Universality of Hecke L-functions in the Grossencharacter-aspect

Hidehiko Mishou

(Nagoya University) Shin-ya Koyama

(Keio University)

Abstract: We consider the Hecke L-function L(s, λ^{m}) of the imaginary quadratic field ℚ(i) with the m-th Grossencharacter λ^{m}. We obtain the universality property of L(s, λ^{m}) as both m and t = Im(s) go to infinity.

Key words: Hecke L-function, universality of zeta functions, Grossencharacter, imaginary quadratic field,

2000 Mathematics Subject Classification: Primary 11M41, Secondary 19R42

1 Introduction

Voronin [V] discovered the universality property of the Riemann zeta function in 1975, which is stated as follows:

Voronin’s Theorem Let C be a compact subset of the strip \{s = σ+iτ ∈ ℂ | \frac{1}{2} < σ < 1\} with connected complement. Let f(s) be a non-vanishing continuous function on C which is analytic in the interior of C. Then for any ε > 0,

\underline{lim}_{T →∞} \frac{μ(\{t ∈ [0, T] | sup_{s∈C}|ζ(s + it) − f(s)| < ε\})}{T}
> 0,

where μ is the Lebesgue measure on ℝ.

This result was extended to various zeta functions. The first author proved it for Hecke L-functions with ideal class characters [M1] and for those with Grossencharacters [M2]. The universality properties are also generalized to various aspects of zeta functions. Recently Nagoshi proved them for automorphic L-functions of GL(2) in the aspect where their weight or level of the cusp forms grows [N1]. Nagoshi also generalized it to Maass cusp forms for GL(2) in the aspect of the Laplace eigenvalues [N2].

In this paper we deal with the Hecke L-functions L(s, λ^{m}) of ℚ(i) with Grossencharacters λ^{m} (m ∈ ℤ), where λ is a fixed generator of Grossencharacters. We consider the universality property as both τ and m grow. More precisely our results are stated as follows:

Let K = ℚ(i), and for an ideal 𝔞 = (α) ∈ K, the m-th Grossencharacter is given by λ^{m}(𝔞) := (\frac{α}{|α|})^{4m} for m ∈ ℤ. The Hecke L-function is defined by L(s, λ^{m}) = ∑_{𝔞} λ^{m}(𝔞)N(𝔞)^{−s} for σ = Re(s) > 1.

Theorem 1.1 Let C be a compact subset in the strip \{s ∈ ℂ | \frac{1}{2} < σ < 1\}. For any function f(s) which is nonzero and continuous on C and which is holomorphic on Int(C), and for any ε > 0, we have

(1.1) \underline{lim}_{T →∞} \frac{1}{T^{2}} μ′(\{(t, m) ∈ [0, T] × \{0, ..., [T]\} |

max_{s∈C}|L(s + it, λ^{m}) − f(s)| < ε}) > 0,

where μ′ is the product measure on ℝ × ℤ.

Remark 1.2 (a) It is possible to extend Theorem 1.1 to any imaginary quadratic field K of class number one, and to general Hecke character χλ^{m} with nontrivial narrow class character χ.

(b) In case K is a general number field of finite degree, (1.1) would be formulated as follows: Let n = [K : ℚ] and λ_{1}, ..., λ_{n−1} be a fixed set of generators of Grossencharacters of K. Put λ^{m} = λ_{1}^{m_{1}} ⋯ λ_{n−1}^{m_{n−1}} for m=(m_{1}, ..., m_{n−1}) ∈ ℤ^{n−1}. Then under the above settings we would have

\underline{lim}_{T →∞} \frac{1}{T^{n}} μ′(\{(t, m) ∈ [0, T]^{n} |

max_{s∈C}|L(s + it, λ^{m}) − f(s)| < ε}) > 0

with μ′ the product measure on ℝ × ℤ^{n−1}. This will be treated in the forthcoming paper [M3].

(c) In Theorem 1.1, it is unfortunate that the range of m and t must be the same. The universality in the m-aspect with t being fixed should also be proved. Difficulty lies in the proof of the mean value theorem for Dirichlet series over O_{K} twisted by λ^{m}. Duke it in [D, Theoreom 1.1], where he takes the average over (m, t) ∈ {0, ..., [T]}×[0, T]. He conjectures that the mean value theorem should hold in case of (m, t) ∈ {0, ..., M} × [0, T]. We see from the proof of our Theorem 1.1 that Duke’s conjecture would imply the universality in the m-aspect.

(d) The Grossencharacter-aspect is also considered in a different context. Petridis and Sarnak [PS] obtain a subconvexity estimate of automorphic L-functions L(s, ϕ) for a Maass cusp form ϕ of SL(2, Z[i]). In order to prove it they consider the twists with Grossencharacters and take an avarage ∑ ∫ |L(\frac{1}{2} + it, ϕ ⊗ λ^{m})|^{2}dt, where the summation and the integration is taken over certain range of (m, t). Consequently they succeed in obtaining subconvexities in the both m and t aspects.

2 Propositions

For describing the proof of our main result, we put for z > 0

L_{z}(s, λ^{m}) := ∏_{N(𝔭)≤z} (1 − \frac{λ^{m}(𝔭)}{N(𝔭)^{s}})^{−1}

where 𝔭 denotes a prime ideal. Theorem 1.1 is an immediate consequence of the following propositions:

Proposition 2.1 For any ε > 0 there exists z_{0} > 0 such that for any z > z_{0}

max_{s∈C} | log L(s + it, λ^{m}) − log L_{z}(s + it, λ^{m})| < ε

holds as T → ∞ for any (t, m) in a subset of [0, T] × {0, ..., [T]} with positive deinsity which is greater than 1 − ε.

Proposition 2.2 For any ε > 0 there exists z_{1} > 0 such that for any z > z_{1}

max_{s∈C} | log L_{z}(s + it, λ^{m}) − log f(s)| < ε

holds as T → ∞ for any (t, m) in a subset of [0, T]× {0, ..., [T]} with positive density which depends only on ε.

Since the intersection of the sets of (t, m) in Propositions 2.1 and 2.2 has a positive density, Theorem 1.1 follows.

3 Proof of Proposition 2.1

Put a_{m}(n) to be the coefficient in the Dirichlet series expansion of L(s, λ^{m}): L(s, λ^{m}) = ∑_{n=1}^{∞} a_{m}(n)n^{−s}. We use the following approximate functional equation of Ramachandra type:

Lemma 3.1 For s = σ + it and x, y > 0, xy = t^{2}, under the conditions that σ < α < 2, 0 < β < σ, 0 < γ < 2, we have

L(s, λ^{m}) = A + B + J_{1} + J_{2} − \frac{W (m)}{2πi}π^{2s−1}(J_{3} + J_{4}),

where |W (m)| = 1 and

A = ∑_{n≤x} \frac{a_{m}(n)}{n^{s}},

B = W (m)π^{2s−1} \frac{Γ(1 − s + 2m)}{Γ(s + 2m)} ∑_{n≤y} \frac{\overline{a_{m}(n)}}{n^{1−s}},

J_{1} = \frac{1}{2πi} ∫_{(−γ)} x^{w} \frac{Γ(1 + \frac{w}{2} )}{w} ∑_{n≤x} \frac{a_{m}(n)}{n^{s+w}}dw,

J_{2} = ∑_{n>x} \frac{a_{m}(n)}{n^{s}}e^{−(n/x)^{2}},

\begin{aligned}J_{3} & = \frac{1}{2πi} ∫_{(β)}(π^{2}x)^{w} \frac{Γ(1 − s − w + 2m)}{Γ(s + w + 2m)} \frac{Γ(1 + \frac{w}{2} )}{w} \\ & × ∑_{n≤y} \frac{\overline{a_{m}(n)}}{n^{1−s−w}} dw, \end{aligned}

\begin{aligned}J_{4} & = \frac{1}{2πi} ∫_{(−α)}(π^{2}x)^{w} \frac{Γ(1 − s − w + 2m)}{Γ(s + w + 2m)} \frac{Γ(1 + \frac{w}{2} )}{w} \\ & × ∑_{n>y} \frac{\overline{a_{m}(n)}}{n^{1−s−w}} dw. \end{aligned}

Let C_{1} be a compact set in \{s ∈ ℂ | \frac{1}{2} < σ < 1\} such that C ⊂ C_{1}. We will compute the integral

I = ∑_{m=T}^{2T} ∫_{T}^{2T} ∫∫ _{C_{1}} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dσdτdt

By changing the order of the integration and the sum, it follows that

I = ∫∫ _{C_{1}} ∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dtdσdτ.

By Lemma 3.1 we have

\begin{matrix}∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dt \\ = ∑_{m=T}^{2T} ∫_{T}^{2T} \\ |\frac{A + B + J_{1} + J_{2} − \frac{W (m)}{2πi}π^{2s−1}(J_{3} + J_{4})}{L_{z}(s + it, λ^{m})} − 1| & 2 \\ dt \\ ≪ ∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{A}{L_{z}(s + it, λ^{m})} − 1|^{2} dt \\ + ∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{B}{L_{z}(s + it, λ^{m})}|^{2} dt \\ + ⋯ \\ + ∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{W (m)}{2πi}π^{2s−1} \frac{J_{4}}{L_{z}(s + it, λ^{m})}|^{2} dt \end{matrix} (3.1)

We will compute each term in (3.1) which we put as I_{A}, I_{B}, I_{J_{1}},...,I_{J_{4}}. By putting x = T we have for some coefficients b_{m}(n) with |b_{m}(n)| < n^{ε} such that

L_{z}(s, λ^{m})^{−1} ∑_{n≤T} \frac{a_{m}(n)}{n^{s}} = 1 + ∑_{z<n<z^{ε}T} \frac{b_{m}(n)}{n^{s}}.

Thus

(3.2) I_{A} = ∑_{m=T}^{2T} ∫_{T}^{2T} |\frac{A}{L_{z}(s + it, λ^{m})} − 1|^{2} dt

= ∑_{m=T}^{2T} ∫_{T}^{2T} | ∑_{z<n<z^{ε}T} \frac{b_{m}(n)}{n^{s}} |^{2} dt.

By the theorem of Montgomery-Vaughn, (3.2) is estimated by

(3.3) T (T ∑_{z<n<z^{ε}T} \frac{1}{n^{2σ−ε}} + ∑_{z<n<z^{ε}T} \frac{1}{n^{2σ−ε−1}} )

≪ T^{2}(z^{1−2σ+ε} + T^{1−2σ+ε})

The contribution I_{B} from the term B to (3.1) is computed by using Stirling’s formula as

I_{B} ≪ T^{3−2σ+ε}. (3.4)

The third term I_{J_{1}} from J_{1} is dealt with by our using the Cauchy inequality as

I_{J_{1}} ≪ T^{3−2σ+ε}. (3.5)

The remaining terms I_{J_{2}},...,I_{J_{4}} are similarly estimated. Taking (3.3), (3.4), (3.5) into account we have

∑_{m=T}^{2T} ∫∫ _{C_{1}} ∫_{T}^{2T} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dtdσdτ

≪_{C_{1}} T^{2}(z^{1−2σ_{1}+ε} + T^{1−2σ_{1}+ε}),

where σ_{1} = min\{σ ∈ C_{1}\}. Since σ_{1} > \frac{1}{2}, by taking z_{0} as z_{0}^{1−2σ_{1}+ε} = ε^{3}, we have

(3.6) \frac{1}{T^{2}} ∑_{m=T}^{2T} ∫_{T}^{2T}

(∫∫ _{C_{1}} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dσdτ) dt < ε^{3}

for z > z_{0}, T > T_{0}(z). It follows from (3.6) that there exists a subset A_{T} of [0, T] × {0, ⋯, [T]} with positive density greater than 1 − ε such that

∫∫ _{C_{1}} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1|^{2} dσdτ < ε^{2}

for any (t, m) ∈ A_{T}. We then have

max_{s∈C} |\frac{L(s + it, λ^{m})}{L_{z}(s + it, λ^{m})} − 1| ≪_{C,C_{1}} ε.

This means that

max_{s∈C} |log L(s + it, λ^{m}) − log L_{z}(s + it, λ^{m})| ≪_{C,C_{1}} ε

for (t, m) ∈ A_{T}.

Remark 3.2 Duke’s conjecture [D] would make it possible to deal with the variables m and t separately.

4 Proof of Proposition 2.2

Lemma 4.1 (Gonek [G]) Let C be a simply connected compact set of the strip \frac{1}{2} < σ < 1. Let h(s) be a continuous function on C which is regular on Int(C). For any y > 0 there exist ν_{0} = ν_{0}(C, h, y) and θ_{p}^{(0)} ∈ [0, 1] such that

max_{s∈C} |h(s) − ∑_{{y<p≤ν \atop p≡1 (mod 4)}} \frac{e(θ_{p}^{(0)})}{p^{s}} | ≪_{C} y^{− \frac{1}{2}}

for any ν > ν_{0}, where p denotes the prime numbers.

Lemma 4.2 ([KV] Theorems 8.1, 8.2) Let a_{n} ∈ ℝ (1 ≤ n ≤ N) be linearly independent over ℚ. Then we have

(i) If we put

I_{A}(T) := {t ∈ [0, T] | ({a_{1}t}, ..., {a_{N}t}) ∈ A}

for any closed Jordan measurable set A ⊂ [0, 1]^{N} and for T > 0, where {x} = x − [x], it holds that lim_{T →∞} \frac{μ(I_{A}(T))}{T} = μ_{N}(A) with μ_{N} the Lebesgue measure on ℝ^{N}.

(ii) Let Ω be a set of continuous functions on A. If Ω is uniformuly bounded and is equicontinuous, it holds uniformly on f ∈ Ω that

lim_{T →∞} \frac{1}{T} ∫_{I_{A}(T)} f(\{a_{1}t\}, ..., \{a_{N}t\})dt

= ∫ ⋯ ∫ _{A} f(x_{1}, ..., x_{N})dx_{1} ⋯ dx_{N}.

Lemma 4.3 Let p ≡ 1 (mod 4) and (p) = 𝔭\overline{𝔭} with 𝔭 a prime ideal in K. We put θ_{p} as λ(𝔭) = e^{iθ_{p}}. Then {θ_{p}}_{p≡1 (mod 4)} is linearly independent over ℚ.

Proof. Putting 𝔭 = (a + bi) (a, b ∈ ℤ), we have |α| = \sqrt{p} and so λ(𝔭) = (\frac{a+bi}{\sqrt{p}})^{4}. Thus cos θ_{p}, sin θ_{p} ∈ ℤ[\frac{1}{\sqrt{p}}]. Assume an algebraic dependence as Mθ_{p} = m_{1}θ_{p_{1}} +⋯+m_{r}θ_{p_{r}} with M, m_{1}, ..., m_{r} ∈ ℤ. Then in the equation cos(Mθ_{p}) = cos(m_{1}θ_{p_{1}} + ⋯ + m_{r}θ_{p_{r}}), the left hand side belongs to ℤ[\frac{1}{\sqrt{p}}], whereas the right hand side is in ℤ[\frac{1}{\sqrt{p_{1}}} , ..., \frac{1}{\sqrt{p_{r}}}]. Hence it holds if and only if cos(Mθ_{p}) ∈ ℤ. Therefore we have M = 0.

Proof of Proposition 2.2 We have

log L_{z}(s, λ^{m}) = ∑_{{p≤z \atop p≡1 (mod 4)}} ∑_{k=1}^{∞} \frac{2 cos(kmθ_{p})}{kp^{s}}

+ ∑_{{p≤z \atop p≡3 (mod 4)}} ∑_{k=1}^{∞} \frac{1}{kp^{2ks}} + ∑_{k=1}^{∞} \frac{1}{k2^{ks}}

We split the sums over p ≤ z into the ones over p ≤ y and y < p ≤ z with 0 < y < z. We also divide the sum over 1 ≤ k < ∞ into k = 1, 2 ≤ k < N, and k ≥ N with N = [σ log_{2} y]. For partial sums we have the estimates ∑_{y<p≤z} ∑_{2≤k<N} \frac{2 cos(kmθ_{p})}{kp^{s}} ≪ y^{1−2σ} and ∑_{p≤y} ∑_{k≥N} \frac{2 cos(kmθ_{p})}{kp^{s}} ≪ y2^{−Nσ} ≪ y^{1−2σ}. Hence

(4.1) log L_{z}(s + it, λ^{m})

= ∑_{{y<p≤z \atop p≡1 (mod 4)}} \frac{2 cos(mθ_{p})}{p^{s}} + l(s + it, y, m)

+ O(y^{1−2σ}),

where

(4.2) l(s, y, m) = ∑_{{p≤y \atop p≡1 (mod 4)}} ∑_{k≤N} \frac{2 cos(kmθ_{p})}{kp^{ks}}

+ ∑_{{p^{2}≤y \atop p≡3 (mod 4)}} ∑_{k≤N} \frac{1}{kp^{2ks}} + ∑_{k≤N} \frac{1}{k2^{ks}} .

We fix sufficiently large y which satisfies y^{1−2σ} < ε and y^{− \frac{1}{2}} < ε. Apply Lemma 4.1 for h(s) = \frac{1}{2}(g(s) − l(s, y, 0)) and fix ν > ν_{0}. Then for any z > ν,

| log L_{z}(s + it, λ^{m}) − g(s)|

≤ | ∑_{{y<p≤ν \atop p≡1 (mod 4)}} \frac{2 cos(mθ_{p})}{p^{s+it}} − ∑_{y<p≤ν} \frac{2e(θ_{p}^{(0)})}{p^{s}} | (4.3)

+ |l(s + it, y, m) − l(s, y, 0)| (4.4)

+ | ∑_{{ν<p≤z \atop p≡1 (mod 4)}} \frac{2 cos(mθ_{p})}{p^{s+it}} | + ε. (4.5)

We first deal with (4.3). It is less than

(4.6) ∑_{{y<p≤ν \atop p≡1 (mod 4)}} \frac{2}{p^{σ}} |\frac{cos(mθ_{p})}{p^{it}} − e(θ_{p}^{(0)})|

= ∑_{{y<p≤ν \atop p≡1 (mod 4)}} \frac{2}{p^{σ}} |cos(mθ_{p})e^{−it log p} − e(θ_{p}^{(0)})|

Hence if we take a sufficiently small δ > 0 and put

V_{T}^{(1)} = {0 ≤ m ≤ T | ∥mθ_{p}∥ < δ (y < p ≤ ν, p ≡ 1 (mod 4))},

U_{T}^{(1)} = \{t ∈ [0, T] | ∥t\frac{log p}{2π} − θ_{p}^{(0)}∥ < δ

(y < p ≤ ν, p ≡ 1 (mod 4))},

then for any (m, t) ∈ V_{T}^{(1)}×U_{T}^{(1)}, it holds that (4.6)< ε. By Lemmas 4.2, 4.3, and the linear indepencence over ℚ of {log p}, we have

lim_{T →∞} \frac{μ′(V_{T}^{(1)} × U_{T}^{(1)})}{T^{2}} = #(V^{(1)}) × μ(U^{(1)}) (4.7)

for some V^{(1)}, U^{(1)} ⊂ ℝ^{π(ν)−π(y)} with π(x) the number of primes not greater than x.

Next we consider (4.4). It is less than

(4.8) ∑_{{p≤y \atop p≡1 (mod 4)}} ∑_{1≤k≤N} \frac{1}{kp^{kσ}} |\frac{2 cos(kmθ_{p})}{p^{ikt}} − 2|

+ ∑_{{p^{2}≤y \atop p≡3 (mod 4)}} ∑_{1≤k≤N} \frac{1}{kp^{2kσ}} |\frac{1}{p^{2ikt}} − 1|

+ ∑_{1≤k≤N} \frac{1}{2^{kσ}} |\frac{1}{2^{ikt}} − 1| .

Again we take a sufficiently small δ′ > 0 and put

V_{T}^{(2)} = {0 ≤ m ≤ T | ∥mθ_{p}∥ < δ′

(p ≤ y, p ≡ 1 (mod 4))},

U_{T}^{(2)} = \{t ∈ [0, T] | ∥t\frac{log p}{2π} ∥ < δ′(p ≤ y)\} .

Then for any (m, t) ∈ V_{T}^{(2)}×U_{T}^{(2)}, it holds that (4.8)< ε.

We put

V_{T} = {0 ≤ m ≤ T |

∥mθ_{p}∥ < δ (y < p ≤ ν, p ≡ 1(mod4)), ∥mθ_{p}∥ < δ′ (p ≤ y, p ≡ 1(mod4))}

and

U_{T} ={t ∈ [0, T] |

∥t\frac{log p}{2π} − θ_{p}^{(0)}∥ < δ (y < p ≤ ν, p ≡ 1(mod4)),

∥t\frac{log p}{2π} ∥ < δ′ (p ≤ y)\}.

Then (4.3) and (4.4) are bounded by ε for any (m, t) ∈ V_{T} × U_{T}, and we have

lim_{T →∞} \frac{#V_{T}}{T} = vol(V) = (2δ)^{\frac{π(ν)−π(y)}{2}} (2δ′)^{\frac{π(y)}{2}}
lim_{T →∞} \frac{μ(U_{T})}{T} = vol(U) = (2δ)^{π(ν)−π(y)}(2δ′)^{π(y)}

where U and V are subsets of [0, 1]^{π(ν)}. Here we have proved that (4.3) and (4.4) are less than ε for any (m, t) in a set with positive density.

Lastly we can check that (4.5) is less than ε for almost all (m, t) ∈ U_{T} × V_{T}. This completes the proof of the theorem.

References

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