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The zero repulsion phenomenon of Deuring and Heilbronn [9, Chapter 10]: if q is • sufficiently large, λ > 0 is sufficiently small, ǫ > 0, and the exceptional zero in the region (1.3) exists and equals 1 λ/ log q, then L(s, χ) has no other zeros − χ (mod q) in the region Q ( 2 ǫ)(log λ−1) (1.5) σ 1 3 − . ≥ − log(q(2 + t )) | | Weiss [30] considered a generalization of (1.2) in the context of a general number field. Let K/Q be a number field with absolute field norm N and absolute discriminant DK , and let q be an integral ideal of K. One considers the (narrow) ray class group I(q)/Pq where I(q) is the group of fractional ideals of K which are coprime to q and Pq is the subgroup of principal ideals (α) with α totally positive and α 1 (mod q). Let H be a subgroup of I(q) ≡ containing Pq; we call any such subgroup a congruence class group of K. Weiss proved that there exist absolute constants c2 > 0 and c3 > 0 such that each coset of H in I(q) contains a prime ideal p satisfying (1.6) Np 2[K : Q]c2[K:Q](D Nq)c3 . ≤ K Consequently, each ideal class of K contains a prime ideal p satisfying Np 2[K : Q]c2[K:Q]Dc3 . ≤ K To prove (1.6), Weiss proved variants of (1.3)-(1.5) for Hecke L-functions with completely effective field uniformity. An even broader generalization of (1.2) lies in the context of the Chebotarev density theorem. Let L/F be a Galois extension of number fields with Galois group G. To each unramified prime ideal p of F , there corresponds a certain conjugacy class of Frobenius automorphisms in G which are attached to the prime ideals of L lying above p. We denote this conjugacy class using the Artin symbol [ L/F ]. For a fixed conjugacy class C G, let p ⊂ L/F π (x) := # p : p is unramified, = C, Np x , C p ≤ n h i o where N = NF/Q is the absolute norm of F . The Chebotarev density theorem asserts that C π (x) | |Li(x). C ∼ G | | In analogy with (1.2), it is natural to bound the quantity L/F P (C, L/F ) := min Np : p is unramified, = C, Np is a rational prime . p n h i o Under GRH for Hecke L-functions, Bach and Sorenson [1] proved that 2 (1.7) P (C, L/F ) (4 log DL +2.5[L : Q]+5) . ≤ 2 We note that if L = Q(e2πi/q) for some integer q 3 and F = Q, then we recover a bound of the same analytic quality as (1.1), though the constants≥ are a bit larger. The first nontrivial, unconditional bound on P (C, L/F ) is due to Lagarias, Montgomery, and Odlyzko [19]; they proved that for some absolute constant c4 > 0, we have that (1.8) P (C, L/F ) 2Dc4 ≤ L Equation (1.8) (up to the computation of c4) is commensurate with the best known bounds when L = Q(√D) for some fundamental discriminant D, F = Q, and C is the nontrivial conjugacy class of G, in which case we are measuring the least quadratic nonresidue modulo D (see Burgess [3]). Recently, the second author [35] proved that one may take c4 = 40 for 2πi/q DL sufficiently large. We observe, however, that if L = Q(e ) and F = Q, then (1.8) is exponential in q, which is significantly worse than (1.2). To explain how (1.6) relates to this Chebotarev setting, we must establish some notation. Let A be any abelian subgroup of G such that A C is nonempty, let A be the character ∩ group of A, let f = f(χ) be the conductor of a character χ A, let IndGχ be a character of χ ∈ A G induced by χ A, and let b ∈ b = (C, L/F ; A) := max D[L:F ]/|A|Nf(IndGχ): χ A irreducible . b Q Q b F A ∈ n o Using the fundamental theorem of class field theory, Deuring’s tricbk [5], and (1.6), Weiss [30, Theorem 6.1] proved that for certain absolute constants c5 > 0 and c6 > 0, (1.9) P (C, L/F ) 2[L : Q]c5[L:Q]/|A| c6 . ≤ Q When A is cyclic, we have from the conductor-discriminant formula that D1/|A| D1/ϕ(|A|). L ≤ Q ≤ L (See [31, Chapter 5, Section 3] for a proof of the upper bound.) Thus Weiss proves a bound on P (C, L/F ) which provides a “continuous transition” from (1.2) to (1.8) with the potential to create significant savings over (1.8) when G has a large abelian subgroup which intersects C. In particular, if L is a cyclotomic extension of F = Q, then (1.2) and (1.9) are equivalent. The fundamental difference between (1.8) and (1.9) is that the proof of (1.8) does not take full advantage of the factorization of the Dedekind zeta function ζL(s) of L into a product of Hecke L-functions; this choice affords one the opportunity to use more elementary tools. The proof of (1.6), and hence the proof of (1.9), takes advantage of the factorization of ζL(s), which requires the use of a log-free zero density estimate as in Linnik’s original work. Our goal in this paper is to prove explicit versions of Weiss’ field-uniform variants of (1.4) and (1.5). In a forthcoming paper, the second author [33] will employ these explicit results to make c2,c3,c5, and c6 explicit. We note that Fogels [6] was the first to prove variants of Principles 2 and 3 for Hecke characters, though his proof did not maintain the necessary field uniformity. Weiss’ results rely critically on his field-uniform variants of Fogels’ work, but Weiss’ results are not explicit. In Section 3, we prove an explicit version of Weiss’ variant of (1.4) for Hecke characters [30, Corollary 4.4]. To state Weiss’ result, we first introduce some notation. Let H (mod q) be a congruence class group of K (that is, H is a subgroup of I(q) containing Pq), let nK = [K : Q], and let hH = [I(q): H]. Define
(1.10) Q = QH := max Nfχ : χ (mod q) satisfying χ(H)=1 , { 3 } and N(σ, T, χ) := # ρ = β + iγ : L(ρ, χ)=0,σ<β< 1, γ T { | | ≤ } where the nontrivial zeros ρ of L(s, χ) are counted with multiplicity. Weiss [30, Corollary 4.4] proved that there exists an absolute constant c > 0 such that if 1 σ < 1 and 7 2 ≤ T n2 h1/nK , then ≥ K H (1.11) N(σ, T, χ) (eO(nK )D2 QT nK )c7(1−σ) ≪ K χ (mod q) χ(XH)=1 with an absolute and computable implied constant. The first main result of this paper exhibits an explicit value of c7.
Theorem 1.1. Let H (mod q) be a congruence class group of K. Let nK = [K : Q] and Q −2/n be as in (1.10). If 1 σ < 1 and T max n D K Q−3/5nK , 1 , then 2 ≤ ≥ { K K } (1.12) N(σ, T, χ) eO(nK )D2 QT nK 81(1−σ) ≪{ K } χ (mod q) χ(XH)=1 where all implied constants are absolute and computable. If 1 10−3 σ < 1, then one may replace 81 with 74. − ≤ Remarks. Theorem 1.1 also contains a noticeable improvement over Weiss’ density estimate • (1.11) in the range of T . One would expect in many applications that the number nK 2 3/5 field K satisfies nK DKQ , in which case Theorem 1.1 holds for T 1. Even for arbitrary K, this≪ will result in appreciable numerical savings in the computation≥ of c2, c3, c5, and c6 in [33] instead of simply following Weiss’ original arguments [30, Sections 5-6]. The appearance of eO(nK ) in (1.12) may seem unusual for an explicit result but it is • always a negligible term. If nK = o(log DKQ) for a certain family of number fields K then O(nK ) 2 nK 2+o(1) 1+o(1) nK e DK QT = DK Q T so we may ignore the contribution of eO(nK ). Recall a classical bound of Minkowski implies nK = O(log DK) so the above scenario is often the case. Otherwise, if nK log(D Q) then (1.12) holds for T n in which case ≫ K ≫ K O(nK ) 2 nK {1+o(1)}nK e DKQT = T so we may again ignore eO(nK ). We prove Theorem 1.1 by constructing a Dirichlet polynomial which is bounded away from zero when in close proximity to a nontrivial zero of a Hecke L-function. This is ensured by using the Tur´an power sum method (cf. Proposition 3.2). The contributions from the detected zeros are summed efficiently using a large sieve inequality for Hecke characters (cf. Theorem 3.1). In order to maintain desirable field uniformity in our large sieve inequality, we use the Selberg sieve instead of the usual duality arguments; see Section 4 for a more detailed discussion. 4 In order to bound sums over integral ideals, we are required to smooth the sums using a kernel which is nK -times differentiable, where nK = [K : Q]. Unfortunately, the smoothing nK introduces the powers of nK (see the comments immediately preceding [30, Section 1]). We note that if nK is small in comparison to log DK/ log log DK (i.e., if the root discriminant of nK K is large), then the powers of nK may be safely absorbed into the powers of DK. On the other hand, if nK is large in comparison to log DK / log log DK (i.e., if the root discriminant nK of K is small), then nK dominates DK; this rare situation happens, for example, when considering the infinite p-class tower extensions studied by Golod and Safareviˇc[8].ˇ We also note that in the case of bounding the least prime in an arithmetic progression, Tur´an’s power sum method does not produce the strongest numerical results. Instead, one typically constructs a suitable mollifier for Dirichlet L-functions relies on cancellation arising from the M¨obius function. However, relying on M¨obius cancellation for Hecke L-functions introduces dependence on DK in the implied constant of Theorem 1.1, which is catastrophic for bounds for the least prime ideal in a congruence class. To the authors’ knowledge, the only device by which one can detect zeros to prove a log-free zero density estimate while maintaining suitable field uniformity is the Tur´an power sum. (The Tur´an power sum method was recently used by Lemke Oliver and the first author [22] to prove an effective log-free zero density estimate for Rankin-Selberg L-functions. Since uniformity in certain parameters was crucial for applications, the Tur´an power sum method was used there as well.) In Section 6, we prove an explicit variant of the zero repulsion phenomenon of Deuring and Heilbronn for Hecke L-functions. Theorem 1.2. Let ψ (mod q) be a real Hecke character and suppose L(s, ψ) has a real zero ′ ′ ′ β1. Let T 1 be given, and χ (mod q) be an arbitrary Hecke character and let ρ = β + iγ be a zero of≥L(s, χ) satisfying 1 (1.13) β′ < 1, γ′ T. 2 ≤ | | ≤ Then c log nK nK ′ (1 β1) log(DK Nq (T + 20) e ) β 1 − · · · ≤ − a1 log DK + a2 log Nq + a3nK log(T +20)+ a4nK + 10 for some absolute, computable constant c> 0 and (51, 54, 26, 74) if ψ is quadratic, (a1, a2, a3, a4)= ((26, 13, 13, 37) if ψ is trivial. Remarks.
Let ǫ> 0. If we consider a sequence of number fields K in which nK = o(log DKNq) • then one may take (48 + ǫ, 48 + ǫ, 24 + ǫ, 0) if ψ is quadratic, (a1, a2, a3, a4)= ((24 + ǫ, 12 + ǫ, 12 + ǫ, 0) if ψ is trivial
when DK Nq is sufficiently large in terms of ǫ. (See the remark at the end of Sec- tion 6.2.1 for details.) One may take q to be the least common multiple of the conductors fχ and fψ. • 5 The proof of Theorem 1.2 is inspired by [19, Theorem 5.1] and its quantitative counterpart [35, Theorem 1.2]. Namely, we apply a power sum inequality and carefully estimate various sums over zeros of Hecke L-functions. Other quantitative versions of Deuring-Heilbronn phenomenon have been established by Kadiri and Ng [17] and the second author [35] for the zeros of the Dedekind zeta function and by the second author [34] for the zeros of Hecke L-functions. The results found in [17, 34] use completely different methods than those used here and have much better explicit constants but, instead of assuming (1.13), one must restrict to an asymptotically smaller range of β′ and γ′ 1. In other words, the key difference between Theorem 1.2 and the aforementioned| results| ≤ is the wide range of validity given by (1.13). Consequently, if the real character ψ has a real zero β1 exceptionally close to 1 (often referred to as a Siegel zero), then Theorem 1.2 allows one to take full advantage of the repulsion effect. The paper proceeds as follows. In Section 2, we introduce the relevant notation and conventions, review some standard results in the theory of Hecke L-functions, prove some explicit estimates involving Hecke L-functions, and bound some standard arithmetic sums over integral ideals of K. In Section 3, we prove Theorem 1.1 under the assumption of Theorem 3.1 (which we prove in Section 4) and Proposition 3.2 (which we prove in Section 5). In Section 6, we prove Theorem 1.2.
Acknowlegements. The authors thank John Friedlander and Robert Lemke Oliver for their comments and suggestions. The first author conducted work on this paper while visiting Centre de Recherches Math´ematiques (hosted by Andrew Granville, Chantal David, and Dimitris Koukoulopoulos) and Stanford University (hosted by Kannan Soundararajan and Robert Lemke Oliver); he is grateful to these departments and hosts for providing a rich and productive work environment.
2. Auxiliary Estimates 2.1. Notation. We will use the following notation throughout the paper: K is a number field. • K is the ring of integers of K. •n O = [K : Q] is the degree of K/Q. • K DK is the absolute value of the discriminant of K. • N = NK is the absolute field norm of K. • Q ζK(s) is the Dedekind zeta function of K. • q is an integral ideal of K. • Cl(q)= I(q)/Pq is the narrow ray class group of K modulo q. • χ, or χ (mod q), is a character of Cl(q), referred to as a Hecke character or ray class • character of K. δ(χ) is the indicator function of the trivial character. • fχ is the conductor of χ; that is, it is the maximal integral ideal such that χ is induced • ∗ from a primitive character χ (mod fχ). Dχ = DKNfχ. • L(s, χ) is the Hecke L-function associated to χ. • H, or H (mod q), is a subgroup of Cl(q), or equivalently of I(q) containing Pq. The • group H is referred to as a congruence class group of K. 6 Q = QH = max Nfχ : χ (mod q) satisfying χ(H)=1 is the maximum analytic • conductor of H. { } fH = lcm fχ : χ (mod q) satisfying χ(H)=1 is the conductor of H. • ∗ { } H (mod fH) is the “primitive” congruence class group inducing H. • h = [I(q): H]. • H We also adhere to the convention that all implied constants in all asymptotic inequalities f g or f = O(g) are absolute with respect to K. If an implied constant depends on a ≪ field-independent parameter, such as ǫ, then we use ǫ and Oǫ to denote that the implied constant depends at most on ǫ. All implied constants≪ will be effectively computable. 2.2. Hecke L-functions. For a more detailed reference on Hecke L-functions, see [20, 11] for example. Strictly speaking, a Hecke character χ is a function on Cl(q) but, by pulling back the domain of χ and extending it by zero, we regard χ as a function on integral ideals of K. We will use this convention throughout the paper. For the entirety of this section, assume that χ is primitive. The Hecke L-function of χ, denoted L(s, χ), is defined as χ(p) −1 (2.1) L(s, χ)= χ(n)Nn−s = 1 − Nps n p X Y for Re s > 1 where the sum is over integral ideals n of K and the product is over prime ideals {p of} K. Define the completed Hecke L-function ξ(s, χ) by (2.2) ξ(s, χ)= s(s 1) δ(χ)Ds/2γ (s)L(s, χ), − χ χ where Dχ = DKNfχ, δ(χ) equals 1 if χ is trivial and 0 otherwise, and γχ(s) is the gamma factor of χ defined by s a(χ) s+1 b(χ) − s − s +1 (2.3) γ (s)= π 2 Γ π 2 Γ . χ 2 · 2 Here a(χ) and b(χ) are certainh non-negative i integersh satisfying i
(2.4) a(χ)+ b(χ)= nK . It is a classical fact that ξ(s, χ) is entire of order 1 and satisfies the functional equation (2.5) ξ(s, χ)= w(χ)ξ(1 s, χ) − where w(χ) C is the root number of χ satisfying w(χ) = 1. The zeros of ξ(s, χ) are the non-trivial zeros∈ ρ of L(s, χ), which satisfy 0 < Re| ρ <| 1. The trivial zeros ω of L(s, χ) are given by { } a(χ) δ(χ) if ω =0, − (2.6) ord L(s, χ)= b(χ) if ω = 1, 3, 5,... s=ω − − − a(χ) if ω = 2, 4, 6,... − − − and arise as poles of the gamma factor of L(s, χ). Since ξ(s, χ) is entire of order 1, it admits a Hadamard product factorization given by s (2.7) ξ(s, χ)= eA(χ)+B(χ)s 1 es/ρ. − ρ ρ 7 Y The zeros ρ of ξ(s, χ) are the non-trivial zeros of L(s, χ) and are known to satisfy 0 < Re ρ < 1. We now collect some standard results on L(s, χ) which follow from Theorems 5.6{ and} Proposition 5.7 of [15].
Lemma 2.1. Let χ be a primitive Hecke character. Then
L′ 1 δ(χ) δ(χ) 1 γ′ Re (s, χ) = log D + Re + Re + Re χ (s) . − L 2 χ s 1 s − s ρ γ ρ χ n o n − o X n − o n o where the sum is over all non-trivial zeros ρ of L(s, χ).
Proof. See [20, Lemma 5.1] for example.
′ By similar arguments, there exists an explicit formula for higher derivatives of L (s, χ). − L Lemma 2.2. Let χ be a Hecke character (not necessarily primitive) and k 1 be a positive integer. Then ≥
( 1)k+1 dk L′ 1 ∞ − (s, χ)= (log Np)(log Npm)kχ(p)(Np)−ms k! dsk L k! p m=1 X X δ(χ) 1 = (s 1)k+1 − (s ω)k+1 ω − X − for Re s > 1, where the first sum is over prime ideals p of K and the second sum is over all zeros{ }ω of L(s, χ), including trivial ones, counted with multiplicity.
Proof. Using the Hadamard product (2.7) of ξ(s, χ), it follows that s (s 1)δ(χ)L(s, χ)= srem1+m2s 1 es/ω − − ω ωY6=0
where m1, m2 are constants depending on χ, the product is over all zeros ω =0 of L(s, χ), including trivial ones, and r = ord L(s, χ). Taking the logarithmic derivative6 of both sides s=0 yields L′ δ(χ) 1 1 r (s, χ)= m + . − L s 1 − 2 − s ω ω − s − Xω6=0 − On the other hand, the Euler product of L(s, χ) implies
L′ ∞ (s, χ)= (log Np)χ(p)(Np)−ms for Re s > 1. − L { } p m=1 X X ′ L k Differentiating k times both of these formulas for L (s, χ) and multiplying by ( 1) /k! yields the desired result. Note that the final sum over− zeros ω of L(s, χ) includes ω−= 0, if it exists. 8 2.3. Explicit L-function estimates. In order to obtain explicit results, we must have explicit bounds on a few important quantities. First, we record a bound for L(s, χ) in the critical strip 0 < Re s < 1 via a Phragmen-Lindel¨of type convexity estimate due to Rademacher. { } Lemma 2.3 (Rademacher [29]). Let χ be a primitive Hecke character and η (0, 1/2]. Then for s = σ + it, ∈ δ(χ) (1+η−σ)/2 1+ s nK Dχ nK L(s, χ) ζQ(1 + η) (3 + t ) | |≪ 1 s (2π)nK | | − uniformly in the strip η σ 1+ η. − ≤ ≤ ′ Next, we record an explicit bound on the digamma function and γχ (s). γχ Lemma 2.4. Let s = σ + it with σ> 1 and t R. Then ∈ Γ′ Re (s) log s + σ−1 Γ ≤ | | and, for any Hecke character χ, γ′ n Re χ (s) K log( s +1)+ σ−1 log π . γ ≤ 2 | | − χ Proof. The first estimate follows from [28, Lemma 4]. The second estimate is a straightfor- ward consequence of the first combined with the definition of γχ(s) in (2.3). Next, we establish some bounds on the number of zeros of L(s, χ) in a circle. Lemma 2.5. Let χ be a Hecke character. Let s = σ + it with σ > 1 and t R. For r > 0, denote ∈ N (r; s) := # ρ = β + iγ :0 <β< 1, L(ρ, χ)=0, s ρ r , χ { | − | ≤ } then, for 0 J1 + J3 ǫ nK . 10≪ For J , consider θ [π/2, 3π/2]. As 1 < σ 1+ ǫ and R< 1, 2 ∈ ≤ 0 < σ + R cos θ 1+ ǫ. ≤ Hence, by Lemma 2.3, log L(s + Reiθ, χ) 1 (1 σ R cos θ + ǫ)+ O(ǫ−1n ) | | ≤ 2 Lχ − − K 1 ( R cos θ + ǫ)+ O(ǫ−1n ). ≤ 2 Lχ − K Thus, 3π/2 J Lχ R cos2 θ + ǫ cos θdθ + O (n ) 2 ≥ 2πR − ǫ K Zπ/2 yielding overall (2.11) J ( 1 + ǫ ) + O (n ). ≥ − 4 πR Lχ ǫ K For the sum over zeros in (2.10), observe that the terms are non-negative so (2.9) follows immediately from (2.10) and (2.11) after taking η 0 which implies R 1. To prove (2.8), 1 → → consider 0 nK nK 1 1 Nn 1 +η 1 Oη(nK ) nK 4 − 2 1 κK log x κK γK e nK DK x . Nn − x − − j ! − ≪ Nn≤x j=1 X X Proof. Without loss, we may assume η (0, 1/2). Observe ∈ 1 nK nK − +i∞ s 1 Nn 1 1 2 x nK! 1 κK log x κK γK = ζK(s + 1) n ds. Nn x j 2πi 1 s K (s + j) − − − − − 2 −i∞ j=1 NXn≤x Xj=1 Z nK Oη(nK ) Q Using Lemma 2.3 and noting ζQ(1 + η) e , it follows that 1 ≪ − 2 +i∞ s 1 x nK ! ζK(s + 1) n ds 2πi 1 s K (s + j) Z− 2 −i∞ j=1 ∞ 1 1 1 Γ( + it) Oη(n ) 4 +η −1/2 Q ( +η)n 2 e K D x n ! (1 + t ) 4 K − dt ≪ K K | | Γ( 1 + n + it) Z−∞ 2 K 1 nK 1 +η 1 Γ( + it) Oη(nK ) 4 −1/2 ( 4 +η)nK 2 e D x nK! (1 + t ) − dt ≪ K | | Γ( 1 + n + it) −nK 2 K 1 Z +η Oη(nK ) 4 −1/2 e D x nK! 1 K nK 4 +η 1 (nK ) ≪ Γ(nK + 2 ) 1 eOη(nK )(nnK D ) 4 +ηx−1/2 ≪ K K as claimed. Corollary 2.9. Let η > 0 and C = C (η) 3 be sufficiently large. If 1 1 ≥ x C eOη(nK ) nnK D )1/2+η, ≥ 1 K K then 1 κ log x. Nn ≫η K n NX≤x Proof. If κ 1/ log x then the claim follows from the trivial bound 1 1. Other- K ≤ Nn≤x Nn ≥ wise, we may assume κ 1/ log x. From Lemma 2.8, it follows K ≥ P n 1 1 K 1 eOη(nK )(nnK D )1/4+η log x log x + γ + O K K . κ Nn ≥ − j K √x K Nn≤x j=1 X X By [13, Proposition 3] , 1 γ + log 2π γ log D + Q n 1 K ≥ −2 K 2 · K − 12 where γ =0.577 ... is the classical Euler’s constant. Bounding j−1 log n +1 Q 1≤j≤nK K and using the condition on x, we deduce from the previous inequality that ≤ P 1 1 1 γ + log 2π (log x) 1+ O(C−1/4) log D + Q n log n 2 κ Nn ≥ { 1 } − 2 K 2 · K − K − K n NX≤x 1 (log x) 1+ O(C−1/4) log D 1 ≥ { 1 } − 2 K − 1 (log x) 1+ O(C−1/4 + (log x)−1) log x ≥ { 1 } − 1+2η (log x) η + O(C−1/4 + (log x)−1) . ≥ { 1 } Since x C = C (η) and C (η) is sufficiently large, the desired bound follows. ≥ 1 1 1 Taking the logarithmic derivative of ζK(s) yields in the usual way ζ′ Λ (n) (2.14) K (s)= K − ζ (Nn)s K n⊆O XK for Re s > 1, where Λ ( ) is the von Mangoldt Λ-function of the field K defined by { } K · log Np if n is a power of a prime ideal p, (2.15) ΛK(n)= (0 otherwise. Using this identity, we prove a simple elementary lemma. Lemma 2.10. For y 2, ≥ Λ (n) K log(D y). Nn ≪ K n NX≤y 1 Proof. Denote σ =1+ log y . From (2.14), it follows Λ (n) Λ (n) ζ′ K e K = K (σ). Nn ≤ Nnσ −ζ Nn≤y n K X X By Lemmas 2.1 and 2.4, the RHS is 1 1 log D + log y +1 Re + O(n ). ≤ 2 K − σ ρ K ρ X n − o As Re (σ ρ)−1 0 and n log D , the claim follows. { − } ≥ K ≪ K Finally, we end this section with a bound for hH in terms of nK ,DK, and Q = QH . Lemma 2.11. Let H be a congruence class group of K. For ǫ> 0, h eOǫ(nK )D1/2+ǫQ1+ǫ. H ≤ K Proof. Observe, by the definitions of Q and fH in Section 2.1, that if χ is a Hecke character satisfying χ(H) = 1 then f f and Nf Q. Hence, χ | H χ ≤ h = 1 1= #Cl(f). H ≤ χ (mod fH ) Nf≤Q χ (mod f) Nf≤Q χ(XH)=1 fX| fH X Xf | fH 13 nK Recall the classical bound #Cl(f) 2 hK Nf where hK is the class number of K (in the broad sense) from [25, Theorem 1.7], for≤ example. Bounding the class number using Minkowski’s bound (see [30, Lemma 1.12] for example), we deduce that 1 h eOǫ(nK )D1/2+ǫNf eOǫ(nK )D1/2+ǫQ1+ǫ . H ≤ K ≤ K (Nf)ǫ Nf≤Q f | fH fX| fH X For the remaining sum, notice 1 1 −1 1 1 exp O eO(ω(fH )), (Nf)ǫ ≤ − Npǫ ≤ Npǫ ≤ Xf | fH pY|fH Xp|fH where ω(fH) is the number of prime ideals p dividing fH . From [30, Lemma 1.13], we have ω(f ) O (n )+ ǫ log(D Q) whence the desired estimate follows after rescaling ǫ. H ≪ ǫ K K 1+ǫ Remark. Weiss [30, Lemma 1.16] achieves a comparable bound with Q replaced by NfH . This seemingly minor difference will in fact play a key role in improving the range of T in Theorem 1.1. 3. Proof of Theorem 1.1 In this section, we deduce Theorem 1.1 from two key results. Without loss, we may assume H (mod q) is a primitive congruence class group of K. Recall n = [K : Q],D = disc(K/Q) K K | | and h = [I(q): H], Q = Q = max Nf : χ(H)=1 . H H { χ } First, we require a mean value theorem for certain Dirichlet polynomials. Theorem 3.1. Let ǫ > 0 be arbitrary. Let b be a complex-valued function on the prime ideals p of K such that b(p) < | | ∞ p X and b(p)=0 when Np y. Let H (mod q) be a primitive congruence class group of K. If T 1 and ≤ ≥ 5nK 3 1 n K +1 1+ǫ (3.1) y h n 4 D 2 Q 2 T 2 eOǫ(nK ) ≥ H K K then 2 T −it 1 2 b(p)χ(p)Np dt ǫ Np b(p) . −T ≪ log y | | χ (mod q) Z p p χ(XH)=1 X X Remark. Weiss proved essentially the same result [30, Corollary 3.8] as Theorem 3.1 but with condition (3.1) replaced by y (h n2nK D QT 2nK )8. ≥ H K K The exponent 8 happens to be large enough so that it inflates c5 and c6 in (1.9). The purpose of Theorem 3.1 is to ensure that the size of y in the required large sieve inequality does not 14 affect the exponents in Theorem 1.1. Our proof mostly follows Weiss’ arguments but with more careful analysis. The second ingredient is a method for detecting zeros of Hecke L-functions. To simplify its statement, define (3.2) := 2 log D + log Q + n log(T +3)+Θn , L K K K where Θ 1 is sufficiently large, and let 1( ) be an indicator function. ≥ · Proposition 3.2. Let χ be a Hecke character satisfying χ(H)=1 and ǫ (0, 1/4) be arbitrary. Suppose L(s, χ) has a non-trivial zero ρ satisfying ∈ 1+ iτ ρ r | − | ≤ for R Φρ,χ(τ) := 1{|1+iτ−ρ|≤r}(τ) so by assumption T r−1 Φ (τ)dτ 1. ρ,χ ≫ Z−T Select y = e2.3φL and x = e122φL. By Proposition 3.2, it follows that T x χ(p) log Np 2 du −73.2φrL −1 4 1 e r Φρ,χ(τ) r 1+iτ + δ(χ) {|τ|<4r}(τ) dτ. ≪ −T L y Np u Z Z y≤Np + δ(χ) 1 (τ)dτ +1 L {|τ|<4r} Z−T since, for τ T , | | ≤ Φρ,χ(τ) Nχ(r;1+ iτ) r ρ ≪ ≪ L L(Xρ,χ)=0 by Lemma 2.5. Summing (3.8) over χ satisfying χ(H) = 1, we obtain (3.9) x T 2 −73.2φrL 4 2 χ(p) log Np du e N(σ, T, χ) r 1+iτ dτ + r · ≪ L y −T Np u L χ(H)=1 Z χ(H)=1 Z y≤Np 0 fixed and sufficiently small, Lemma 2.11 implies y = e2.3φL D4.6φQ2.3φT 2.3φnK e2.3φΘnK h n1.25nK D1.5Q0.5T 0.5nK +1 1+νeΘnK ≥ K ≥{ H K K } −2/nK −3/5nK since T max nKDK Q , 1 , nK 2, and Θ 1 is sufficiently large. Thus y satisfies the≥ conditions{ of Theorem 3.1,} so the≥ RHS of (3.9)≥ is x 1 (log Np)2 du (3.10) r4 2 + r . ≪ L log y Np u L y y≤Np TheX log R/q savings, which arises from forcing an = 0 when n has a small prime factor, turns out to be decisive in certain applications, such as Bombieri’s proof of (1.2) in [2]. The key ingredients in proof of Theorem 4.1 are the duality argument, properties of Gauss sums, and the fact that the Farey fractions up to height R are R−2-well-spaced (cf. [15, Sections 7.3-7.4]); apart from the duality argument, sufficiently strong analogues of these results over number fields for the purpose of replacing the Dirichlet characters in Theorem 4.1 with Hecke characters do not exist yet. In order to circumvent these deficiencies, we use the Selberg sieve to prove a variant of Theorem 4.1 where the log R/q term on the left hand side is translated to a (log R)−1 savings on the right hand side. The use of the Selberg sieve introduces several sums over integral ideals whose evaluation requires smoothing. Ultimately, this introduces the factor of nK the lower bound for T in Theorem 1.1. 4.1. Preparing for the Selberg Sieve. To apply the Selberg sieve, we will require several weighted estimates involving Hecke characters. Before we begin, we highlight the necessary properties of our weight Ψ. Lemma 4.2. For T 1, let A = T √2n . There exists a weight function Ψ(x) C ((0, )) ≥ K ∈ c ∞ with Mellin transform Ψ(s) such that: (i) 0 Ψ(x) A/2 and Ψ(x) vanishes outside the interval ≤ ≤ b e−2nK /A x e2nK /A ≤ ≤ 2n sinh(s/A) K (ii) Ψ(s) is an entire function and further Ψ(s)= s/A . 17 h i b b (iii) For all complex s = σ + it, A 2nK Ψ(s) e|σ|/A. | | ≤ s | | (iv) For s A, b | | ≤ 2 s 2nK Ψ(s) 1+ | | . | | ≤ 5A2 (v) Uniformly for σ A/√2n , | | ≤ K b Ψ(s) 1. | |≪ (vi) Let bm m≥1 be a sequence of complex numbers with m bm < . Then { } b | | ∞ T 2 ∞ 2 −it P x dx bmm dt bmΨ −T ≪ 0 m x Z m Z m X X Proof. See [30, Lemma 3.2 and Corollary 3.3]; in his notation, Ψ( x)= H2nK (x) with param- eter A = T √2nK . For the remainder of this section, assume: H (mod q) is an arbitrary primitive congruence class group of K. • 0 <ǫ< 1/2 and T 1 is arbitrary. • ≥ Ψ is the weight function of Lemma 4.2. • Next, we establish improved analogues of [30, Lemmas 3.4 and 3.6 and Corollary 3.5]. Lemma 4.3. Let χ (mod q) be a Hecke character satisfying χ(H)=1. For x> 0, 1 nK 1 nK χ(n) x ϕ(q) 4 +1 1+ǫ Oǫ(nK ) 2 2 2 Ψ δ(χ) κK e nK DKQ T Nn · Nn − Nq ≤ · n X Proof. We have χ(n) x ϕ(q) 1 −1+i∞ Ψ δ(χ) κ = L(s +1, χ)Ψ(s)xsds. Nn · Nn − Nq K 2πi n −1−i∞ X Z ∗ If χ (mod q) is induced by the primitive character χ (mod fχ), then b L(s, χ)= L(s, χ∗) (1 χ∗(p)Np−s) − p|q Yp∤fχ implying L(it, χ) 2ω(q) L(it, χ∗) | | ≤ | | where ω(q) is the number of distinct prime ideal divisors of q. Since H (mod q) is primitive, ω(q) 6e4/ǫn + ǫ log(D Q), ≤ K 2 K by [30, Lemma 1.13]. Hence, for Re s = 1, { } − Oǫ(nK ) ǫ/2 ∗ L(s +1, χ) e (DKQ) L(s +1, χ ) | |≪ 18 | | Thus, by Lemma 2.3, we have 1 −1+i∞ L(s +1, χ)Ψ(s)xsds 2πi Z−1−i∞ 1 ∞ +ǫ 1 eOǫ(nK )(D Q) 2 x−1 b (1 + t )( 2 +ǫ)nK Ψ( 1+ it) dt ≪ K | | | − | Z0 as Dχ DKQ. By Lemma 4.2(iii) and (iv), it follows that b ≤ ∞ 1 (1 + t )( 2 +ǫ)nK Ψ( 1+ it) dt | | | − | Z0 A ∞ 2 1 b 1 = (1 + t )( 2 +ǫ)nK Ψ( 1+ it) dt + (1 + t )( 2 +ǫ)nK Ψ( 1+ it) dt | | | − | A | | | − | Z0 Z 2 1 eO(nK )A( 2 +ǫ)nK +1. b b ≪ Collecting the above estimates, the claimed bound follows upon recalling A = T √2nK. Corollary 4.4. Let C be a coset of the primitive congruence class group H (mod q), and let d be an integral ideal coprime to q. For all x> 0, we have 1 nK 1 nK 1 x ϕ(q) κK 1 4 +1 1+ǫ 1 Ψ eOǫ(nK ) n D 2 Q 2 T 2 . Nn Nn − Nq h · Nd ≤ · K K · x n∈C H Xd|n Proof. The proof is essentially the same as that of [30, Corollary 3.5], except for the fact that we have an improved bound in Lemma 4.3. We now apply the Selberg sieve. For z 1, define ≥ 1 (4.1) S = n : p n = Np > z and V (z)= . z { | ⇒ } Nn n NX≤z Lemma 4.5. Let C be a coset of the primitive congruence class group H (mod q). For x> 0 and z 1, ≥ 1 x κ Mz2+2ǫ Ψ K + O , Nn Nn ≤ h V (z) x n H ∈XC∩Sz where 1 nK 1 nK 4 +1 1+ǫ (4.2) M = eOǫ(nK ) n D 2 Q 2 T 2 . · K K Proof. The proof is essentially the same as that of [30, Lemma 3.6], except for the fact that we have an improved bound in Lemma 4.3. 4.2. Proof of Theorem 3.1. Let z be a parameter satisfying 1 z y, which we will specify later. Applying Lemma 4.2 and writing ≤ ≤ bm = b(n)χ(n), Nn=m X19 for each Hecke character χ satisfying χ(H) = 1, it follows that 2 2 T ∞ x dx b(n)χ(n)Nn−it dt b(n)χ(n)Ψ . −T ≪ 0 Nn x χ(H)=1 Z n Z χ(H)=1 n X X X X By the orthogonality of characters and the Cauchy-Schwarz inequality, 2 x 2 x 1 x b(n)χ(n)Ψ hH Nn b(n) Ψ Ψ Nn ≤ | | Nn Nn Nn χ(H)=1 n C∈I(q)/H n∈C n∈C∩Sz X X X X X since z y and b(n) is supported on prime ideals with norm greater than y. By Lemma 4.5, the RHS≤ is x κ h M ′ Nn b(n) 2Ψ K + H ≤ | | Nn V (z) x n∈C C∈XI(q)/H X x κ h M ′ Nn b(n) 2Ψ K + H , ≤ | | Nn V (z) x n X where M ′ = Mz2+2ǫ. Combining the above estimates yields 2 T b(p)χ(p)Np−it dt −T p χ(XH)=1 Z X ∞ ∞ κK x dx 1 x dx Nn b(n) 2 Ψ + h M ′ Ψ ≪ | | V (z) Nn x H x Nn x n 0 0 X Z Z κ h M ′ Nn b(n) 2 K Ψ(0) + H Ψ(1) ≪ | | V (z)| | Nn | | n X κ b h M ′eO(nK ) b Nn b(n) 2 K + H . ≪ | | V (z) Nn n X by Lemma 4.2. Since b(n) is supported on prime ideals whose norm is greater than y, the above is κ h Mz2+2ǫeOǫ(nK ) (4.3) Np b(p) 2 K + H ≪ | | V (z) y p X as M ′ = Mz2+2ǫ with M defined by (4.2). Now, select z satisfying (4.4) y = h MeB1nK z2+4ǫ, H · where B = B (ǫ) > 0 is sufficiently large. From (3.1), it follows that 1 z y and further, 1 1 ≤ ≤ 1 2 n +ǫ z eB nK (n K D ) 2 ≥ K K where B2 = B2(ǫ) > 0 is sufficiently large. Hence, after inputting this choice of z into (4.3), it follows by Corollary 2.9 that 2 T −it 1 2 b(p)χ(p)Np dt ǫ Np b(p) . −T ≪ log z | | χ(H)=1 Z p p X X X 20 Finally, from (3.1) and (4.4), one can verify that log z ǫ log y which completes the proof after rescaling ǫ> 0 appropriately. ≫ 5. Detecting the Zeros of Hecke L-functions 5.1. Setup. The objective of this section is to prove Proposition 3.2 so we fix some notation to be used throughout this section. Let H (mod q) be a congruence class group and let χ (mod q) be a Hecke character, satisfying χ(H) = 1, induced from the primitive character χ∗ (mod f ). Define Q = Q by (1.10), and for T 1, χ H ≥ (5.1) := 2 log D + log Q + n log(T +3)+Θn L K K K 1 where Θ 1 is sufficiently large. Let R 1 be sufficiently large and 0 Theorem 5.1. For any integer M 0 and complex numbers z1,...,zN , there is an integer k with M +1 k M + N such that≥ ≤ ≤ N N zk + + zk 1.007 z k. | 1 ··· N | ≥ 4e(M + N) | 1| N N Remark. One can verify that the expression 4e(M+N) is a decreasing function of N. Any improvement on the constant 4e in Theorem 5.1 would lead to a reduction of the exponent 73.2 in Proposition 3.2, but 4e has been shown by Makai [24] to be best possible. 5.2. A Large Derivative. Denote L′ (5.4) F (s) := (s, χ∗) L and ξ := 1 + r + iτ. Using Theorem 5.1, the goal of this subsection is to show F (s) has a large high order derivative, which we establish in the following lemma. Lemma 5.2. Keeping the above notation, if ǫ (0, 1/4) and r < ǫ/3.8 then ∈ 0 ( 1)krk+1 exp( 16.6 φr ) δ(χ) 1 (τ)+ − F (k)(ξ) − · L · {|τ|<4r} k! · ≫ 2k+1 where φ =1+ 4 ǫ + 16ǫ2 and for some integer k satisfying π (5.5) 25.0 φr k 28.8 φr · L ≤ 21 ≤ · L Proof. By [30, Lemma 1.10], δ(χ) 1 F (s)+ = + G(s) s 1 s ρ − |1+iτX−ρ|<1/2 − uniformly in the region 1+ iτ s < 1/2, | − | where G(s) is analytic and G(s) in this region. Differentiating the above formula k times and evaluating at ξ =1+| r|≪L+ iτ, we deduce ( 1)k δ(χ) 1 (5.6) − F (k)(ξ)+ = + O(4k ) k! · (ξ 1)k+1 (ξ ρ)k+1 L − |1+iτX−ρ|<1/2 − (k) for η > 0 and 0 k+1 δ(χ) 1 (5.8) r δ(χ) 1{|τ| To lower bound the remaining sum over zeros, we wish to apply Theorem 5.1. Denote N = N (Ar;1+ iτ) = # ρ : L(ρ, χ)=0, 1+ iτ ρ Ar . χ { | − | ≤ } Let ǫ (0, 1/4) be fixed. Provided ∈ ǫ (5.10) r < 0 A then by Lemma 2.7 and the definition of in (5.1) it follows that L (5.11) N φAr +4+4δ(χ) ≤ L as Θ is sufficiently large (depending on ǫ). We require a choice of M which depends on the fixed absolute parameters α (0, 1), ǫ (0, 1/4) and A 1, all of which will be specified later. Define ∈ ∈ ≥ φAr +4+4δ(χ) (5.12) M := L α so N αM by (5.11). Thus, from Theoreml 5.1 and (5.3),m ≤ 1 α αM 1 (5.13) (ξ ρ)k+1 ≥ 4e(1 + α) (2r)k+1 |1+iτ−ρ|≤Ar − X for some M +1 k (1 + α)M. To simplify the error term in (5.9), notice r M k so ≤ ≤ L≪ ≪ (4r)k+1 k(4r)k kA−k. L≪ ≪ 1 provided 1 (5.14) r0 < . 4A1 Moreover, select A 1 so that A = √A2 + 1 is given by ≥ 1 4e(1 + α) α (5.15) A =2 (1 + η) 1 α where η (0, 1) is fixed. This choice implies ∈ α αM 1 A−(k+1) 1 ≤ 4e(1 + α) 2k+1(1 + η)k+1 since αk αM. Incorporating (5.13) and the subsequent observations into (5.9) yields ≥ ( 1)krk+1 δ(χ)1 τ < Ar + − F (k)(ξ) (5.16) {| | } k! · α φArL+8 1 k 1 O ≥ 4e(1 + α) · 2k+1 − (1 + η)k+1 n o after bounding N by (5.11) and assuming (5.10) and (5.14) hold. Since k M r R, we may impose R to be sufficiently large, depending on η (0, 1), so that the≫ above≫ errorL≫ term is negligible. Finally, we select α =0.15 and η = 10−4 yielding∈ A =3.752 ... by (5.15). With these choices, conditions (5.10) and (5.14) are automatically satisfied as r0 < ǫ/3.8 < 1/16 by assumption. The desired result follows after inputting these values into (5.16) and recalling M +1 k (1 + α)M. ≤ ≤ 23 Remark. Let us motivate our choice of α =0.15. Ultimately, we will wish to maximize the righthand side of (5.16) when k is large; that is, supposing A k (1 + α)M (1 + α) φr ≈ ≈ α · L 2 by (5.12). By (5.15), notice A 4Cα 1 for η (0, 1) sufficiently small and where α ≈ − ∈ C = 4e(1+α) . Therefore, we select α (0, 1) which minimizes the quantity α α p ∈ 2 4Cα 1 − log Cα +(1+ α) log 2 p α and this turns out to be roughly α =0.15. 5.3. Short Sum over Prime Ideals. Defining ΛK by (2.15), it follows by the Euler produt for L(s, χ∗) that L′ F (s)= (s, χ∗)= χ∗(n)Λ (n)(Nn)−s L − K n X for Re s > 1. Differentiating the above formula k times, we deduce { } ( 1)k+1rk+1 Λ (n)χ∗(n) (5.17) − F (k)(ξ)= K rE (r log Nn) k! · Nn1+iτ · k n X for any integer k 1, where ξ =1+ r + iτ and ≥ uke−u (5.18) E (u)= . k k! As a preliminary observation, notice from Stirling’s formula in the form kke−k√2πk k! kke−k√2πke1/12k ≤ ≤ (see [27]), one can verify k (1 + η)−k if u , (5.19) Ek(u) ≤ e(1 + η) 2(1+η) ≤ (1 + η)−ke−δu if u 2 log k, ≥ 1−δ 1−δ for k 1,η > 0 and δ (0, 1). The goal of this subsection is to bound the infinite sum in (5.17)≥ by an integral average∈ of short sums over prime ideals. Lemma 5.3. Keeping the above notation, assume the integer k satisfies (5.5). Then ∗ x ∗ χ (n)ΛK (n) 2 χ (p) log Np du −16.8φrL −k 1+iτ rEk(r log Nn) r 1+iτ + O e (2.01) Nn · ≤ y Np u n Z y≤Np log Np S r4−k | 3| ≤ (Np)1+0.01r Np≥x X ζ′ log Np r4−k K (1+0.01r) ≤ − ζ − (Np)1+0.01r K Np We prepare for the application of this result by establishing a few preliminary estimates and then end this section with the proof of Theorem 1.2. 26 6.1. Preliminaries. Lemma 6.2. Let χ (mod q) be a Hecke character. For σ 1 and t R, ≥ ∈ 1 1 1 2σ + σ2 nK if χ is primitive, 2 1 1 · 1 2 σ + it ω ≤ + 2 n + + 2 log Nq unconditionally, ω trivial ( 2σ σ K 2σ σ log 2 X | − | · · where the sum is over all trivial zerosω of L(s, χ) counted with multiplicity. ∗ Proof. Suppose χ (mod q) is induced by the primitive character χ (mod fχ). Then χ∗(p) L(s, χ)= P (s, χ)L(s, χ∗) where P (s, χ)= 1 − Nps p|q Yp∤fχ for all s C. Thus, the trivial zeros of L(s, χ) are either zeros of the finite Euler product P (s, χ) or∈ trivial zeros of L(s, χ∗). We consider each separately. From (2.6) and (2.4), observe 1 ∞ 1 ∞ 1 a(χ) + b(χ) σ + it ω 2 ≤ (σ +2k)2 + t2 (σ +2k + 1)2 + t2 ω trivial k=0 k=0 L(ω,χX∗)=0 | − | X X ∞ 1 1 1 n + n ≤ K (σ +2k)2 ≤ 2σ σ2 K Xk=0 Now, if χ is primitive then P (s, χ) 1 and hence never vanishes. Otherwise, notice the zeros of each p-factor in the Euler product≡ of P (s, χ) are totally imaginary and are given by 2πiZ a (p)i + χ log Np for some 0 aχ(p) < 2π/ log Np. Translating these zeros ω ω + it amounts to choosing another representative≤ 0 b (p; t) < 2π/ log Np. Therefore, 7→ ≤ χ 1 ∞ 1 2 σ + it ω 2 ≤ σ2 + (2πk/ log Np)2 ω trivial | − | p|q k=0 P (ω,χX)=0 Xp∤fχ X 1 ∞ 1 2 2 + 2 2 dx ≤ σ 0 σ + (2πx/ log Np) p|q Z Xp∤fχ log Np 1 2 + ≤ 4σ σ2 p|q Xp∤fχ 1 2 + log Nq, ≤ 2σ σ2 log 2 as required. 27 Lemma 6.3. Suppose ψ (mod q) is real and χ (mod q) is arbitrary. For σ = α+1 with α 1 and t R, ≥ ∈ 1 1 1 1 + + + σ ρ 2 σ ρ 2 σ + it ρ 2 σ + it ρ 2 ρ | − | ρ | − | ρ | − | ρ | − | ζKX(ρ)=0 L(ρ,ψX)=0 L(Xρ,χ)=0 L(ρ,ψχX)=0 1 1 2 log(D3 Nq2D )+ log(α +2)+ 2 log π n ≤ α · 2 K ψ α +1 − K h 4 4 + n log(α +2+ t )+ + , K | | α α +1 i where the sums are over all non-trivial zeros of the corresponding L-functions. Remark. If ψ is trivial, notice that the LHS equals 1 1 2 + . σ ρ 2 σ + it ρ 2 ρ | − | ρ | − | ζKX(ρ)=0 L(Xρ,χ)=0 This additional factor of 2 will be useful to us later. Proof. Suppose ψ and χ are induced from the primitive characters ψ∗ and χ∗ respectively. From the identity 0 (1 + ψ∗(n))(1 + Re χ∗(n)(Nn)−it ), ≤ { } it follows that ζ′ L′ L′ L′ 0 Re K (σ)+ (σ, ψ∗)+ (σ + it, χ∗)+ (σ + it, ψ∗χ∗) . ≤ − ζK L L L n o Applying Lemmas 2.1 and 2.4 to each term yields 0 1 log(D3 Nq2D )+ n log(σ +1+ t ) + (log(σ +1)+2σ−1 2 log π)n ≤ 2 K ψ K | | − K 1+ δ(ψ) 1+ δ(ψ) δ(χ)+ δ(χψ) δ(χ)+ δ(χψ) + + + Re + (6.2) α α +1 α + it α +1+ it 1 1n 1 o 1 Re + + + − σ ρ σ ρ σ + it ρ σ + it ρ ρ − ρ − ρ − ρ − n ζKX(ρ)=0 L(ρ,ψX)=0 L(ρ,χX)=0 L(ρ,ψχX)=0 o Since 0 <β< 1, we notice 1 α +1 β α Re = − σ + it ρ σ + it ρ 2 ≥ σ + it ρ 2 n − o | − | | − | and 1 1 1 1 Re + + . α + it α +1+ it ≤ α α +1 n o Rearranging (6.2) and employing these observations gives the desired conclusion. 6.2. Proof of Theorem 1.2. We divide the proof according to whether ψ is quadratic or trivial. The arguments in each case are similar but require some minor differences. 28 6.2.1. ψ is quadratic. Let m be a positive integer, α 1 and σ = α + 1. From the identity ≥ ′ 0 (1 + ψ∗(n))(1 + Re χ∗(n)(Nn)−iγ ) ≤ { } and Lemma 2.2 with s = σ + iγ′, it follows ∞ 1 1 δ(χ)+ δ(ψχ) δ(χ)+ δ(ψχ) (6.3) Re zm + Re n ≤ αm − (α +1 β )2m (α + iγ′)2m − (α +1+ iγ′ β )2m n=1 1 1 n X o − n − o where z = z (γ′) satisfies z z ... and runs over the multisets n n | 1|≥| 2| ≥ (σ ω)−2 : ω is any zero of ζ (s) , { − K } −2 ∗ (σ ω) : ω = β1 is any zero of L(s, ψ ) , (6.4) { − 6 } (σ + iγ′ ω)−2 : ω = β is any zero of L(s, χ∗) , { − 6 1 } (σ + iγ′ ω)−2 : ω = β is any zero of L(s, ψ∗χ∗) . { − 6 1 } Note that the multisets includes trivial zeros of the corresponding L-functions and ψ∗χ∗ is a Hecke character (not necessarily primitive) modulo the least common multiple of fχ and fψ. With this choice, it follows (6.5) (α +1/2)−2 (α +1 β′)−2 z α−2. ≤ − ≤| 1| ≤ The RHS of (6.3) may be bounded via the observation 1 1 1 α−2m 1 α−2m−1m(1 β ), 2m 2m 1−β1 1 (α + it) − (α + it +1 β1) ≤ − (1 + )2m ≪ − − α+it whence ∞ (6.6) Re zm α−2m−1m(1 β ). n ≪ − 1 n=1 n X o On the other hand, by Theorem 6.1, for ǫ> 0, there exists some m0 = m0(ǫ) with 1 m0 (12 + ǫ)M such that ≤ ≤ ∞ Re zm0 ǫ z m0 ǫ (α +1 β′)−2m0 ǫ α−2m0 exp( 2m0 (1 β′)), n ≥ 50 | 1| ≥ 50 − ≥ 50 − α − n=1 n X o where M = z −1 ∞ z . Comparing with (6.6) for m = m , it follows that | 1| n=1 | n| 0 (6.7)P exp( (24+2ǫ) M (1 β′)) M (1 β ). − α − ≪ǫ α − 1 Therefore, it suffices to bound M/α and optimize over α 1. By (6.4), the quantity M is a sum involving non-trivial≥ and trivial zeros of certain L- functions. For the non-trivial zeros, we employ Lemma 6.3 with Dψ DKNq since ψ is qua- ≤ ∗ ∗ dratic. For the trivial zeros, apply Lemma 6.2 in the “primitive” case for ζK(s), L(s, ψ ), L(s, χ ) 29 and in the “unconditional” case for L(s, ψ∗χ∗). Then from (6.5), it follows that M (α +1/2)2 3 α 2α 2 log D + + + log Nq α ≤ α2 · K 2 2α +2 (α + 1)2 log 2 h 4α (6.8) + log(α +2)+2 2 log π + n − (α + 1)2 K 4 4 + n log(α +2+ T )+ + , K α α +1 for α 1. Selecting α = 18, inputting the resulting boundi into (6.7), and fixing ǫ > 0 sufficiently≥ small completes the proof for ψ quadratic. Remark. The final choice of α was somewhat arbitrary because the coefficients of log DK , log Nq and n in (6.8) cannot be simultaneously minimized. As α , it is apparent that the K → ∞ coefficients of log DK and log Nq decrease and converge to a minimum but the coefficient of nK grows arbitrarily large. Hence, in the interest of having relatively small coefficients of comparable size for all quantities, we chose the value α = 18. 6.2.2. ψ is trivial. Now, for ψ trivial, we begin with the identity ′ 0 1+Re χ∗(n)(Nn)−iγ . ≤ { } This similarly implies ∞ 1 1 δ(χ) δ(χ) (6.9) Re zm + Re n ≤ αm − (α +1 β )2m (α + iγ′)2m − (α +1+ iγ′ β )2m n=1 1 1 n X o − n − o for a new choice z = z (γ′) satisfying z z ... and which runs over the multisets n n | 1|≥| 2| ≥ −2 (σ ω) : ω = β1 is any zero of ζK(s) , (6.10) { − 6 } (σ + iγ′ ω)−2 : ω = β is any zero of L(s, χ∗) . { − 6 1 } Following the same arguments as before, we may arrive at (6.7) for the new quantity M = z −1 ∞ z . To bound the non-trivial zeros arising in M, apply Lemma 6.3 with D = | 1| n=1 | n| ψ DK since ψ is trivial. For the trivial zeros, apply Lemma 6.2 in the “primitive” case for both P ∗ ζK(s) and L(s, χ ). 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