Subconvexity of Hecke L-functions in the Grossencharacter-aspect

Subconvexity of Hecke L-functions in the Grossencharacter-aspect Shin-ya Koyama 1. Introduction. Let K be a number field with [K : Q] = n = r1 + 2r2 . The real and complex conjugates of K are denoted by K (q) (q = 1, ..., r1) and K (p) (p = r1 +1, ..., n) with K (p+r2 ) = K (p) . The conjugate of µ ∈ K in K (j) (j = 1, ..., n) is denoted by µ(j) . Put N (µ) = µ(1) · · · µ(n) . Let JK and PK be the idele group and the principal idele group of K, respectively. Grossencharacter λ of K is a continuous character of the idele class group CK = JK /PK . We use the same symbol λ for the induced character of JK . We decompose as JK = J∞ × J0 with the infinite part J∞ and the finite part J0 . Let U0 be the unit group of J0 , and put Um,0 = {u ∈ U0 | u ≡ 1 (mod m)}, where m is an integral ideal of K. The set {Um,0 } is a fundamental neighbourhood system of 1 in J0 . A map from Jm,0 = {a ∈ J0 | ap = 1 (∀p|m)} to an ideal class group G(m) = {ã | (a, m) = 1} defined by Jm,0 ∋ a 7→ ã = Y p pνp (a) ∈ G(m) is a homomorphism whose kernel is a subset of Um,0 . By continuity a Grossencharacter λ satisfies that λ(Um,0 ) = 1 for some m. The greatest common divisor f of such m’s are called the conductor of λ. For a ∈ Jf,0 , the value λ(a) is determined by the ideal class ã ∈ G(f). Thus we define a character λ̃ of G(f) as λ̃(ã) := λ(a). The Hecke L-function with respect to the Grossencharacter λ is defined by X λ̃(ã) L(s, λ) = . (Re(s) > 1) N (ã)s ã∈G(f) When we restrict λ on J∞ = R×r1 × C×r2 , we have for x = (x(1) , ..., x(r1 +r2 ) ) ∈ J∞ λ(x) = r1 Y (sgn x v=1 (v) av ) r1Y +r2 v=1 |x (v) ibv | r1Y +r2 v=r1 +1  x(v) |x(v) | av with av ∈ Z and bv ∈ R, and a1 , ..., ar1 , which may take only the values 0 or 1, depend only on the narrow class character mod f. In what follows we fix a1 , ..., ar1 so that the narrow class character χ mod f is fixed. We will make the remaining part in the Grossencharacter vary, which is parametarized by an n-dimensional vector a = (b1 , ..., br1+r2 , ar1 +1 , ..., ar1+r2 ) with P a a v bv = 0. We denote such Hecke character λ by χλ . If χλ is a normalized Hecke ma character, then χλ is also normalized for any m ∈ Z. The chief concern of this Typeset by AMS-TEX 1 paper is to estimate the size of L( 21 + it, χλma ) as |m| → ∞ with χ, a and t being fixed. We may call this problem the m-aspect in the estimate of Hecke L-functions. In what follows we write χλm := χλma . Duke [D] computed the convexity bound n 1 L( + it, χλm ) ≪t,ε |m| 4 +ε 2 (1.1) which can be proved by applying the Phragmén-Lindelöf principle to the lines Re(s) = 1 + δ and Re(s) = −δ with δ > 0. We improve this bound by using the approximate functional equation in the m-aspect, which will be proved in the next section. In Section 3 we express the exponential sum over certain range of norms in terms of the sum over algebraic integers in certain cubes. The main theorem is proved using the van der Corput method in Section 4. It asserts that Theorem 1.1. In the above notation we have as |m| → ∞ 1 n 1 L( + it, λma ) ≪t,ε |m| 4 − 18 +ε 2 for any ε > 0. The Grossencharacter-aspect is recently considered in a different context. In their paper [PS], Petridis and Sarnak obtain a subconvexity estimate of automorphic L-functions L(s, φ) for a Maass cusp form φ of SL(2, Z[i]). In the proof they consider the twists with Grossencharacters and take an avarage in the form XZ 1 |L( + it, φ ⊗ λm )|2 dt, 2 where the summation and the integration are taken over certain range of (m, t). Consequently they succeed in obtaining the subconvexities in the both m and t aspects. 2. Approximate Functional Equation. We require the functional equation for the Hecke L-function given by Mordell [Mo] (5.14): L(s, χλ) = W (χλ)A1−2s G(1 − s, χ̂λ̄)L(1 − s, χ̄λ̄), (2.1) 1/2  with dK the discriminant of K, χ̂ is where |W (χλ)| = 1 and A = dπKn N(f) 22r2 the real character induced by χ on the group of narrow ideal classes lying over the broad class of principal ideals, and G(s, χ̂λ) is defined by r1Y +r2 Γ(s + 12 |av | + ibv ) Γ( 12 (s + av + ibv )) G(s, χ̂λ) = . Γ( 21 (1 − s + av − ibv )) v=r +1 Γ(1 − s + 12 |av | − ibv ) v=1 r1 Y 1 We denote each factor by Gv , namely, G(s, χ̂λ) = 2 Qr1 +r2 v=1 Gv (s, χ̂λ). Lemma 2.1. The function G(s, χ̂λ) is holomorphic except at the following simple poles: s = −2k − av − ibv (1 ≤ v ≤ r1 ) (−1)k 22(k+av )+1 (k + av )! Y √ Gv ′ (−2k − av − ibv ) πk!(2(k + av ))! ′ with residue v 6=v s = −k − |av | − ibv (r1 < v ≤ r1 + r2 ) 2 Y |av | (−1)k Gv ′ (−k − − ibv ) k!(k + |av |)! ′ 2 with residue v 6=v P Ress=ρ G(s,χ̂λ) with k ≥ 0, k ∈ Z, and the sum ρ over all poles of G(s, χ̂λ) absoρ lutely converges. In particular, G(s, χ̂λ) is holomorphic in Re(s) > 0. Proof. We first treat the case 1 ≤ v ≤ r1 . The numerator of Γ( 12 (s + av + ibv )) Gv (s, χ̂λ) = Γ( 12 (1 − s + av − ibv )) has poles at s = −2k − av − ibv , none of which agrees to any zero coming from a pole of the denominator, since k ≥ 0. By the fact that the residue of Γ(z) at k z = −k is (−1) k! , the residue of Gv (s, χ̂λ) at s = −2k − av − ibv is computed as 2(−1)k k!Γ
= 1+2k+2av 2 (−1)k 22(k+av )+1 (k + av )! √ , πk!(2k + 2av )! where we used the identity  1 Γ N+ 2  = √ π(2N )! 22N N ! for any nonnegative integer N . The case r1 + 1 ≤ v ≤ r1 + r2 can be treated similarly. Huxley’s expression of Gv (s, χ̂λ) (See (2.7) below) and Stirling’s formula show that Gv ′ (−2k − av − ibv ) and Gv ′ (−k − |a2v | − ibv ) decay exponentially as k → ∞. It gives the absolute convergence of the sum.  In what follows we define for z ∈ C \ R− z z := exp(z log z) = exp(z(log |z| + i arg(z))) with | arg(z)| < π. 3 Lemma 2.2. Put z = s + iy ∈ C with s = σ + it, σ, t ∈ R being fixed. Then we have     1 πiz z s+iy s+iy 1+O z = (s + iy) = |y| exp s + sgn(y) 2 y as |y| → ∞. Proof. Putting |z| = r and arg(z) = θ give z z = r σ exp(−θ(t + y) + i((t + y) log r + θσ)). By putting ϕ = π2 − |θ| = π2 − sgn(t + y)θ, we have sin ϕ = ϕ + O(ϕ3 ). On the other σ hand we have sin ϕ = √ 2σ 2 = |y| + O( y12 ). Thus θ(t + y) ≥ 0, and σ +y σ ϕ= +O |y|  1 y2  , (2.3) which leads to     π exp(−θ(t + y)) = exp −θt + sgn(t + y) − + ϕ y 2   1 π = exp −θt − |y| + σ + O 2 y2      1 π . 1+O = exp −θt − |y| + σ 2 y2 (2.4) Next we easily see that σ 2 2 σ/2 r = (σ + (t + y) )    1 = |y| 1 + O . y σ (2.5) By the facts that ! 2    2 σ t t 1 1 + 2 = log |y| + + O 1+ log r = log |y| + log 2 y y y y2 and that |θ| = π 2 −ϕ= π 2 +O   1 y which is deduced from (2.3), we compute that π (t + y) log r + θσ = (t + y) log |y| + sgn(y) σ + t + O(1). 2 Combining (2.4), (2.5) and (2.6) gives that z z = r σ e−θ(t+y)+i((t+y) log r+θσ)    1 s+iy s+sgn(y) πiz 2 = |y| e 1+O . y The proof is complete.  4 (2.6) Lemma 2.3. As |y| → ∞, it holds that Γ(s + iy) = |y| z− 12      √  1 π π 2π 1 + O exp i sgn(y) z − (2 − sgn(y)) − y 2 4 y with s ∈ C being fixed and z = s + iy. Proof. Stirling’s formula and the previous lemma show that √    1 2π 1 + O Γ(z) = e z |z|    √ 1 z i(sgn(y) πz −y) −1/2 2 = |y| e 2π 1 + O . z |z| −z z− 12 By y = e 1−sgn(y) πi 2 |y| and thus (s + iy)−1/2 = |y|−1/2 e−(2−sgn(y))πi/4 (1 + O(y −1 )). We have the conclusion.  Lemma 2.4. As |m| → ∞, m G(s, χ̂λ ) = 2 −r1 s |m| n(s− 21 ) n Y v=1 c̃v (m)|bv | s− 21  1+O  1 m  , where c̃v (m) is given by c̃v (m) = (−1) for 1 ≤ v ≤ r1 , and by av 2 (1 − sgn(mbv )i)  |mbv | 2e imbv c̃v (m)2 = −sgn(mbv )icv (m) for r1 + 1 ≤ v ≤ r1 + r2 with cv (m) = ( 1 − (−1) 1 − (−1) mav 2 v (2s − 1) log av2ib +2ibv mav −1 2 v (2s − 1) log av2ib +2ibv (mav ∈ 2Z) (mav ∈ 1 + 2Z). Proof. Huxley [H] p.115 computes that Gv (s, χ̂λ)  v )π  Γ(s + ibv ) 21−s−ibv π −1/2 cos (s+ib  2  v )π = 21−s−ibv π −1/2 sin (s+ib Γ(s + ibv ) 2   1  sin(s+ 2 |av |+ibv )π Γ(s + |a2v | + ibv )Γ(s − π 5 (1 ≤ v ≤ r1 , av = 0) |av | 2 (1 ≤ v ≤ r1 , av = 1) − ibv ). (r1 + 1 ≤ v ≤ r1 + r2 ) (2.7) When 1 ≤ v ≤ r1 , the cos or sin part in (2.7) is asymptotically equal to av +1 2 (−1) 2 π esgn(bv ) 2 (bv −i(s+1)) up to an exponentially small error term. Hence from Lemma 2.3 we compute that    av (−1) 2 1 m s+imbv − 21 −(sgn(mbv ) π +mbv )i 4 Gv (s, χ̂λ ) = s+imb − 1 |mbv | 1+O e . v m 2 2 π Since e−sgn(mbv ) 4 i = 2−1/2 (1 − sgn(mbv )i), we have    1 m −s s− 21 Gv (s, χ̂λ ) = 2 c̃v (m)|mbv | 1+O . m (2.8) When r1 + 1 ≤ v ≤ r1 + r2 , (2.7) can further be calculated as follows. We iterate the formula of the Gamma function Γ(z + 1) = zΓ(z) to get Gv (s, χ̂λ)  |av | |av |  2 Q  (−1) 2 sin(s+ibv )π s+ibv +k−1  Γ(s + ib )Γ(s − ib )  v v  π s−ibv −k k=1 = |av |−1  |av |−1  2 Q  (−1) 2 cos(s+ibv )π 1 1   Γ(s + + ib )Γ(s − − ib ) v v π 2 2 k=1 (av ∈ 2Z) s+ibv +k− 12 s−ibv −k− 12 (av ∈ 1 + 2Z) (2.9) We estimate the product over k in (2.9) for Gv (s, χ̂λm ) as |m| → ∞. When mav ∈ Z, it is |mav | 2 |mav |   2 Y s + imbv + k − 1 Y 2s − 1 −1 + = s − imbv − k s − imbv − k k=1 k=1 = (−1) = (−1) |mav | 2 |mav | 2 |mav | 2   1 1 − (2s − 1) +O s − imbv − k m k=1   2ibv 1 . − (2s − 1) log +O av + 2ibv m X When mav ∈ 1 + 2Z, we have |av |−1 2 Y s + ibv + k − s − ibv − k − k=1 1 2 1 2 = (−1) |mav |−1 2 2ibv − (2s − 1) log +O av + 2ibv  1 m  . Next by applying Lemma 2.3 we estimate the product of the gamma functions in (2.9) as |m| → ∞ as follows:    1 2s−1 −π|mbv | 1+O Γ(s + imbv )Γ(s − imbv ) = −2π|mbv | e m 6 and    1 1 1 2s−1 −π|mbv | Γ(s+imbv + )Γ(s−imbv − ) = −2πisgn(mbv )|mbv | 1+O . e 2 2 m Then (2.9) is written as m 2 −sgn(mbv )isπ Gv (s, χ̂λ ) = c̃v (m) e |mbv | 2s−1    1 1+O . m Multiplying (2.8) and (2.10) for all v leads to the lemma. Corollary 2.5. Put G̃(s, χ̂λ̄m ) = |m| −n(s− 12 ) (2.10)  G(s, χ̂λ̄m ), and we have G̃(s, χ̂λ̄m ) = O(1) as |m| → ∞. Proof. By the definition of c̃v (m) in Lemma 2.4, we have c̃v (m) = O(1) as |m| → ∞.  We follow the method of Luo-Sarnak [LS] for proving the approximate functional equation. Put for σ > 1/2 Z ds 1 Γ(s + l)X −s , F (X) = 2πi (σ) s 1 F (w, X) = 2πi Z Γ(−s + l)G(w̄ + s, χ̂λ̄m )X −s (σ) ds s where w ∈ C is fixed with Re(w) = 1/2. Lemma 2.6. Both F (X) and F (w, X) rapidly decrease as X → ∞, and we have F (X) = O(1) for X ≪ 1 F (w, X) = O(1) for X ≪ |m|n Proof. By integrating by parts Z Z ∞ −ξ l−1 −X l−1 e ξ dξ = e X + (l − 1) F (X) = ∞ e−ξ ξ l−2 dξ. X X It is O(1) as X → 0, and is estimated as −X F (X) = e X l−1  1+O  1 X  as X → ∞. Next we change the order of integrals to get F (w, X) = Z ∞ −ξ l−1 e 0 ξ 1 2πi Z (σ)  |m|n ξ X 7 s+it0  ξ X −it0 G̃(w̄ + s, χ̂λ̄m ) ds dξ, s where we put w = 12 + it0 . We first compute the inner integral on s. When X > |m|n ξ, we shift the contour from (σ) to (+∞). By Lemma 2.1 there are no poles in the region. and we deduce that the integral goes to 0. When X < |m|n ξ, by shifting the contour to (−∞), we have 1 2πi −it0 ξ ds G(w̄ + s, χ̂λ̄m ) X s (σ)   −it0  ρ+it 0 X |m|n ξ Ress=ρ G̃(w̄ + s, χ̂λ̄m ) ξ m = G(w, χ̂λ̄ ) + X ρ X ρ Z  |m|n ξ X s+it0  (2.11) where the sum is taken over the poles ρ of G(w̄+s, χ̂λ̄m ). By Lemma 2.1 we see that 1 the sum absolutely converges. We also have G(w, χ̂λ̄m ) = O(|m|n(w− 2 ) ) = O(1) since Re(w) = 21 . Therefore we have F (w, X) ≪ Z ∞ e−ξ ξ l−1 dξ, X |m|n which can be estimated again by integrating by parts.  Proposition 2.7. Let am (k) be the k-th coefficient in the Dirichlet series expansion of the Hecke L-function L(w, χλm ), that is, m L(w, χλ ) = ∞ X am (k) kw k=1 . Then with Re(w) = 1/2 and w, χ, λ being fixed, it holds that m L(w, χλ )Γ(l) = ∞ X am (k) k=1 kw m F (kx) − W (χλ )A 1−2w ∞ X am (k) k=1 k w̄ F (w, k/xA2 ), where x > 0 and l ∈ Z, l > 0. Proof. Consider the integral 1 I= 2πi Z L(w + s, χλm )Γ(s + l)x−s (σ) ds , s (2.12) where σ > 1/2. We compute I= ∞ X am (k) k=1 kw F (kx). On the other hand, since the integrand in (2.12) has a pole at s = 0, Cauchy’s theorem shows Z 1 ds m I = L(w, χλ )Γ(l) + L(w + s, χλm )Γ(s + l)x−s . 2πi (−σ) s 8 By changing the variable s 7→ −s, the second term I − L(w, χλm )Γ(l) is equal to 1 − 2πi Z (σ) L(w − s, χλm )Γ(−s + l)xs ds . s We deduce by the functional equation (2.1) that I−L(w, χλm )Γ(l) Z 1 ds =− L(w̄ + s, χ̄λ̄m )Γ(−s + l)W (χλm )A1−2(w−s) G(w̄ + s, χ̂λ̄m )xs 2πi (σ) s   Z ∞ −s ds k W (χλm )A1−2w X am (k) m Γ(−s + l)G( w̄ + s, χ̂ λ̄ ) . =− w̄ 2 2πi k A x s (σ) k=1 This gives the conclusion.  Corollary 2.8. For fixed w ∈ C with Re(w) = 1/2, we have   L(w, χλm ) = O  X b:ideal N (b)<|m|n/2  χλm (b)   N (b)w Proof. Choose x = |m|−n/2 , and the corollary follows from |W (χλm )| = 1.  3. Reduction to Cubes. Let A, B ∈ R satisfy 0 < A < B ≤ 2A. Let L(a) be the set of principal ideals divisible by a. In this section we estimate the exponential sum X χλm (b) Sa (A, B) = (3.1) N (b)it b⊂L(a) A≤N (b)<B as |m| → ∞. Lemma 3.1. Let ε1 , ..., εr (r = r1 + r2 − 1) be the fundamental units of K. Let (i) (αik ) be the inverse of the square matrix (log |εk |)1≤i,k≤r . Let M be the set of µ ∈ a ∩ OK such that A ≤ |N (µ)| < B and that 0 ≤ tk (µ) < 1 where for k = 1, ..., r, r X   1 (i) tk (µ) = αik log |µ | − log |N (µ)| . n i=1 Then Sa (A, B) = 1 X χλm (µ) , w N (µ)it µ∈M where w is the number of roots of unity in K. Proof. We follow Q the method of Mitsui [M] Lemma 1. Put ak = [tk (µ)] (k = 1, ..., r). r × ) belongs to M if and We have µ1 := µ k=1 εk−ak ∈ M. The product µ1 ε (ε ∈ OK only if ε is a root of unity. Therefore the number of generators of the ideal (µ) which belongs to M is equal to w.  9 Let γ1 , ...γn be the basis of a. Let J be a subset of {1, ..., r1}. We define a subset XJ of Rn as   n P (q)   x γ > 0 for q ∈ J  i i      i=1     n P (q) x γ < 0 for q ∈ 6 J, 1 ≤ q ≤ r . XJ := (x1 , ..., xn) i i 1   i=1     n   P (q)    xi γi 6= 0 for r1 + 1 ≤ q ≤ n  i=1 n A map fJ : XJ → R is defined as follows: fJ (x1 , ..., xn) = (y0 , y1 , ..., yr , θ1 , ..., θr2 ) where y0 = n X log q=1 yk = r X n X i=1 αqk log q=1 θp = arg (q) xi γ i n X (q) xi γ i i=1 n X (p+r1 ) xi γ i 1 − y0 n ! (k = 1, ..., r) (p = 1, ..., r2). i=1 We see that fJ is injective. Put   V = (y0 , y1 , ..., yr , θ1 , ..., θr2 )   log A ≤ y0 < log B  0 ≤ yk < 1 (k = 1, ..., r) .  0 ≤ θp < 2π (p = 1, ..., r2) Pn Then µ = i=1 mi γi ∈ a ∩ OK belongs to M if and only if (m1 , ..., mn) ∈ fJ−1 (V ) for some J. Putting for µ ∈ K × we have F (µ) := exp(−it log |N (µ)| + log χ(µ) + m log λ(µ)), 1 Sa (A, B) = w X X F n X . (3.2) = {(x1 , ..., xn) | kq M ≤ xq < (kq + 1)M (q = 1, ..., n)}, (3.3) J⊂{1,2,...,r 1} mi γi ! i=1 −1 (m1 ,...,mn )∈f (V ) J mj ∈Z Lemma 3.2. For k1 , ..., kn ∈ Z, we define a cube Q ⊂ Rn as Q = Q(k1 , ..., kn; M ) where M = [A8/9n ]. Then X −1 (m1 ,...,mn )∈f (V ) J mj ∈Z F n X i=1 mi γi ! = X X Q⊂fJ−1 (V ) (m1 ,...,m n )∈Q F n X i=1 ! 1 mi γi +O(A1− 9n ). Proof. This is a corollary of the proof of Mitsui’s lemma ([M] Lemma 2). His lemma 1 is stated in the case when M = [A6/7n ] with remainder O(A1− 7n ), but the proof does not rely on the exponent of A. Actually in his proof he shows in page 237 1 that the remainder term is bouded by δA ≪ M A1− n with δ the maximum of the diameters of the fJ (Q) having points with V in common.  10 4. Main Theorem. Lemma 4.1 ([T] Lemma 5.11). Let f (x) be real and have continuous derivatives up to the third order, and let λ3 ≤ f ′′′ (x) ≤ hλ3 , or λ3 ≤ −f ′′′ (x) ≤ hλ3 , and b − a ≥ 1. Then 1/6 X a<n≤b −1/6 e2πif (n) = O(h1/2 (b − a)λ3 ) + O((b − a)1/2 λ3 ). Lemma 4.2. For fixed k1 , ..., kn ≥ 0 and the cube Q = Q(k1 , ..., kn) defined in Lemma 3.2, we have X F n X mi γi i=1 (m1 ,...,m n )∈Q ! 1 1 1 ≪ M n− 2 |m| 6 + M n |m|− 6 . Proof. The left hand side of the lemma is X m1 ··· X X mn−1 mn r1Y +r2 exp −it log |N (µ)| + log χ(µ) + m log v=1 |µ (v) ibv | r1Y +r2 v=r1 +1  µ(v) |µ(v) | av !! , where mq runs through the integers in kq M ≤ mq < (kq + 1)M (q = 1, ..., n) with µ = m1 γ1 + · · · + mn γn . We will trivially estimate the sum over m1 , ..., mn−1, and will obtain a nontrivial estimate for the sum over mn by van der Corput’s technique. (v) (v) In what follows we write µ(v) = x1 γ1 + · · · + xn γn with real variables x1 , ..., xn. Fix x1 , ..., xn−1 and put f (xn ) = − × t n X v=1 1 2π log |µ(v) | − arg χ(µ) − m Let A (v) = n−1 X (v) xj Re(γj ) n X v=1 bv log |µ(v) | − i and B j=1 (v) = n−1 X rX 1 +r2 v=r1 µ(v) av log (v) |µ | +1 (v) xj Im(γj ). j=1 Then we have (v) Re(µ )= n X (v) xj Re(γj ) = xn Re(γn(v) ) + A(v) j=1 Im(µ(v) ) = n X (v) xj Im(γj ) = xn Im(γn(v)) + B (v) . j=1 11 !! . We compute ∂ µ(v) ∂ −i log (v) = ∂xn ∂xn |µ | =  tan −1 Im(µ(v) ) Re(µ(v) ) (v)  (v) A(v) Im(γn ) − B (v) Re(γn ) (v) (v) (xn Re(γn ) + A(v) )2 + (xn Im(γn ) + B (v) )2 . Since A(v) and B (v) are linear in x1 , ..., xn−1, the condition (x1 , ..., xn) ∈ Q implies that ∂ µ(v) log (v) ≈ M −1 , ∂xn |µ | where ≈ means ≪ and ≫. Therefore !! rX n n (v) (v) 1 +r2 (v) X X 1 b γ γ µ ∂ n n v t log (v) f ′ (xn ) = − −m − iav 2π ∂xn |µ(v) | µ(v) |µ | v=1 v=1 v=r +1 1 ≈ mM −1 Similarly we compute ∂2 µ(v) log ∂x2n |µ(v) | = and (v) (v) (v) (v) (v) −2(A(v) Im(γn ) − B (v) Re(γn ))(|γn |2 xn + A(v) Re(γn ) + B (v) Im(γn )) (v) (v) ((xn Re(γn ) + A(v) )2 + (xn Im(γn ) + B (v) )2 )2 ≈ M −2 ∂3 µ(v) log ∂x3n |µ(v) | = (v) (v) −2(A(v) Im(γn ) − B (v) Re(γn )) (v) (v) ((xn Re(γn ) + A(v) )2 + (xn Im(γn ) + B (v) )2 )3 × (|γn(v)|2 |µ(v) |2 − 4(|γn(v)|2 xn + A(v) Re(γn(v)) + B (v) Im(γn(v)))2 ) These give ≈ M −3 . f ′′ (xn ) ≈ By Lemma 4.1 we have X m M2 2πif (mn ) e kn M ≤mn <(kn +1)M and f ′′′ (xn ) ≈ m . M3  m 1/6  m −1/6 1/2 ≪M +M M3 M3 = M 1/2 m1/6 + M m−1/6 . By estimating the sums over mq (1 ≤ q ≤ n − 1) trivially, we have ! n X X mi γi ≪ M n−1 (M 1/2 |m|1/6 + M |m|−1/6 ) F (m1 ,...,m n )∈Q i=1 1 This gives the conclusion.  1 1 = M n− 2 |m| 6 + M n |m|− 6 . 12 Corollary 4.3. Put Sa (A) := Sa (A, 2A) which is defined by (3.1). Then 4 1 1 Sa (A) ≪ A1− 9n |m| 6 + A|m|− 6 . Proof. The number of cubes Q such that Q ⊂ fJ−1 (V ) in (3.3) is O(AM −n ) = O(A1/9 ). The number of subsets J in (3.2) is a constant 2r1 . Then from (3.2), Lemma 3.2 and Lemma 4.2 with B = 2A, we reach the conclusion.  Theorem 4.4. The Hecke L-functions for arbitrary number field K of degree n satisfies the following estimate for fixed vector a and as |m| → ∞: n 1 1 L( + it, λma ) ≪t,ε |m| 4 − 18 +ε 2 Proof. Let a1 , ..., ah be the representatives of ideal classes of K. By Corollary 2.8 it suffices to estimate the sum X b:ideal 0<N (b)≤|m|n/2 χλm (b) 1 N (b) 2 +it = h X χλm (aj ) X 1 j=1 N (aj ) 2 +it b∈L(aj ) χλm (b) 1 N (b) 2 +it . 0<N (b)≤|m|n/2 The inner sum over principal ideals b = (µ) with µ ∈ OK is estimated by the standard diadic decomposition ν X Sa (2k ) k=1 with ν = n 2 2k/2 n 1 ≪ |m| 4 − 18 +ε log2 |m|.  References [D] W. Duke, Some problems in multidimensional analytic number theory, Acta Arith. 52 (1989), 203-228. [H] M.N. Huxley, The large sieve inequality for algebraic number fields. II: Means of moments of Hecke zeta-functions, Proc. London Math. Soc. 21 (1970), 108-128. [LS] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on P SL2 (Z)\H 2 , Publ. I.H.E.S 81 (1995), 207-237. [M] T. Mitsui, On the prime ideal theorem, J. Math. Soc. Japan 20 (1968), 233-247. [Mo] L.J. Mordell, On Hecke’s modular functions, zeta functions and some other analytic functions in the theory of numbers, Proc. London Math. Soc. 32 (1931), 501-556. [N] W. Narkiewicz, Elementary and analytic theory of algebraic numbers (1990), Springer. [PS] Y. Petridis and P. Sarnak, Quantum unique ergodicity for SL2 (O)\H 3 and estimates for L-functions,, J. Evol. Equ. 1 (2001). [T] E.C. Titchmarsh, The theory of the Riemann zeta function (1986), Oxford. ✁✂✄☎✆✝✁✞✆ ✟✠ ✡ ✄✆☛✁✝✄✆☞✌✍✎ ✏ ✁☞✟ ✑✞☞✒✁☎✍☞✆✓ ✎ ✔ ✕☞✓✟✍☛☞✎ ✏✟☛✟✖✗ ✎ ✘ ✟✖✟☛✄✝✄ ✙✙✔✚ ✛✜✙✙ ✎ ✢✄✂✄✞ 13