{m}. Then it follows from the above that

|∑_{j=0}^{m} α_{p}^{m−j}β_{p}^{j}|^{2} = O(p^{1+ε}).

Since |α_{p}β_{p}| = 1, we must have one of |α_{p}| or |β_{p}|, bounded. Suppose without loss, β_{p} = O(1). Then, we see that

α_{p} = O(p^{1/2m+ε})

follows from the preceding discussion. In an analogous fashion, considering L(s, π_{p}, r_{m}⊗ \overline{r}_{m}) when p = ∞, we find that |Re r| ≤ 1/m as L(s, π_{p}, r_{m} ⊗ \overline{r}_{m}) is analytic for Re s > 1. Letting m → ∞, we deduce both the Ramanujan-Petersson conjecture and the Selberg conjecture. These considerations are due to Langlands [9]. Analogous ideas were utilised by Deligne [2] in his proof of the Ramanujan-Petersson conjecture for the case of holomorphic forms. In that case, the p-Euler factor can be identified as an L-function of a certain algebraic variety over the rational function field over the finite field of p-elements. The necessary analytical properties are deduced by the etale cohomology theory of such varieties.

It is a result of Jacquet, Piatetskii-Shapiro and Shalika (see H. Jacquet, From GL_{2} to GL_{n}, 1975 U.S.-Japan Seminar on Number Theory, Ann Arbor) that L(s, π, r_{2} ⊗ \overline{r}_{2}) has an analytic continuation to the entire complex plane except for a simple pole at s = 1 arising from ζ(s). By our earlier remarks, it follows that a_{p} = O(p^{1/4+ε}) and that λ ≥ 3/16.

Utilising a general lemma on Dirichlet series with functional equations, one can improve the above exponent of 1/4 to 1/5, as was noticed by Serre. Such a lemma was first proved by Landau [7] and generalized by Chandrasekharan and Narasimhan [1]. We state a specialization of their theorem for future reference.

Lemma 1. Let a_{n} ≥ 0 and set f(s) = ∑_{n=1}^{∞} a_{n}/n^{s}. Suppose f(s) is convergent in some half-plane and that it has analytic continuation, except for a simple pole at s = 1 of order k, to the entire complex plane and it satisfies a functional equation

c^{s}Δ(s)f(s) = c^{1−s}Δ(1 − s)f(1 − s)

where c is a certain positive constant and Δ(s) = ∏_{i=1}^{N} Γ (α_{i}s + β_{i}). Then

∑_{n≤x} a_{n} = xP_{k−1}(log x) + O(x^{(2A−1)(2A+1)} log^{k−1} x),

where A = ∑_{i=1}^{N} α_{i}, and P_{k−1} is a polynomial of degree k − 1.

Remark . If, instead of a_{n} ≥ 0, we assume that ∑_{x≤n≤x′}|a_{n}| = O(x^{θ}), then the conclusion of the lemma still holds with an error term of

O(x^{(2A−1)(2A+1)} log^{k−1} x + x^{θ}).

(See [1] or [7] for a proof.)

It is known that if we set Γ (s, π, r_{m}) to be the product of the Γ -factors appearing in L(x, π, r_{m}), then

Γ (s, π, r_{m}) = ∏_{j=0}^{m} π^{−(s−λ_{j})/2}Γ (\frac{s − λ_{j}}{2} ) ,

where λ_{j} are certain real numbers. If π corresponds to a Maass wave form with eigenvalue λ = 1/4 + r^{2}, then λ_{j} = i(m − 2j)r in the above product. With this definition of the Γ -factors, we have the functional equation

L(s, π, r_{m}) = L(1 − s, π, \overline{r}_{m}),

where \overline{r}_{m} is the contragredient of r_{m}, for m = 2, 3 and 4.

Applying the above lemma to the Dirichlet series f(s) = L(s, π, r_{2} ⊗ \overline{r}_{2}), we find that

|a_{p}| ≤ p^{1/5} + p^{−1/5}

since f(s) has a simple pole at s = 1.

We can ask if there is a corresponding improvement at the infinite prime. That is, can we obtain a better lower bound for λ_{1}?

To this end, let us define

L_{m}(s) = ∏_{p≠∞} L(s, π_{p}, r_{m})

and write

L_{2}(s)L_{4}(s) = ∑_{1}^{∞} \frac{a_{n}}{n^{s}} ,

for Re s > 1. We have the following Theorem 1. The Dirichlet series ∑_{1}^{∞} a_{n}/n^{s} converges for Re s > 9/11 and therefore the function

g(s) = L_{2}(s)L_{4}(s)

has an analytic continuation for Re s > 9/11. If g(s) has no real zeroes for s > 9/11, then we have λ_{1} ≥ 403/1936 = .208 . . ..

Remark . If, in addition, we assume the Ramanujan-Petersson conjecture, we obtain λ_{1} > .21.

Proof. We have that if

f(s) = ζ(s)g(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} ,

then a_{n} ≥ 0 and

∑_{n≤x} a_{n} = cx + O(x^{4/5}),

by our earlier remarks. We would like to apply Lemma 1 to g(s) and obtain information concering partial sums of the coefficients. In view of the remark following Lemma 1, we need to obtain an estimate for

∑_{x<n<x+x^{4/5}}|c_{n}|.

Since c_{n} = ∑_{d | n} μ(d)a_{n/d}, where μ is the familiar Möbius function, we have the majorization

|c_{n}| ≤ ∑_{d | n} a_{d}.

Applying Lemma 1 to the series ζ(s)f(s), we find that

∑_{x<n<x+x^{4/5}}|c_{n}| = O(x^{9/11}).

Therefore, applying Lemma 1 to g(s) yields

∑_{n≤x} c_{n} = O(x^{9/11}).

This establishes that the Dirichlet series defined by g(s) converges for Re s > 9/11 and hence defines an analytic function there. If now, we suppose that g(s) has no real zeroes in this half-plane, then

\frac{L(s, π, r_{2} ⊗ \overline{r}_{2})}{ζ(s)g(s)} = Γ (s, π, r_{2} ⊗ \overline{r}_{2})

is analytic for Re s > 9/11. Therefore, the Γ -factors are analytic there. It follows that

|Re r| ≤ \frac{9}{11},

and the corresponding lower bound for λ_{1} follows from this.

3. Ω-Theorems for Fourier Coefficients

We begin with the following lemma.

Lemma 2. Suppose that

(1) f(s) is an analytic function for Re s ≥ 1, except for a simple pole at = 1;

(2) log f(s) = ∑_{n=1}^{∞} c_{n}/n^{s} with c_{n} ≥ 0. Then f(1 + it) ≠ 0 for any real t.

Proof. Suppose f(1 + it) = 0. Then we consider the classical combination

g(s) = f^{3}(s)f^{4}(s + it)f(s + 2it),

and note that g(s) has a zero at s = 1. Then, as usual, log g(s) is a Dirichlet series with non-negative coefficients and so log g(σ) ≥ 0, for σ > 1. Thus, |g(σ)| ≥ 1, for σ > 1. Letting σ → 1^{+} gives a contradiction.

We apply the lemma to the function f(s) = ζ(s)L_{2}(s)L_{4}(s). Since L(s, π, r_{2} ⊗ \overline{r}_{2}) = Γ (s, π, r_{2} ⊗ \overline{r}_{2})f(s), we find that f(s) has a simple pole at s = 1 and is analytic for Re s ≥ 1, s ≠ 1. By the lemma, it does not vanish on the line σ = 1. The Wiener-Ikehara Tauberian theorem imples

Theorem 2. ∑_{p≤x}|a_{n}|^{4} log p = (2 + o(1))x, as x → ∞.

Proof. From the Tauberian theorem, we have

∑_{p≤x}|α_{p}^{2} + α_{p}β_{p} + β_{p}^{2}|^{2} = (1 + o(1))x/ log x.

Now, a_{p} = α_{p} + β_{p} and so

|a_{p}|^{4} = |α_{p}^{2} + 2α_{p}β_{p} + β_{p}^{2}|^{2}
= |α_{p}^{2} + α_{p}β_{p} + β_{p}^{2}|^{2} + 1 + 2 Re(\overline{χ}(p)(α_{p}^{2} + β_{p}^{2} + α_{p}β_{p})).

Now, L(s, π, r_{2} ⊗ \overline{χ}) and L(s, π, \overline{r}_{2} ⊗ χ) do not vanish on the line σ = 1 since ζ(s)L(s, π, r_{2} ⊗ \overline{χ}) has non-negative coefficients in its defining Dirichlet series and an easy application of Lemma 2 gives the non-vanishing result. It therefore follows, by the Tauberian theorem, that

∑_{p≤x} Re(\overline{χ}(p)(α_{p}^{2} + α_{p}β_{p} + β_{p}^{2})) = o(x/ log x)

as x → ∞. Combining this with our earlier derivation gives us the desired result.

Corollary 1. For a positive proportion of the primes, |a_{n}| ≥ 1.189 . . ..

Corollary 2. a_{n} = Ω(exp(c log n)/(log log n)) for some constant c > 0.

Proof. As a_{n} is multiplicative, we set N to be the product of the primes p ≤ x such that |a_{p}| ≥ 1.189. Then, by a familiar argument,

|a_{N}| ≥ exp (\frac{c log N}{log log N} ) ,

for some c > 0, which proves the result.

Remarks . A version of Theorem 2 was proved for holomorphic modular forms in [11], without Nebentypus. The non-vanishing of L_{4}(s) on σ = 1 and the Ω-result for the Fourier coefficients was established there only in this case also.

Corollary 2 has an interesting application to modular forms of half integral weight.

Corollary 3. Let g(z) = ∑_{1}^{∞} c_{n}e^{2πinz} be a modular form of weight k + 1/2 for Γ_{0}(4N), with character χ and k an integer ≥ 3. Then

c_{n} = Ω(n^{k/2−1/4} exp(c log n/ log log n)),

for some c > 0.

Proof. Shimura and Niwa have constructed a lifting of cusp forms of weight k +1/2 on Γ_{0}(4N), with character χ, to cusp forms of weight 2k on Γ_{0}(2N), with character χ^{2}, for each squarefree integer N. If g is an eigenfunction of T_{k+1/2}(p^{2}), then defining, for every D,

c_{D}a_{n} = ∑_{d | n} d^{k−1}c_{n^{2}D/d^{2}} (\frac{D}{d} )

we find f(z) = ∑_{1}^{∞} a_{n}e^{2πinz} is a modular form of weight 2k. If the stated Ω-result is false, then we would get

|c_{n}| ≤ εn^{k/2−1/4} exp(c log n/ log log n)

for all n sufficiently large. Substituting into the above formula yields

|a_{n}| = εn^{k−1/2} exp(c log n/ log log n) ∑_{d | n} d^{−1/2}.

The last sum is O(exp(c(log n)^{1/2})) and, therefore, the above estimate contradicts Corollary 2. This completes the proof.

4. The Selberg-Linnik Conjecture

Let us define the Kloostermann sum

S(n, m, c) = ∑_{\begin{matrix}a( mod c) \\ a\overline{a}≡1(c) \end{matrix}} e^{2πi/c(ma+n\overline{a})}.

The estimate

|S(n, m, c)| ≤ d(c)c^{1/2}(m, n, c)^{1/2}

is well-known and due to Weil [17]. Many problems in additive number theory reduce to estimating sums of the type

G(x) = ∑_{c≤x} \frac{S(m, n, c)}{c}.

One expects considerable cancellation to occur in the sum. Selberg [13] and Linnik [10] independently conjectured that for x ≥ (m, n)^{1/2+ε},

∑_{c≤x} \frac{S(m, n, c)}{c} = O(x^{ε}).

Our Ω-result for the Fourier coefficients gives us that G(x) = Ω(exp(c log x)/(log log x)) for some c > 0. This was proved in [11].

Selberg’s original motivation for this conjecture was that it could yield a proof of the Ramanujan-Petersson conjecture in the holomorphic case corresponding to the full modular group. With this in view, we prove the following result, which was certainly known to Selberg.

Theorem 3. Suppose that

f(z) = ∑_{x=1}^{∞} a(n)e^{2πinz}

is a modular form of weight k for the full modular group. If G(x) = O(x^{β+ε}), then

a(n) = O(n^{k−1+β/2+ε}).

Remark . If in particular β = 0, then the Ramanujan conjecture follows. It was shown by Kuznetsov [6] that G(x) ≪_{ε,m,n} x^{1/6+ε}, but, unfortunately, the constant depends on m and n.

Proof. Let us define the t-th Poincaré series

G_{t}(z) = \frac{1}{2} ∑_{(c,d)=1}(cz + d)^{−k}e^{2πit(az+b)/(cz+d)},

for 1 ≤ t ≤ r, where r is the dimension of the space of cusp forms of weight k for the full modular group. It is a result of Petersson that G_{1}(z), …, G_{r}(z) span this space of cusp forms. Expanding G_{t}(z) in a Fourier series, we find that the n-th Fourier coefficient of G_{t}(z) is given by

(\frac{n}{t})^{(k−1)/2} \{δ_{tn} + π ∑_{c=1}^{∞} \frac{S(t, n, c)}{c}J_{k−1} (\frac{4π\sqrt{nt}}{c})\} ,

where δ_{tn} is the Kronecker delta function and J_{k} is the Bessel function defined by

J_{k}(z) = \frac{(z/2)^{k}}{\sqrt{π}Γ (k + \frac{1}{2})} ∫_{0}^{∞} sin^{2k} θ cos(z cos θ) dθ.

An easy partial summation reveals that

H(x) = ∑_{c≤x} S(n, m, c) = O(x^{1+β+ε}).

In estimating the Fourier coefficient, it suffices to estimate

∑_{c=1}^{∞} \frac{S(t, n, c)}{c}J_{k−1} (\frac{4π\sqrt{nt}}{c}) .

We easily find that

∑_{c>\sqrt{n}} \frac{S(t, n, c)}{c}J_{k−1} (\frac{4π\sqrt{nt}}{c}) = ∑_{c>\sqrt{n}} G(c) (J_{k−1} (\frac{4π\sqrt{nt}}{c + 1}) − J_{k−1} (\frac{4π\sqrt{nt}}{c}))

≪ \sqrt{n} ∑_{c>\sqrt{n}} \frac{G(c)}{c^{2}}J′_{k−1}(ξ_{c}).

for some ξ_{c} ∈ (4π\sqrt{nt}/(c + 1), 4π\sqrt{nt}/c). Since J′_{k−1}(x) ≪ x^{−1/2}, we find that the above sum is

≪ n^{1/4} ∑_{n>\sqrt{n}} \frac{|G(c)|}{c^{3/2}} ≪ n^{(β+ε)/2}.

It remains only to show that

∑_{c≤\sqrt{n}} \frac{S(t, n, c)}{c}J_{k−1} (\frac{4π\sqrt{nt}}{c}) = O(n^{β+ε/2}).

We show this by induction on k. For k = 10, there are no cusp forms of weight 10, therefore the estimate is valid in view of our earlier bound. By induction, we must therefore show

∑_{c≤\sqrt{n}} \frac{S(t, n, c)}{c} (J_{k+1} (\frac{4π\sqrt{nt}}{c}) + J_{k−1} (\frac{4π\sqrt{nt}}{c})) = O(n^{(β+ε)/2}).

In view of the identity

\frac{2kJ_{k}(x)}{x} = J_{k+1}(x) + J_{k−1}(x),

it suffices to estimate

\frac{1}{\sqrt{n}} ∑_{c≤\sqrt{n}} S(t, n, c)J_{k} (\frac{4π\sqrt{nt}}{c}) .

By partial summation and the estimate J′_{k}(x) ≪ x^{−1/2}, we find that the above sum is

≪ \frac{1}{n^{1/4}} ∑_{c≤\sqrt{n}} \frac{|H(c)|}{c^{3/2}} ≪ n^{(β+ε)/2},

as desired.

Remark . It would be of interest to have an alternate proof of the above which avoids the induction step, for this would have an application to other congruence subgroups of the modular group. In connection with λ_{1}, Selberg [13] has shown that the Dirichlet series ∑_{c=1}^{∞} S(m, n, c)/c^{2} admits an analytic continuation to Re s > σ_{0} > 1/2 if and only if λ_{1} ≥ σ_{0}(1 − σ_{0}). Weil’s estimate therefore gives λ_{1} ≥ 3/16.

5. Concluding Remarks and Conjectures

In this section, we explore the possibility of improving the error terms in Lemma 1. Certainly this would lead to better extimates for the eigenvalues. But, as we shall discover, the method has limitations.

Suppose f(s) is an analytic function for Re s > 0, except for a pole of order k at s = 1, and that, for Re s > 1, it is given by the Dirichlet series f(s) = ∑_{n=1}^{∞} a_{n}/n^{s}. Define

ν(σ) = inf {ν : ∫_{T}^{2T} |f(σ + it)|^{2} dt = O(T^{1+ν})} .

Theorem 4. Suppose that, for some σ > 0, ν(σ) > 0. Let

E(x) = ∑_{n≤x} a_{n} − xP_{k−1}(log x)

be o(x) for a suitable polynomial of degree k − 1. Then

E(x) = Ω(x^{σ}/ log x).

Proof. It is clear that

E(x) = \frac{1}{2π} ∫_{−∞}^{∞} \frac{f(σ + it)}{σ + it}x^{σ+it} dt,

for any σ > 0. By Parseval’s formula,

∫_{0}^{∞} (\frac{E(x)}{x^{σ+1/2}} )^{2} dx = \frac{1}{2π} ∫_{−∞}^{∞} |\frac{f(σ + it)}{σ + it}|^{2} dt.

If E(x) = O(x^{σ}/ log x), then

\frac{1}{2π} ∫_{T}^{2T} |\frac{f(σ + it)}{σ + it}|^{2} dt < ∞,

for all T. Therefore, ν(σ) ≤ 1, contrary to our assumption. This completes the proof.

Now, suppose that f(s) satisfies a suitable functional equation, i.e., c^{s}Δ(s)f(s) = c^{1−s}Δ(1 − s)f(1 − s), where Δ(s) = ∏_{i=1}^{N} Γ (α_{i}s + β_{j}). By Stirling’s formula and the functional equation, we deduce

ν(σ) = p(1 − 2σ) + ν(1 − σ),

where p = ∑_{j=1}^{N} α_{j}. Thus, we find p(1 − 2σ) > 1 is σ < (p − 1)/2p, and so ν((p − 1)/2p) > 1. It follows therefore from the theorem that

E(x) = Ω(x^{(p−1)/2p}/ log x).

In analogy with the classical divisor and circle problems, we conjecture

E(x) = O(x^{(p−1)/2p+ε}).

This would lead to an estimate of a_{p} = O(p^{1/9+ε}), for the eigenvalues.

The above theorem, together with Lemma 1, easily yields the following corollary. In the holomorphic case, (1) was proved by Rankin [12] and (2) was proved by Walfisz [16].

Corollary 4. (1) ∑_{n≤x} a_{n} = O(x^{1/3+ε}), and

(2) ∑_{n≤x} a_{n} = Ω(x^{1/4−ε})

for every ε > 0.

Indeed, it is easy to see that

\frac{E(e^{u+δ}) − E(e^{u})}{e^{uδ}} = \frac{1}{2πi} ∫_{(c)} \frac{f(s)}{s}(e^{δs} − 1)e^{uit} ds.

Parseval’s formula gives that

∫_{0}^{∞} (\frac{E(e^{u+δ}) − E(e^{u})}{e^{uδ}})^{2} du = \frac{1}{2π} ∫_{−∞}^{∞} |\frac{f(σ + it)}{σ + it}|^{2} |e^{δ(σ+it)} − 1|^{2} dt.

Utilising the fact that for δ = log(1 + 1/T), we have, for |t| ≤ T, the estimate |e^{δ(σ+it)} − 1| ≤ |t|/T, we easily find, from the above formula,

∫_{0}^{∞} (\frac{E(e^{u+δ}) − E(e^{u})}{e^{uδ}})^{2} du ≪ T^{−1+ν(σ)+ε} + ∑_{k=1}^{∞} ∫_{2^{k}T}^{2^{k+1}T} |\frac{f(σ + it)}{σ + it}|^{2} dt
≪ T^{−1+ν(σ)+ε}.

If we let S(x) = ∑_{n≤x} a_{n}, we deduce that

∫_{0}^{∞} (S (y + \frac{y}{T} ) − S(y) − \frac{y}{T} )^{2} y^{−2σ−1} dy ≪ T^{−1+ν(σ)+ε}.

Choosing T = n and considering only the interval [n, n + 1] in the integral immediately gives that a_{n} ≪ n^{σ+ν(σ)/2+ε}. In particular, if we assume the mean value conjecture that ν(σ) = 0, for σ ≥ 1/2, then we deduce that a_{n} = O(n^{1/2+ε}). For the eigenvalues, we find, on the mean value conjecture, for the L-function L(s, π, r_{2} ⊗ \overline{r}_{2}), that a_{p} = O(p^{1/8+ε}). Under this general situation, one should be able to show that ν(σ) = 0, for σ ≥ p/(p + 1).

In the case of the eigenvalues, it is not difficult to see that for the corresponding L-function, we indeed have that ν(σ) = 0, for σ ≥ (p − 1)/(p + 1). This fact equally implies the Serre bound of p^{1/5}. Consequently, any improvement in the range of vanishing of the ν-function, yields a corresponding improvement in the eigenvalue bound.

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Department of Mathematics, McGill University, Montreal, Quebec, H3A 2K6