A zero density estimate for Dedekind zeta functions
aa r X i v : . [ m a t h . N T ] S e p A ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS
JESSE THORNER AND ASIF ZAMAN
Abstract.
Given a finite group G , we establish a zero density estimate for Dedekindzeta functions associated to Galois extensions K/ Q with Gal( K/ Q ) ∼ = G without assumingunproven progress toward the strong form of Artin’s conjecture. As an application, westrengthen a recent breakthrough of Pierce, Turnage–Butterbaugh, and Wood on the averagesize of the error term in the Chebotarev density theorem. Statement of the main result
The study of zeros of L -functions associated to families of automorphic representations hasstimulated much research over the last century. Zero density estimates, which show that few L -functions within a family can have zeros near the edge of the critical strip, are especiallyuseful. They often allow one to prove arithmetic results which are commensurate with whatthe generalized Riemann hypothesis (GRH) predicts for the family under consideration.Classical triumphs include Linnik’s log-free zero density estimate for L -functions associatedto Dirichlet characters modulo q [14] and the Bombieri–Vinogradov theorem for primes inarithmetic progressions [2]. More recently, Kowalski and Michel [10] proved a zero densityestimate for the L -functions of cuspidal automorphic representations of GL m over Q whichsatisfy the generalized Ramanujan conjecture. See also [4, 5, 13] for related results.All of the aforementioned zero density estimates require an inequality of large sieve type,which indicates that certain Dirichlet polynomials indexed by the representations in thefamily under consideration are small on average. Such results for Dirichlet characters areclassical, while the problem of averaging over higher-dimensional representations (as consid-ered in [4, 10, 13], for example) has relied decisively on the automorphy of the representationsover which one averages (so that one might form the Rankin–Selberg convolution of two rep-resentations, whose L -function will have good analytic properties). In this paper, we prove alarge sieve inequality and corresponding zero density estimate for the L -functions associatedto certain families of representations which are not yet known to be automorphic.Given a family F of Galois extensions K/ Q , we study the zeros of the quotients ζ K ( s ) /ζ ( s ),where ζ K ( s ) is the Dedekind zeta function of K and ζ ( s ) is the Riemann zeta function. This L -function is not known to be automorphic over Q apart from some special cases. For T ≥ σ ≥
0, define N K ( σ, T ) := { ρ = β + iγ ∈ C : ζ K ( ρ ) /ζ ( ρ ) = 0 , β ≥ σ, | γ | ≤ T } . Our main result is
Theorem 1.1.
Let G be a finite nontrivial group of order m + 1 . Let F = F G be a setof number fields K such that K/ Q is a Galois extension with Gal( K/ Q ) ∼ = G . Define F ( Q ) := { K ∈ F : D K ≤ Q } and (1.1) m F ( Q ) := max K ∈ F ( Q ) { K ′ ∈ F ( Q ) : K ∩ K ′ = Q } . If Q, T ≥ and / ≤ σ ≤ , then X K ∈ F ( Q ) N K ( σ, T ) ≪ m m F ( Q )( QT ) m (1 − σ ) (log QT ) m . Remark. If F ( Q ) is non-empty, then m F ( Q ) ≥ ζ K ( s ) /ζ ( s ) may be expressed as a product of L -functions, each of which is automorphic over Q . Assuming this conjecture, Pierce, Turnage–Butterbaugh, and Wood [20, Theorem 6.5] used Kowalski and Michel’s result to establisha weaker variant of Theorem 1.1. The unconditional nature of Theorem 1.1 was motivatedby recent work of Lai and Silberman [12] which extends the Aramata–Brauer theorem;while we do not use their results directly, we appeal to a related theorem of Brauer [3] (seeProposition 4.3). Theorem 1.1 leads to applications on the average error in the Chebotarevdensity theorem, subconvexity of Dedekind zeta functions, and torsion in class groups. Wedescribe these consequences in the next section. Acknowledgements.
We thank Kannan Soundararajan for helpful discussions as well asNicholas Lai and Lior Silberman for bringing [3] to our attention. Work on this paper beganwhile the authors were postdoctoral researchers at Stanford University. Jesse Thorner waspartially supported by a NSF Postdoctoral Fellowship. Asif Zaman was partially supportedby an NSERC fellowship.2.
Applications to the Chebotarev density theorem and subconvexity
Let K/ Q be a Galois extension of number fields with Galois group isomorphic to G , andlet C ⊆ Gal( K/ Q ) be a conjugacy class. Let π ( x ) := { p ≤ x } , π C ( x, K/ Q ) := n p : p ∤ D K , h K/ Q p i = C, p ≤ x o , where the Artin symbol [ K/ Q p ] denotes the conjugacy class of Frobenius automorphisms at-tached to the prime ideals of K which lie over p . It follows from work of Lagarias–Odlyzko[11] that under GRH for the Dedekind zeta function ζ K ( s ) associated to K , we have aneffective form of the Chebotarev density theorem:(2.1) (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ | C || G | √ x log( D K x [ K : Q ] ) , x ≥ (log D K ) . Thus one can accurately count prime ideals of small norm with a given Artin symbol. With-out GRH, one may hope that something similar to (2.1) holds when averaging over a set F of Galois extensions K over Q . We say a set F has a positive level of distribution if, for any A > B = B ( A, F ) > F ( Q ) X K ∈ F ( Q ) max C ⊆ Gal( K/ Q ) (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ A, F x (log x ) A , x ≥ (log Q ) B .This implies that for almost most fields K ∈ F , primes equidistribute in the Chebotarevdensity theorem when x is merely a power of log D K . This is commensurate with what GRHpredicts.If K/ Q ranges over abelian extensions, then class field theory implies that the Artin L -functions of K/ Q are in fact Dirichlet L -functions, which brings us to the setting of [2]. ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 3
An extension of this abelian situation was attained by M. R. Murty and V. K. Murty [15].For a fixed (possibly nonabelian) Galois extension k/ Q , they exhibited a positive level ofdistribution for the family of fields K q = k ( e πi/q ) over Q indexed by integers q ≥ k ∩ Q ( e πi/q ) = Q . While the extension K q / Q with Galois group Gal( k/ Q ) × ( Z /q Z ) × ispossibly nonabelian, the varying large subextension K q /k is always abelian. The key analysisreduces to the study of Dirichlet characters.A recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood [20, Theorems 1.1 and1.4] allows one to control the average size of the error term in the Chebotarev density theoremacross families of extensions K/ Q with a fixed nonabelian Galois group. For example, theyconsider the symmetric group S n , the alternating group A n , and the dihedral group D n oforder 2 n for n ≥
3. We describe part of their work in an exemplary case, namely G = S n with n ≥
5. Assume the strong Artin conjecture for S n (that all irreducible Galois representationsover Q with image isomorphic to S n are in fact cuspidal automorphic representations). Fora number field k , let e k denote the Galois closure over Q . Define(2.3) F n ( D ) := { k : [ k : Q ] = n, Gal( e k/ Q ) ∼ = S n , D k squarefree , D k ≤ D } . If there exists a fixed ε > D ′ ≤ D { k ∈ F n ( D ) : D k = D ′ } ≪ n,ε D − ε F n ( D ) , then for every A > c = c A,n > k with[ k : Q ] = n , squarefree absolute discriminant D k ≤ D , and Gal( e k/ Q ) ∼ = S n satisfy(2.5) (cid:12)(cid:12)(cid:12) π C ( x, e k/ Q ) − | C || G | π Q ( x ) (cid:12)(cid:12)(cid:12) ≤ | C || G | x (log x ) A , x ≫ A,n (log D k ) c (log log D k ) / o (1) with at most O n,ε ( D − ε F n ( D )) exceptions. Therefore, subject to well-believed conjecturesthat do not include GRH, they show that for the Galois closures of most degree n S n -fields,one can obtain highly accurate counts for primes with a given Artin symbol, nearly providinga positive level of distribution per (2.2). For dihedral groups D n ( n ≥
3) and small groupslike S and A , their results are unconditional for two key reasons: they succeed in provinganalogues of (2.4), and the strong Artin conjecture holds for these groups. In these situations,Pierce, Turnage-Butterbaugh, and Wood prove, (for all integers ℓ ≥
1) nontrivial bounds for ℓ -torsion in the class groups of almost all fields in the family under consideration.As an application of Theorem 1.1, we establish a positive level of distribution for F withoutrecourse to unproven progress toward the strong Artin conjecture when m F ( Q ) is suitablysmall. Our results improve when one restricts consideration to the primes that split com-pletely. Theorem 2.1.
Let G be a nontrivial finite group of order m + 1 . Let F be a set of numberfields K such that K/ Q is a Galois extension with Gal( K/ Q ) ∼ = G . For Q ≥ , let F ( Q ) = { K ∈ F : D K ≤ Q } . If there exists a fixed ε ∈ (0 , such that (2.6) m F ( Q ) ≪ m,ε Q − ε F ( Q ) , where m F ( Q ) is given by (1.1) , then all of the following hold with δ = ε/ (10 m ) .(i) If ε − A log log x ≤ log Q ≤ x δ for some A > and x ≥ , then F ( Q ) X K ∈ F ( Q ) max C ⊆ Gal( K/ Q ) (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ m,ε,A x (log x ) A , JESSE THORNER AND ASIF ZAMAN where C ranges over the conjugacy classes in G .(ii) If log x ≤ log Q ≤ x δ , then F ( Q ) X K ∈ F ( Q ) (cid:12)(cid:12)(cid:12) π { } ( x, K/ Q ) − | G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ m,ε x − δ . (iii) For all fields K ∈ F ( Q ) with at most O m,ε ( Q − ε F ( Q )) exceptions, the Dedekind zetafunction ζ k ( s ) of each subfield k ⊆ K whose Galois closure e k over Q equals K satisfies | ζ k (1 / | ≪ [ k : Q ] ,ε D − δ k . (iv) Let η > , and let ℓ ≥ be a positive integer. For all fields K ∈ F ( Q ) with at most O m,ε ( Q − ε F ( Q )) exceptions, the ℓ -torsion subgroup of the class group Cl k [ ℓ ] of eachsubfield k ⊆ K whose Galois closure e k over Q equals K satisfies | Cl k [ ℓ ] | ≪ [ k : Q ] ,ℓ,η D − ℓ ([ k : Q ] − + ηk . Remark.
Parts (i) and (ii) follow from the more general Theorem 8.8, which is unconditional.Estimate (i) improves upon the unconditional results in [23] in the stated range of Q . Finally,note that (2.6) implicitly requires F ( Q ) ≫ m,ε Q ε since m F ( Q ) ≥ K ∩ K ′ = Q for Galois extensions K, K ′ of Q then,as K ∩ K ′ is Galois over Q , we have K N = K ∩ K ′ = ( K ′ ) N ′ for some non-trivial normalsubgroups N ⊆ Gal( K/ Q ) and N ′ ⊆ Gal( K ′ / Q ). Thus if G is simple, thenmax K ∈ F ( Q ) { K ′ ∈ F ( Q ) : K ∩ K ′ = Q } = 1 . With field counting results of Pierce, Turnage-Butterbaugh, and Wood [20] and Bhargava,Shankar, and Wang [1], these observations imply the following result.
Corollary 2.2.
For n ≥ , let G = A n or S n . Let D ≥ and define F = F G = { e k : k/ Q degree n with Gal( e k/ Q ) ∼ = G, D k ≤ D } , where e k is the Galois closure of k over Q . Then (2.6) , and hence Theorem 2.1(i)–(iv), holdswith Q = D | G | / when:(a) G = A n for n ≥ and ε = 1 / (10 n ) unconditionally.(b) G = S n for n ≥ , provided that for some ε ∈ (0 , , one has (2.7) max d { e k ∈ F : e k A n = Q ( √ d ) } ≪ n,ε D + n − ε . Remark.
For the case G = A n , this appears to be the first unconditional instance whereinfinitely many unsolvable extensions of arbitrarily large degree possess any of the following:a positive level of distribution in the Chebotarev density theorem, subconvexity for theirDedekind zeta functions, and non-trivial bounds on ℓ -torsion in their class groups.In addition to removing the hypothesis of Artin’s conjecture, Corollary 2.2 strengthensthe conclusion of [20, Theorems 1.13]. However, for S n -fields, the relation between (2.7) and(2.4) from [20, Theorem 1.15] may not be entirely obvious, so we explain the connection ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 5 here. From our earlier discussion, if Gal( e k/ Q ) ∼ = Gal( e k ′ / Q ) ∼ = S n then e k ∩ e k ′ = Q if and onlyif there exists a fundamental discriminant d such that e k A n = Q ( √ d ) = ( e k ′ ) A n . Thus,max e k ∈ F |{ e k ′ ∈ F : Gal( e k ′ / Q ) ∼ = S n , e k ∩ e k ′ = Q , D k ≤ D }| = max d |{ e k ∈ F : e k A n = Q ( √ d ) , D k ≤ D }| . Restricting e k ∈ F to squarefree discriminants D k allows Pierce, Turnage-Butterbaugh, andWood to show that the second maximum is(2.8) ≪ n max D ′ ≤ D |{ k ∈ F n ( D ) : D k = D ′ }| , where F n ( D ) is given by (2.3). A deep “discriminant multiplicity conjecture” of Duke[6, §
3] suggests that (2.8) is ≪ n,ε D ε for any fixed ε >
0. In light of the lower bound | F n ( D ) | ≫ n D / /n that follows from work of Bhargava, Shankar, and Wang [1, Corollary1.3], sufficient progress toward the discriminant multiplicity conjecture would yield a resultakin to Corollary 2.2 for degree n S n -fields with squarefree discriminant. The problem ofobtaining sufficient progress toward the discriminant multiplicity conjecture appears to bequite difficult; see [20, 19] for recent comprehensive accounts.For other groups such as G = S , A , S , and D n ( n ≥ G -fields and make sufficient progress on the arising discriminant multiplicity problem. Theirstrategy produces unconditional results such as bounds on ℓ -torsion in the correspondingclass groups. A level of distribution for each of those families was also demonstrated byBrumley, Thorner, and Zaman [4]. As illustrated with S n above, Theorem 2.1 is flexibleenough to use those same families; in those cases, (2.6) holds by the results of [20].2.1. Outline of the paper.
Sections 3 and 4 review facts about Artin L -functions. Sec-tions 5 and 6 develop a large sieve inequality for Dedekind zeta functions which leads to theproof of Theorem 1.1 in Section 7. Finally, Sections 8 and 10 focus on our application to theChebotarev density theorem, with the proof of Theorem 2.1 in Section 9.3. Artin L -functions We briefly recall the definition of an Artin L -function from [16, Chapter 2, Section 2]. Let K/ Q be a Galois extension of number fields with Galois group G . For each prime p , and aprime ideal p of K lying above p , we define the decomposition group D p to be Gal( K p / Q p ),where K p (resp. Q p ) is the completion of K (resp. Q ) at p (resp. p ). We have a map D p toGal( k p /k p ) (the Galois group of the residue field extension), which is surjective by Hensel’slemma. The kernel of this map is the inertia group I p . We thus have the exact sequence1 → I p → D p → Gal( k p /k p ) → . The group Gal( k p /k p ) is cyclic with generator x x p , where p is the cardinality of k p . Wecan choose an element σ p ∈ D p whose image in Gal( k p /k p ) is this generator. We call σ p a Frobenius element at p ; it is well-defined modulo I p . We have that I p is trivial for allunramified p , and for these p , σ p is well-defined. For p unramified, we denote by σ p theconjugacy class of Frobenius elements at primes p above p . JESSE THORNER AND ASIF ZAMAN
Let ρ : G → GL n ( C ) be a representation of G , and let χ denote its character. Let V bethe underlying complex vector space on which ρ acts, and let V I p be the subspace of V onwhich I p acts trivially. We now define L p ( s, χ ) = ( det( I n − ρ ( σ p ) p − s ) − if p is unramified in L, det( I n − ρ ( σ p ) | V I p p − s ) − if p is ramified in L. Note that the matrix ρ ( σ p ) | V I p remains the same if one changes the prime p lying above p . Indeed, if p is unramified, then ρ ( σ p ) | V I p = ρ ( σ p ). We then define(3.1) L ( s, χ ) = Y p L p ( s, χ ) . We have that | α j,χ ( p ) | ≤ j and p , so L ( s, χ ) is an absolutely convergent Dirichletseries and Euler product for Re( s ) > Q , we let Γ R ( s ) = π − s/ Γ( s/ L ∞ ( s, χ ) = Γ R ( s ) a ( χ ) Γ R ( s + 1) χ (1) − a ( χ ) , where a ( χ ) is the dimension of the +1 eigenspace of complex conjugation. Let q χ denote theconductor of χ over Q . The function(3.2) Λ( s, χ ) := q s/ χ L ( s, χ ) L ∞ ( s, χ )has a meromorphic continuation to C , and there exists W ( χ ) ∈ C of modulus one such thatΛ( s, χ ) = W ( χ )Λ(1 − s, χ ) for all s ∈ C at which Λ( s, χ ) is holomorphic. Here, χ is thecomplex conjugate of χ , and L ( s, χ ∞ ) = L ( s, χ ∞ ). Conjecturally, Λ( s, χ ) is entire when χ isnontrivial. Define(3.3) C ( χ ) := q χ χ (1) − a ( χ ) . Note the trivial character χ = 1 corresponds to the Riemann zeta function L ( s, χ ) = ζ ( s )which has trivial Artin conductor q χ = 1 and C ( χ ) = 1.4. Dedekind zeta function for the compositum of linearly disjoint fields
The Aramata–Brauer theorem.
For a finite group G , let reg G (resp. 1 G ) denotethe character of the regular (resp. trivial) representation of G . For a Galois extension K/ Q of number fields, let χ K denote the character of Gal( K/ Q ) defined by(4.1) χ K = reg Gal( K/ Q ) − Gal( K/ Q ) . Its associated Artin L -function is given by(4.2) L ( s, χ K ) = ζ K ( s ) /ζ ( s ) . Using the notation of Section 3, the Artin L -function L ( s, χ K ) may be expressed as(4.3) L ( s, χ K ) = Y p [ K : Q ] − Y j =1 (cid:16) − α j,K ( p ) p s (cid:17) − = X n a K ( n ) n − s . The Dirichlet series coefficients a K ( n ) are defined in terms of the local roots α j,K ( p ) via thisidentity. We make the following observations: • For each p , the set of local roots { α j,K ( p ) } j is invariant under conjugation. • q χ K = D K and C ( χ K ) ≍ [ K : Q ] D K . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 7
The first observation follows from (4.1) because the local roots are those of a characteristicpolynomial of the regular representation of Gal( K/ Q ) excluding the trivial root { } . Since D K is the conductor of ζ K ( s ) over Q , the second observation follows from (4.2). Theorem 4.1 (Aramata–Brauer) . The Artin L -function L ( s, χ K ) is entire of order one. Composite fields.
Let K/ Q (resp. K ′ / Q ) be a Galois extension of number fields withGalois group G (resp. G ′ ). Write m = [ K : Q ] − m ′ = [ K ′ : Q ] −
1. Assume that K ∩ K ′ = Q . This condition yields a natural isomorphismGal( K ′ K/ Q ) ∼ = −→ Gal( K/ Q ) × Gal( K ′ / Q ) = G × G ′ which sends σ ( σ | K , σ | K ′ ). Moreover, via this isomorphism, every irreducible characterof Gal( K ′ K/ Q ) is an (external) tensor product of the shape χ ⊗ χ ′ , where χ (resp. χ ′ ) isan irreducible character of G (resp. G ′ ). Thus, as characters, reg G × G ′ = reg G ⊗ reg G ′ =1 G × G ′ + χ K ⊗ G ′ + 1 G ⊗ χ K ′ + χ K ⊗ χ K ′ . In terms of Artin L -functions, this yields(4.4) ζ K ′ K ( s ) = ζ ( s ) L ( s, χ K ) L ( s, χ K ′ ) L ( s, χ K ⊗ χ K ′ ) . The Artin L -function L ( s, χ K ⊗ χ K ′ ) is defined by the character χ K ⊗ χ K ′ of Gal( K ′ K/ Q ).The following lemma provides useful bounds for the conductor of q χ K ⊗ χ K ′ . Lemma 4.2. If K/ Q and K ′ / Q are Galois extensions and K ∩ K ′ = Q , then D K ′ K | D [ K ′ : Q ] K D [ K : Q ] K ′ .Proof. We appeal to standard facts in [18, Part III, Section 2]. As K ∩ K ′ = Q , the relativediscriminant satisfies D K ′ K = D [ K ′ : Q ] K N K/ Q d KK ′ /K . By definition, d KK ′ /K is the ideal of O K generated by all bases of K ′ K/K contained in O K ′ K . Since K ∩ K ′ = Q , any basis of K ′ / Q contained in O K ′ is a basis for K ′ K/K contained in O K ′ K . Hence d KK ′ /K contains D K ′ O K ,so d KK ′ /K divides D K ′ O K . Since K ∩ K ′ = Q , we have that N K/ Q ( D K ′ O K ) = D [ K : Q ] K ′ so weconclude that D K ′ K divides D [ K ′ : Q ] K D [ K : Q ] K ′ , as desired. (cid:3) Proposition 4.3 (Brauer) . Let K/ Q and K ′ / Q be Galois extensions of number fields withrespective degrees m + 1 and m ′ + 1 . If K ∩ K ′ = Q , then the Artin L -function L ( s, χ K ⊗ χ K ′ ) = ζ K ′ K ( s ) ζ ( s ) ζ K ( s ) − ζ K ′ ( s ) − is entire. Further, its Artin conductor divides D m ′ K D mK ′ , and C ( χ K ⊗ χ K ′ ) ≪ m,m ′ D m ′ K D mK ′ ≪ m,m ′ C ( χ K ) m ′ C ( χ K ′ ) m . Proof.
Brauer [3] proved that L ( s, χ K ⊗ χ K ′ ) is entire. The expression for L ( s, χ K ⊗ χ K ′ ) inProposition 4.3 as a ratio of Dedekind zeta functions follows from (4.2) and (4.4). We seefrom the functional equations of these Dedekind zeta functions that the Artin conductor of L ( s, χ K ⊗ χ K ′ ) equals D K ′ K D − K D − K ′ . By Lemma 4.2, the Artin conductor indeed divides D m ′ K D mK ′ as m + 1 = [ K : Q ] and m ′ + 1 = [ K ′ : Q ]. Since χ K ⊗ χ K ′ is a representation ofdimension m ′ m , it follows from Lemma 4.2 that C ( χ K ⊗ χ K ′ ) ≪ m,m ′ D m ′ K D mK ′ . (cid:3) Preparations for the large sieve
Let G be a finite group, let m = | G | −
1, and let
K, K ′ ∈ F = F G . We define L p ( s, χ K × χ K ′ ) = m Y j =1 m Y j ′ =1 (cid:16) − α j,K ( p ) α j ′ ,K ′ ( p ) p s (cid:17) − (5.1) JESSE THORNER AND ASIF ZAMAN and L ∗ ( s, χ K × χ K ′ ) = Y p ∤ D K D K ′ L p ( s, χ K × χ K ′ ) = X n ≥ a K × K ′ ( n ) n s . When K ∩ K ′ = Q , Lemma 4.2 tells us that p ∤ D K D K ′ if and only if p is unramified in K ′ K . This facilitates a straightforward description of the Euler factors L p ( s, χ K ⊗ χ K ′ ) for p ∤ D K D K ′ , namely L p ( s, χ K ⊗ χ K ′ ) = L p ( s, χ K × χ K ′ ). Thus for Re( s ) >
1, we have(5.2) L ∗ ( s, χ K × χ K ′ ) = L ( s, χ K ⊗ χ K ′ ) Y p | D K D K ′ L p ( s, χ K ⊗ χ K ′ ) − . When K ∩ K ′ = Q , the Dirichlet series L ∗ ( s, χ K × χ K ′ ) still converges absolutely for Re( s ) > | α j,K ( p ) | ≤
1. However, it is now unclear whether L ( s, χ K × χ K ′ ) has an analytic continuation past the line Re( s ) = 1.5.1. The coefficients a K × K ′ ( n ) . A partition λ = ( λ ( i )) ∞ i =1 is a sequence of nonincreasingnonnegative integers λ (1) ≥ λ (2) ≥ · · · with only finitely many nonzero entries. For apartition λ , let ℓ ( λ ) be the length of λ (number of nonzero λ ( i )), and let | λ | = P ∞ i =1 λ ( i ).For a set { α , . . . , α m } and a partition λ with ℓ ( λ ) ≤ m , let s λ ( α , . . . , α m ) = det[( α λ ( j )+ m − ji ) ij ] / det[( α m − ji ) ij ]be the Schur polynomial associated to λ . If | λ | = 0, then s λ ( α , . . . , α m ) ≡ L p ( s, χ K ) = m Y j =1 (1 − α j,K ( p ) p − s ) − = ∞ X k =0 s ( k, ,... ) ( A K ( p )) p − ks , which one obtains by expanding the left hand side as a product of geometric sums. Whenthe above identity is applied to (3.1), we arrive at the identity a K ( p k ) = s ( k, ,... ) ( A K ( p )). Foran integral ideal n with factorization n = Q p p ord p ( n ) (where ord p ( n ) = 0 for all but finitelymany p ), the multiplicativity of a K ( n ) tells us that(5.3) a K ( n ) = Y p s (ord p ( n ) , ,... ) ( A K ( p )) . By Cauchy’s identity, if p ∤ D K D K ′ , then by (5.1), L p ( s, χ K × χ K ′ ) = ∞ X j =0 a K × K ′ ( p j ) p js = X ℓ ( λ ) ≤ n s λ ( A K ( p )) s λ ( A K ′ ( p )) p −| λ | s , Re( s ) > . A comparison of coefficients of p − js for j ≥ a K × K ′ ( p j ) = X ℓ ( λ ) ≤ m | λ | = j s λ ( A K ( p )) s λ ( A K ′ ( p )) . By the multiplicativity of a K × K ′ ( n ), we see that if ( n, D K D K ′ ) = 1, then a K × K ′ ( n ) = Y p a K × K ′ ( p ord p ( n ) ) = X ( λ p ) p ∈ λ [ n ] h Y p s λ p ( A K ( p )) ih Y p s λ p ( A K ′ ( p )) i , (5.4)where ( λ p ) p denotes a sequence of partitions indexed by the primes p and λ [ n ] := { ( λ p ) p : ℓ ( λ p ) ≤ m and | λ p | = ord p ( n ) for each p } . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 9
Sums of the coefficients a K × K ′ ( n ) . Let φ be a smooth test function which is sup-ported in a compact subset of [ − , b φ ( s ) = Z R φ ( y ) e sy dy be its Laplace transform. Then b φ ( s ) is an entire function of s , and integration by parts tellsus that for any fixed integer k ≥ b φ ( s ) ≪ φ,k e | Re( s ) | | s | − k . Let T ≥
1. By Fourier inversion, for any x > c ∈ R , we have the identity φ ( T log x ) = 12 πiT Z c + i ∞ c − i ∞ b φ ( s/T ) x − s ds. Lemma 5.1.
Let
T, x ≥ . If K ∩ K ′ = Q and K, K ′ ∈ F ( Q ) , then for every fixed ε > , (cid:12)(cid:12)(cid:12) X ( n,D K D K ′ )=1 a K × K ′ ( n ) φ (cid:16) T log nx (cid:17)(cid:12)(cid:12)(cid:12) ≪ m,φ,ε √ xQ m + ε T m . Proof. If K ∩ K ′ = Q , then by Proposition 4.3 and (5.2), the sum we want to estimate equals (cid:12)(cid:12)(cid:12) πiT Z + i ∞ − i ∞ L ( s, χ K ⊗ χ K ′ ) Q p | D K D K ′ L p ( s, χ K ⊗ χ K ′ ) b φ ( s/T ) x s ds (cid:12)(cid:12)(cid:12) . Since | α j,K ( p ) | ≤ | L ( + it, χ K ⊗ χ K ′ ) | ≪ m Q m (2 + | t | ) m . Since | α j,K ( p ) | ≤ ε > Y p | D K D K ′ | L p ( + it, χ K ⊗ χ K ′ ) | − ≤ Y p | D K D K ′ (1 + p − ) m ≪ m,ε Q ε . Thus by (5.6), the integral on the line Re( s ) = 1 / ≪ m √ xT Q m + ε Z ∞−∞ (2 + | t | ) m (cid:12)(cid:12)(cid:12) b φ (cid:16) T (cid:16)
12 + it (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) dt ≪ m,φ √ xT Q m + ε Z ∞−∞ (2 + | t | ) m min n , T m +2 (2 + | t | ) m +2 o dt, which is bounded as claimed. (cid:3) We also require an estimate corresponding to Lemma 5.1 when K ∩ K ′ = Q . Lemma 5.2.
Let
T, x ≥ . If K, K ′ ∈ F ( Q ) , then X ( n,D K D K ′ )=1 | a K × K ′ ( n ) | φ (cid:16) T log nx (cid:17) ≪ m,φ x (log x ) m T + √ xT m . Proof.
Since each complex number in the sets A K ( p ) , A K ′ ( p ) have modulus at most 1, itfollows from (5.4) that if ( n, D K D K ′ ) = 1 then | a K × K ′ ( n ) | ≤ d m ( n ), where d m ( n ) is theDirichlet coefficient at n for ζ ( s ) m . A standard calculation yields X ( n,D K D K ′ )=1 | a K × K ′ ( n ) | φ (cid:16) T log nx (cid:17) ≪ m,φ X n ∈ ( x,xe /T ] d m ( n ) ≪ m,φ x (log x ) m T + √ xT m . (cid:3) A large sieve inequality for Artin representations
We use the combinatorial identities for a K ( n ) and a K × K ′ ( n ) from the previous section toprove a large sieve inequality for the Dirichlet coefficients of L ( s, χ K ). We then apply ourlarge sieve to bound the mean value of a certain Dirichlet polynomial which naturally arisefrom the method of detecting zeros of L -functions studied in [21]. For convenience, we define q ( n ) to be the indicator function of the condition that ( n, q ) = 1, where n and q are positiveintegers. Recall m = | G | − F = F G is a set of fields K with Gal( K/ Q ) ∼ = G . Theorem 6.1.
Let b : Z → C be a function. If Q, T, x ≥ , then X K ∈ F ( Q ) (cid:12)(cid:12)(cid:12) X n ∈ ( x,xe /T ]( n,D K )=1 a K ( n ) b ( n ) (cid:12)(cid:12)(cid:12) ≪ m (cid:16) x (log x ) m T m F ( Q ) + √ xQ m +1) T m (cid:17) X n ∈ ( x,xe /T ] | b ( n ) | . Proof.
It suffices to consider b such that P n ∈ ( x,xe /T ] | b ( n ) | = 1. By duality, we have(6.1) X K ∈ F ( Q ) (cid:12)(cid:12)(cid:12) X n ∈ ( x,xe /T ]( n,D K )=1 a K ( n ) b ( n ) (cid:12)(cid:12)(cid:12) ≤ sup k β k =1 X n ∈ ( x,xe /T ] (cid:12)(cid:12)(cid:12) X K ∈ F ( Q ) a K ( n ) D K ( n ) β ( K ) (cid:12)(cid:12)(cid:12) , where β ranges over the functions from F ( Q ) to C which satisfy P π ∈ F ( Q ) | β ( π ) | = 1. Using(5.3), we can restate the right hand side of (6.1) as(6.2) sup k β k =1 X K ∈ F ( Q ) (cid:12)(cid:12)(cid:12) X n ∈ ( x,xe /T ] h Y p s (ord p ( n ) , ,... ) ( A K ( p )) i D K ( n ) β ( K ) (cid:12)(cid:12)(cid:12) . We bound (6.2) by embedding it in the “completed sum”(6.3) sup k β k =1 X n ∈ ( x,xe /T ] X ( λ p ) p ∈ λ [ n ] (cid:12)(cid:12)(cid:12) X K ∈ F ( Q ) h Y p s λ p ( A K ( p )) i D K ( n ) β ( K ) (cid:12)(cid:12)(cid:12) . Fix a nonnegative smooth function φ , compactly supported inside of [ − , φ ( T log tx ) is a pointwise upper bound for the indicator function of ( x, xe /T ]. Then (6.3) is(6.4) ≤ sup k β k =1 X n X ( λ p ) p ∈ λ [ n ] (cid:12)(cid:12)(cid:12) X K ∈ F ( Q ) h Y p s λ p ( A K ( p )) i D K ( n ) β ( K ) (cid:12)(cid:12)(cid:12) φ (cid:16) T log nx (cid:17) . We expand the square, swap the order of summation, and apply (5.4) to see that (6.4) equals(6.5) sup k β k =1 X K,K ′ ∈ F ( Q ) β ( K ) β ( K ′ ) h X ( n,D K D K ′ )=1 a K × K ′ ( n ) φ (cid:16) T log nx (cid:17)i . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 11
The innermost sum is evaluated using Lemma 5.1 or Lemma 5.2, depending on whether K ∩ K ′ = Q or not. Using the inequality of arithmetic and geometric means, the trivialbound m F ( Q ) ≤ F ( Q ), and the assumption that k β k = 1, we bound (6.5) by ≪ m,φ x (log x ) m T m F ( Q ) + √ xQ m + ε T m F ( Q ) . It remains to estimate F ( Q ). Since Q ≥ m ≥
1, a slight modification in the proofof [7, Theorem 1.1] due to Ellenberg and Venkatesh shows that F ( Q ) ≪ m Q m +1) (take r = 1 and c = m in [7, Equation 2.6]). This yields the estimate. (cid:3) Corollary 6.2.
Let
Q, T ≥ . If y ≫ m ( QT ) m +1) and u ∈ [ y, y ] , then X K ∈ F ( Q ) Z T − T (cid:12)(cid:12)(cid:12) X y
Gallagher [8, Theorem 1] proved that if P n | c ( n ) | < ∞ , then Z T − T (cid:12)(cid:12)(cid:12) X n c ( n ) n − it (cid:12)(cid:12)(cid:12) dt ≪ T Z ∞ (cid:12)(cid:12)(cid:12) X n ∈ ( x,xe /T ] c ( n ) (cid:12)(cid:12)(cid:12) dxx . We choose c ( n ) = a K ( n ) b ( n ) D K ( n ), in which case X K ∈ F ( Q ) Z T − T (cid:12)(cid:12)(cid:12) X ( n,D K )=1 a K ( n ) b ( n ) n − it (cid:12)(cid:12)(cid:12) dt ≪ T Z ∞ X K ∈ F ( Q ) (cid:12)(cid:12)(cid:12) X n ∈ ( x,xe /T ]( n,D K )=1 a K ( n ) b ( n ) (cid:12)(cid:12)(cid:12) dxx . We apply Theorem 6.1 and bound the above display by ≪ m X n | b ( n ) | n (cid:16) (log n ) m m F ( Q ) + Q m +1) T m n / (cid:17) . Choose y ≫ m Q m +1) T m . Let u ∈ [ y, y ] and choose b ( n ) = ( (log n ) /n if n is prime and n ∈ [ y, u ],0 otherwise.Then we may conclude that X K ∈ F ( Q ) Z T − T (cid:12)(cid:12)(cid:12) X y
In this section, we use Corollary 6.2 to prove Theorem 1.1. Our approach closely followsthe approach in [21]. Indeed, by the results in Section 3, it follows that for each K ∈ F , the L -function L ( s, χ K ) is in the class S ( m ) described in [21, Subsections 1.1-1.4]. We will relyheavily on the results in [21] with some minor modifications.Let K ∈ F ( Q ). We have the Dirichlet series identity − L ′ L ( s, χ K ) = X n λ K ( n )Λ( n ) n s , Re( s ) > , where Λ( n ) is the usual von Mangoldt function for ideals of Q and(7.1) λ K ( n ) = (P mj =1 α j,K ( p ) k if n = p k , k ≥ p a prime ideal,0 otherwise.Since we have the uniform bound | α j,K ( p ) | ≤
1, it is straightforward to prove that(7.2) X n | λ K ( n )Λ( n ) | n η ≤ mη , η > . By hypothesis, Λ( s, χ K ) is entire of order 1 and has the Hadamard product representationΛ( s, χ K ) = e a K + b K s Y ρ (cid:16) − sρ (cid:17) e s/ρ , where ρ ranges over the nontrivial zeros of L ( s, χ K ). Such zeros ρ = β + iγ satisfy 0 < β < Lemma 7.1. If < η ≤ and t ∈ R , then X ρ η − β | η + it − ρ | ≤ m log C ( χ K ) + m log(2 + | t | ) + 2 nη + O ( m ) and { ρ : | ρ − (1 + it ) | ≤ η } ≤ ηn log C ( χ K ) + 5 ηn log(2 + | t | ) + O ( m ) .Proof. This is [21, Lemma 3.1] applied to L ( s, χ K ). (cid:3) Detecting zeros.
Recall K ∈ F ( Q ) so C ( χ K ) ≤ Q . Let k ≥ QT ) − < η ≤ (220 m ) − . By [21, Equation 4.2], we find that(7.4) (cid:12)(cid:12)(cid:12) ( − k k ! (cid:16) L ′ L ( s , χ K ) (cid:17) ( k ) − X | s − ρ | < η s − ρ ) k +1 (cid:12)(cid:12)(cid:12) ≪ m log( QT )(200 η ) k . Lemma 7.2.
Let τ ∈ R satisfy | τ | ≤ T , and let η satisfy (7.3) . If L ( s, χ K ) has a zero ρ satisfying | ρ − (1 + it ) | ≤ η and M > ⌈ ηm log( QT ) + O ( m ) ⌉ , then for some integer k ∈ [ M, M ] , one has (recall s = 1 + η + iτ ) (cid:12)(cid:12)(cid:12) X ρ | s − ρ |≤ η s − ρ ) k +1 (cid:12)(cid:12)(cid:12) ≥ η ) k +1 . Proof.
This is [21, Lemma 4.2] applied to L ( s, χ K ). (cid:3) We proceed to the upper bound.
Lemma 7.3.
Let K ∈ F ( Q ) . Let T ≥ . Let τ ∈ R satisfy | τ | ≤ T , and let η satisfy (7.3) . Let M ≥ and k ∈ [ M, M ] be integers, and put N = exp( M/ (300 η )) and N =exp(40 M/η ) . Then (cid:12)(cid:12)(cid:12) η k +1 k ! (cid:16) L ′ L ( s , χ K ) (cid:17) ( k ) (cid:12)(cid:12)(cid:12) ≤ η Z N N (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) duu + O (cid:16) ηm log( QT )(110) k (cid:17) . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 13
Proof.
This is very similar to [21, Lemma 4.3]. Since η >
0, one has (cid:12)(cid:12)(cid:12) η k +1 k ! (cid:16) L ′ L ( s, χ K ) (cid:17) ( k ) (cid:12)(cid:12)(cid:12) = η (cid:12)(cid:12)(cid:12) X n λ K ( n )Λ( n ) n η + iτ ( η log n ) k k ! (cid:12)(cid:12)(cid:12) . In the proof of [21, Lemma 4.3], it is shown that ( η log n ) k /k ! ≤ n − η/ (110) − k for n / ∈ [ N , N ].Thus by (7.2),(7.5) (cid:12)(cid:12)(cid:12) X n/ ∈ [ N ,N ] λ K ( n )Λ( n ) n η + iτ ( η log n ) k k ! (cid:12)(cid:12)(cid:12) ≪ k X n | λ K ( n )Λ( n ) | n η/ ≪ m log( QT )(110) k . To handle the composite n with n ∈ [ N , N ], note that since (log u ) k ≤ k ! u for all k, u ≥ η log n ) k k ! = (2 η ) k (log n ) k k ! ≤ (2 η ) k n ≤ n (110) k . Since | λ K ( n ) | ≤ n , the above bound implies that (cid:12)(cid:12)(cid:12) X n ∈ [ N ,N ] n composite λ K ( n )Λ( n ) n η + iτ ( η log n ) k k ! (cid:12)(cid:12)(cid:12) ≪ m (110) k X p X r ≥ log pp r ( + η ) ≪ m (110) k X p log pp η . By (7.2), we conclude that(7.6) (cid:12)(cid:12)(cid:12) X n ∈ [ N ,N ] n composite λ K ( n )Λ( n ) n η + iτ ( η log n ) k k ! (cid:12)(cid:12)(cid:12) ≪ m log( QT )(110) k . Note that λ K ( p )Λ( p ) = a K ( p ) log p for all p . Hence by partial summation and (7.2), (cid:12)(cid:12)(cid:12) X p ∈ [ N ,N ] λ K ( p )Λ( p ) p η + iτ ( η log n ) k k ! (cid:12)(cid:12)(cid:12) ≤ η Z N N (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) duu + X N ≤ p ≤ N | λ K ( p ) | p η k (log p ) k +1 k ! ≤ η Z N N (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) duu + 1(110) k X n | λ K ( n )Λ( n ) | n η/ = η Z N N (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) duu + O (cid:16) m log( QT )(110) k (cid:17) . (7.7)The lemma follows once we sum (7.5), (7.6), and (7.7). (cid:3) Counting zeros.
Our work in the previous subsection produces an upper bound forthe count of zeros of L ( s, χ K ) close to the line Re( s ) = 1. Lemma 7.4.
Under the hypotheses of Lemma 7.3, if M ≥ m η log( QT ) + O m (1) , N K (1 − η/ , T ) ≪ m (101) M M η Z T − T Z N N (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) duu dτ. Proof.
We mimic the proof of [21, Equation 4.5], the key distinction being the use ofLemma 7.3 instead of [21, Lemma 4.3]. (cid:3)
We use Corollary 6.2 and Lemma 7.4 to prove Theorem 1.1.
Proof of Theorem 1.1.
Choose M = 2 · m η log( QT )+ O m (1). Recall that N = exp( M/ (300 η ))and N = exp(40 M/η ), and the range of η is given by (7.3). By Lemma 7.4, X K ∈ F ( Q ) N K (1 − η/ , T ) ≪ m (101) M M η Z N N X K ∈ F ( Q ) Z T − T (cid:12)(cid:12)(cid:12) X N ≤ p ≤ u a K ( p ) log pp iτ (cid:12)(cid:12)(cid:12) dτ duu . (7.8)We apply Corollary 6.2 with y = N which is valid as (2 · m ) / ≥ m + 1) for m ≥
1. It follows from Corollary 6.2 that (7.8) is ≪ m m F ( Q )(101) M M (log QT ) m . Usingour particular choices of M and η and writing σ = 1 − η/
2, we conclude that X K ∈ F ( Q ) N K ( σ, T ) ≪ m m F ( Q )( QT ) m (1 − σ ) (log QT ) m , provided that 1 − / (440 m ) ≤ σ < − / (2 log( QT )). The above bound also holds when σ ≥ − (2 log( QT )) − (compare N K ( σ, T ) with N K (1 −
12 log( QT ) , T )). On the other hand, anapplication of Lemma 7.1 with η = 1 shows that if σ < − (440 m ) − , then our estimate istrivial since N K (1 / , T ) ≪ m T log( QT ) for all K ∈ F ( Q ). (cid:3) Bounds for central L -values.Lemma 7.5. Let k be any number field over Q and ≤ α < , then log | ζ k (1 / | ≤ (cid:16) − α (cid:17) log D k + α { ρ = β + iγ : ζ k ( ρ ) = 0 , β ≥ − α, | γ | ≤ } + O [ k : Q ] (1) . Proof.
This is [21, Theorem 1.1] applied to ζ k ( s ). (cid:3) Average error in the Chebotarev density theorem
The remainder of the article is dedicated to proving Theorem 2.1 as an application ofTheorem 1.1. In fact, we prove Theorem 8.8, which is a more robust version of Theorem 2.1.The argument is divided into three subsections.8.1.
Counting primes with any zero-free region.
First, given an arbitrary zero-freeregion for an L -function L ( s ), we wish to formulate a prime number theorem for L ( s ) wherethe error term depends on the zero-free region in an explicitly computable fashion. Definition 8.1.
For any entire function L ( s ) , the zero-free region data for L is thefunction ∆ : [3 , ∞ ) → [0 , ] such that L ( s ) is zero-free in the region ∆( t + 3) = sup { α ∈ [0 , ] : L ( σ ± it ) = 0 for σ > − α } for t ≥ . The error term data for L is the associated function η : [3 , ∞ ) → [0 , ∞ ) given by (8.1) η ( x ) = inf t ≥ h ∆( t ) log x + log t i . The zero-free region data and error term data for ζ K ( s ) /ζ ( s ) are denoted ∆ K/ Q and η K/ Q ,respectively. The zero-free region data and error term data for ζ ( s ) are denoted ∆ Q and η Q ,respectively. We will establish two key technical propositions which are refined versions of the Cheb-otarev density theorem. The first is completely unconditional.
ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 15
Proposition 8.2.
Let K/ Q be a Galois extension of number fields with G = Gal( K/ Q ) . Re-call the data defined in Definition 8.1. For any conjugacy class C of G and x ≥ (log( eD K )) , (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ | C || G | x log x (cid:16) e − η K/ Q ( x ) log( eD K ) + e − η Q ( x ) (cid:17) + | C || G | x / log x . The second proposition imposes some conditions on the conjugacy classes which we packageinto a definition for later use.
Definition 8.3.
Let K/ Q be a Galois extension of number fields. A conjugacy class C of Gal( K/ Q ) is ( K/ Q ) -admissible if there exists a subgroup H ⊆ Gal( K/ Q ) satisfying all ofthe following conditions:(i) H ∩ C is non-empty.(ii) For every non-trivial irreducible Artin character χ of Gal(
K/K H ) , the Artin L -function L ( s, χ, K H ) is entire.(iii) The L -function ζ K H ( s ) /ζ ( s ) is entire.Remark. By class field theory and the Aramata–Brauer theorem, the conjugacy classes C = { g } for g in the center of G are always ( K/ Q )-admissible by taking H = h g i . Remark.
Condition (iii) is the most stringent and its truth is a conjecture of Dedekind. IfArtin’s holomorphy conjecture is true for Gal( K/ Q ), i.e. (ii) holds with H = G , then (iii) istrivially satisfied in which case all conjugacy classes are ( K/ Q )-admissible. Unconditionally,(iii) is known to hold provided H is normal in G by the Aramata–Brauer theorem.This definition will allow us to execute the “base change” step in the proof of theseChebotarev density theorems while also isolating the main term with minimal loss. Proposition 8.4.
Let K/ Q be a Galois extension of number fields with G = Gal( K/ Q ) .Recall the data defined in Definition 8.1. If C is a ( K/ Q ) -admissible conjugacy class thenfor x ≥ (log( eD K )) , (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ | C || G | x log x e − η K/ Q ( x ) log( eD K ) + | C || G | x / log x . The proofs of Propositions 8.2 and 8.4 are postponed to Section 10. The fundamentalideas are classical but require careful execution.8.2.
Leveraging zero-free region data.
Next, we demonstrate how to estimate the errorterm data from Definition 8.1. We begin with a special case: the standard zero-free regionfor the Dedekind zeta function along with Stark’s effective bound for an exceptional zero.This will also be helpful in later arguments.
Theorem 8.5.
Let c > be a sufficiently small absolute constant. For any number field E ,the Dedekind zeta function ζ E ( s ) has at most one zero β in the region Re( s ) > − c log D E + [ E : Q ] log( | Im( s ) | + 3) . Further, if this exceptional zero β exists then it is real and simple and, for any ε > , onehas β ≤ − c ( ε ) D − εE . The constant c ( ε ) is effective if ε ≥ E : Q ] and ineffective otherwise.Proof. This follows from results of Lagarias–Odlyzko [11, Lemmas 8.1 and 8.2] and of Stark[22, Theorem 1 and 1’, p. 148]. (cid:3)
This theorem reduces the estimation of error term data η E ( x ) for the Dedekind zetafunction of E to a straightforward optimization problem of a single-variable function. Lemma 8.6.
Let E/ Q be a Galois extension of number fields. There exists an effectivelycomputable constant c = c ([ E : Q ]) > such that for log D E ≤ [ E : Q ] log log x , e − η E/ Q ( x ) log( eD E ) ≪ [ E : Q ] e − c √ log x . Proof.
By Theorem 8.5 and Definition 8.1, the zero-free region data ∆ E of ζ E ( s ) satisfies∆ E (3) ≥ c ( ε ) D − εE , ∆ E ( t ) ≥ c log D E + [ E : Q ] log t for t > . Using the substitution t = e u in Definition 8.1 for η E , we see for x ≥ η E/ Q ( x ) ≥ min n c ( ε ) log xD εE , inf u ≥ (cid:16) c log x log D E + [ E : Q ] u + u (cid:17)o . The infimum is minimized at u = max { , ( c log x ) / [ E : Q ] / − log D E [ E : Q ] } . With ε = E : Q ] , it follows that η E/ Q ( x ) ≫ [ E : Q ] min n log xD / [ E : Q ] E , log x log D E , p log x o . The constraint log D E ≤ [ E : Q ] log log x implies that η E ( x ) ≫ [ E : Q ] √ log x , so e − η E/ Q ( x ) log( eD E ) ≪ [ E : Q ] e − c √ log x for some constant c = c ([ E : Q ]) > (cid:3) Now, we use Theorem 1.1 to construct large zero-free regions for most fields K in a set offields F . Using Definition 8.1, this naturally translates into strong estimates for η K/ Q ( x ). Proposition 8.7.
Let G be a finite group with m + 1 = | G | ≥ . Let F = F G be any non-empty family of fields which are Galois over Q with Galois group isomorphic to G . Recallthe definition of m F ( Q ) in (1.1) . Let < ε < and Q ≥ be arbitrary and set δ = ε m .For all fields K ∈ F ( Q ) , with at most O m,ε ( m F ( Q ) Q ε ) exceptions, one has for x ≥ that e − η K/ Q ( x ) ≤ x − δ + exp (cid:16) − (20 δ log Q log x ) / − log Q (cid:17) + exp (cid:16) − (cid:16) c log xn (cid:17) / − Q ε/ (cid:17) . Proof.
For 2 ≤ j ≤ Q ε/ + 1, we appeal to the zero density estimate (Theorem 1.1) with T = T j := e j − σ = σ j := 1 − δ log Q log Q + log( T j + 3) . By iteratively applying Theorem 1.1, we throw out O m,ε ( m F ( Q ) Q m · δ ) exceptions atmost Q ε/ times. As 10 m · δ = ε/ ≤ ε/
2, this dyadically builds a zero-free region asidefrom at most O m,ε ( m F ( Q ) Q ε ) exceptional fields. Thus for all except at most O m,ε ( m F ( Q ) Q ε )of the fields K ∈ F ( Q ), the ratio of Dedekind zeta functions ζ K ( s ) /ζ ( s ) = 0 in the regionRe( s ) > − δ log Q log Q + log( | Im( s ) | + 3) , | Im( s ) | ≤ exp( Q ε/ ) . Theorem 8.5 and the upper bound D K ≤ Q also implies that ζ K ( s ) /ζ ( s ) = 0 in the regionRe( s ) > − c Q + [ K : Q ] log( | Im( s ) | + 3) , | Im( s ) | > . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 17
Therefore, the zero-free region data ∆ K/ Q ( t ) satisfies∆ K/ Q ( t ) ≥ ( δ log Q log Q +log t ≤ t ≤ exp( Q ε/ ) , c Q +[ K : Q ] log t t ≥ exp( Q ε/ ) . Using the same change of variables t = e u from Lemma 8.6 in (8.1), one has for x ≥ η K/ Q ( x ) ≥ min n inf u ≥ Q ε/ φ ( u, x ) , inf ≤ u ≤ Q ε/ φ ( u, x ) o , where φ ( u, x ) = c log x Q + [ K : Q ] u + u, φ ( u, x ) = 20 δ (log Q )(log x )log Q + u + u. For a given x ≥
3, we solve each optimization problem. The global minimum of φ ( u, x ) onthe interval − Q [ K : Q ] < u < ∞ is attained at u = u := ( c log x ) / [ K : Q ] / − Q . Thus, in therestricted interval u ≥ Q ε/ , the function φ ( u, x ) is minimized at u = max { u , Q ε/ } so(8.3) inf u ≥ Q ε/ φ ( u, x ) ≥ ( c log x ) / [ K : Q ] / + Q ε/ . Similarly, the global minimum of φ ( u, x ) on the interval − log Q < u < ∞ is attained at u = u := (20 δ log Q log x ) / − log Q . In the interval 0 ≤ u ≤ Q ε/ , the function φ ( u, x )achieves its minimum at u = 0 , u , or Q ε/ . Observe φ (0 , x ) = 20 δ log x . Also, note that u ≥ x ≥ log Q δ , in which case φ ( u , x ) ≥ (20 δ log Q log x ) / ≥ (20 δ log Q log x ) / / + log Q . If u ≥
0, then φ ( u , x ) ≤ φ ( Q ε/ , x ) since u is the global minimum. Thus for x ≥ ≤ u ≤ Q ε/ φ ( u, x ) ≥ min { δ log x, (20 δ log Q log x ) / / + log Q } . Proposition 8.7 follows from combining (8.2), (8.3), and (8.4) with the trivial observationthat e − min { A,B } ≤ e − A + e − B for A, B ≥ (cid:3) A level of distribution.
We may finally prove the main result of this section.
Theorem 8.8.
Let G be a finite group with n + 1 = | G | ≥ . Let F = F G be any family offields K which are Galois over Q with Gal( K/ Q ) ∼ = G . Recall the definition of m F ( Q ) from (1.1) . Let < ε < , and set δ = ε m . Both of the following hold:(a) If ≤ log Q ≤ x δ and x ≥ e , then X K ∈ F ( Q ) max C ⊂ Gal( K/ Q ) (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ m,ε F ( Q ) xe c √ log x + m F ( Q ) Q ε x log x . The constant c = c ( | G | ) > is effectively computable, and the maximum is overconjugacy classes of Gal( K/ Q ) . (b) Recall the definition of ( K/ Q ) -admissibility in Definition 8.3. If ≤ log Q ≤ x δ , then X K ∈ F ( Q ) max C ( K/ Q ) − admissible (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ m,ε F ( Q ) (cid:0) x − δ + Q − e − √ δ log Q log x + e − Q ε/ e − q c x m (cid:1) + m F ( Q ) Q ε x log x . The absolute effectively computable constant c > is from Theorem 8.5.Proof. We have the trivial bound(8.5) | π C ( x, K/ Q ) − | C || G | Li( x ) | ≪ x log x . The contribution from any exceptional fields that arise from an application of Proposition 8.7will be estimated in this trivial manner. Note that if Q ≪ m,ε m F ( Q ) ≥ ≪ F ( Q ) x log x ≪ m,ε x log x ≪ m,ε m F ( Q ) x log x . This yields (a) and (b), so the claim follows. Thus we may assume Q is sufficiently largedepending only on m and ε .Now, assume Q ≥ Q m,ε is large. By Proposition 8.7, for all fields K ∈ F ( Q ) with at most O m,ε ( m F ( Q ) Q ε ) exceptions, we have that e − η K/ Q ( x ) log( eD K ) ≤ (cid:16) x − δ + e − (20 δ log Q log x )1 / / − log Q + e − q c x m − Qε/ (cid:17) log( eD K ) ≪ ε x − δ + e − ( δ log Q log x )1 / / − log Q + e − q c x m − Qε/ because log D K ≤ log Q ≤ x δ . Result (b) now follows from Proposition 8.4. For (a), since Q ≥ Q m,ε is large, we have ( δ log Q ) / ≥ c / m . As c is absolute, the above display andLemma 8.6 imply that e − η K/ Q ( x ) log( eD K ) + e − η Q ( x ) ≪ m,ε e − q c x m + e − c (log x ) / . Result (a) now follows from Proposition 8.2. (cid:3) Proof of Theorem 2.1 and Corollary 2.2
Finally, using Theorems 1.1 and 8.8, we establish Theorem 2.1 and Corollary 2.2. Theremaining proofs of Propositions 8.2 and 8.4 will be addressed in the final section.
Proof of Theorem 2.1.
Part (i) follows from Theorem 8.8(a) and (2.6).For (ii), we have from Theorem 8.8(b) and (2.6) that1 F ( Q ) X K ∈ F ( Q ) max C ( K/ Q ) − admissible (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | π ( x ) (cid:12)(cid:12)(cid:12) ≪ m,ε x − δ for 4 log x ≤ log Q ≤ x δ . Since C = { } is always admissible per Definition 8.3 with H = { } ,Theorem 2.1(ii) now follows. ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 19
For (iii), our starting point is Lemma 7.5. Since K is Galois over k , the function ζ K ( s ) /ζ k ( s )is entire by the Aramata–Brauer theorem. Since ζ ( s ) has no nontrivial zeros β + iγ with | γ | ≤
14, we conclude that(9.1) { ρ = β + iγ : ζ k ( ρ ) = 0 , β ≥ − α, | γ | ≤ } ≤ N K/ Q (1 − α, . By assumption (2.6) and Theorem 1.1, all except at most O m,ε ( Q − ε F ( Q )) of the K ∈ F ( Q )satisfy N K/ Q (1 − α,
6) = 0 for α = ε · m = 2 δ . Part (iii) follows from Lemma 7.5 and (9.1).Finally, (iv) is an immediate consequence of Theorem 2.1(i) with the choice x = Q o (1) anda lemma of Ellenberg and Venkatesh [7, Lemma 3]. (cid:3) Proof of Corollary 2.2.
For G = A n , a result of Pierce, Turnage-Butterbaugh, and Wood [20,Theorem 2.7] implies | F ( D | G | / ) ≫ n D (1 − n ! ) / (4 n − . Since A n is simple for n ≥
5, condition(2.6) holds with ε = 1 / (10 n ). For G = S n , a result of Bhargava, Shankar, and Wang [1]gives | F ( D | G | / ) | ≫ n D / /n . As A n is the only proper non-trivial normal subgroup of S n ,(2.6) becomes the condition in (b). (cid:3) Proofs of Propositions 8.2 and 8.4
All that remains is to establish Propositions 8.2 and 8.4. These are extensions of [4,Proposition 8.1] and borrow from the analysis in [23, Sections 2 and 4]. The exposition hereis essentially self-contained but, since many of the arguments are standard, we will omit sometedious details which can be found in [23]. We first introduce an explicit weight function.
Lemma 10.1.
For all x ≥ and ε ∈ (0 , / , there exists a continuous real-variable function f ( t ) = f x,ε ( t ) such that:(i) ≤ f ( t ) ≤ for all t ∈ R , and f ( t ) ≡ for ≤ t ≤ .(ii) The support of f is contained in the interval [ − ε log x , ε log x ] .(iii) Its Laplace transform F ( z ) = R R f ( t ) e − zt dt is entire and is given by F ( z ) = e − (1+ ε log x ) z · (cid:16) − e ( + ε log x ) z − z (cid:17)(cid:16) − e εz x − εz x (cid:17) . (iv) Let s = σ + it, σ > , and t ∈ R . Then | F ( − s log x ) | ≤ e σε x σ min n , x − σ/ | s | log x (cid:16) ε | s | (cid:17) o . Moreover, / < F (0) < / and (10.1) F ( − log x ) = x log x + O (cid:16) εx + x / log x (cid:17) . (v) Let s = − + it with t ∈ R . Then | F ( − s log x ) | ≤ x − / log x (cid:16) ε (cid:17) (1 / t ) − . Proof.
This lemma and its proof can be found by taking ℓ = 2 in [23, Lemma 2.2]. (cid:3) We will also require a general lemma that allows us to change the base field.
Lemma 10.2 (Murty-Murty-Saradha) . Let K/ Q be a Galois extension of number fields withGalois group G , and let C ⊆ G be a conjugacy class. Let H be a subgroup of G such that C ∩ H is nonempty, and let K H be the fixed field of K by H . Let g ∈ C ∩ H , and let C H ( g ) denote the conjugacy class of H which contains g . If x ≥ , then (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | | H || C H | π C H ( x, K H /F ) (cid:12)(cid:12)(cid:12) ≤ | C || G | (cid:16) [ K : Q ] x / + 2log 2 log D K (cid:17) . Proof.
This is carried out during the proof of [17, Proposition 3.9]. (cid:3)
Proof of Proposition 8.2.
To start, we record a basic observation that will be often used:(10.2) [ K : Q ] ≪ log D K ≤ x / . The first bound Minkowski’s inequality; the second bound holds by assumption.Select the weight function f ( · ) = f x,ε ( · ) from Lemma 10.1 for any x ≥ ε = x − / + min { , e − η K/ Q ( x ) / } . Since C is ( K/ Q )-admissible, we may take g ∈ H ∩ C by Definition 8.3(i). Let C H = C H ( g )be the conjugacy class of H containing g . Define the weighted prime sum e ψ C H ( x ; f ) by e ψ C H ( x ; f ) := | C H || H | X χ χ ( C H ) log x πi Z i ∞ − i ∞ − L ′ L ( s, χ, K H ) F ( − s log x ) ds, where χ runs over all the irreducible Artin characters of K/K H . This definition matches(2.13) in [23]. By Definition 8.3(ii), L ( s, χ, K H ) is assumed to be entire for non-trivial χ sowe may shift the contour to Re( s ) = − /
2. Since ζ K H ( s ) corresponds to χ trivial, this shiftpicks up the simple pole at s = 1 of ζ K H ( s ), the non-trivial zeros of all the L -functions, andthe trivial zero at s = 0 at most P χ [ K H : Q ] = [ K : Q ] times. From Lemma 10.1(iv), thecontribution at s = 0 is therefore at most [ K : Q ] | F (0) | log x ≪ [ K : Q ] log x . The remainingcontour integral is estimated using Lemma 10.1(v) and the standard bound − L ′ L ( s, χ, K H ) ≪ log( D K H N K H / Q q χ ) + [ K H : Q ] log( | Im( s ) | + 3) , Re( s ) = − . Since ε ≥ x − / , it follows by (10.2) and the conductor-discriminant formula that(10.3) | H || C H | e ψ C H ( x ; f )log x = F ( − log x ) − X χ χ ( C H ) X ρ χ F ( − ρ χ log x ) + O ( x / log x ) , where the inner sum runs over all non-trivial zeros ρ χ of L ( s, χ, K H ) counted with multiplic-ity. The details of this calculation can be found in [23, Lemma 4.3].From the factorization ζ K ( s ) = Q χ L ( s, χ, K H ), all the zeros in (10.3) are the non-trivialzeros ρ of ζ K ( s ) counted with multiplicity. Moreover, ζ K H ( s ) /ζ ( s ) is entire by Defini-tion 8.3(iii), so the zeros of ζ ( s ) are a subset of those ρ χ in (10.3) belonging to the trivialcharacter χ = 1. Since ζ K ( s ) /ζ ( s ) is entire by Theorem 4.1, we partition the zeros ρ of ζ K ( s )according to whether ζ ( ρ ) = 0 or not. Collecting all of these observations, it follows from(10.3) that(10.4) | H || C H | e ψ C H ( x ; f )log x − F ( − log x ) + X ζ ( ρ )=0 F ( − ρ log x ) ≪ X ζ K ( ρ ) /ζ ( ρ )=0 | F ( − ρ log x ) | + x / log x . ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 21
Let Λ( n ) be the usual von Mangoldt function. Note that(10.5) F ( − log x ) − X ζ ( ρ )=0 F ( − ρ log x ) = 1log x X n ≥ Λ( n ) f (cid:16) log n log x (cid:17) + O (cid:16) x / log x (cid:17) via (10.2) and Mellin inversion (apply (10.3) with H = { } and K = Q ). Define ψ ( x ) := P n ≤ x Λ( n ). Using Lemma 10.1 and the trivial bound P n ≤√ x Λ( n ) ≪ x / , we deduce from(10.5) the equality(10.6) F ( − log x ) − X ζ ( ρ )=0 F ( − ρ log x ) = ψ ( x )log x + O (cid:16) εx log x + x / log x (cid:17) . For the sum over zeros of ζ K ( s ) /ζ ( s ) in the error term of (10.4), we first observe byLemma 10.1(iv) that the contribution of the zeros ρ with | ρ | ≤ / ≪ P | ρ |≤ / x / ≪ x / log D K ≪ x / by (10.2). For the zeros ρ = β + iγ of ζ K ( s ) /ζ ( s ) with | ρ | ≥ / K/ Q for ζ K ( s ) /ζ ( s ) given in Definition 8.1 impliesRe( ρ ) ≥ − ∆ K/ Q ( | γ | + 3) in which case x − (1 − β ) ( | γ | +3) ≤ e − η K/ Q ( x ) by definition of η K/ Q . HenceLemma 10.1(iv) and our choice of ε yields the estimate(log x ) | F ( − ρ log x ) | ≪ x β ( | γ | + 3) · ε − ( | γ | + 3) ≪ xe − η K/ Q ( x ) · e η K/ Q ( x ) / ( | γ | + 3) for | ρ | ≥ /
4. Thus, summing over all zeros ρ of ζ K ( s ) /ζ ( s ), it follows that X ζ K ( ρ ) /ζ ( ρ )=0 | F ( − ρ log x ) | ≪ xe − η K/ Q ( x ) / log x X ζ K ( ρ ) /ζ ( ρ )=0 | γ | + 3) + x / . All the zeros of ζ K ( s ) /ζ ( s ) are zeros of ζ K ( s ). Thus, by standard estimates for zeros of theDedekind zeta function [23, Lemma 2.5] and (10.2), the above expression is ≪ xe − η K/ Q ( x ) / log x ∞ X T =1 X T − ≤| Im( ρ ) |≤ T log D K + log( T + 3) T + x / ≪ xe − η K/ Q ( x ) / log( eD K )log x + x / . Combined with (10.4) and (10.6), this implies (by our choice of ε and (10.2)) that(10.7) | H || C H | e ψ C H ( x ; f )log x = ψ ( x )log x + O (cid:16) x log x e − η K/ Q ( x ) / log( eD K ) + x / log x (cid:17) . By [23, Lemma 2.3], we may replace the weighted version e ψ C H ( x ; f ) by the usual ψ C H ( x, K/K H )given by [23, Equation 2.1] at the cost of introducing an error of size O ( εx + x / ). This isagain absorbed into the existing error term above by (10.2). Therefore, by partial summation[23, Lemma 2.1 and Equation 5.3], it follows that | H || C H | π C H ( x, K/K H ) = π ( x ) + O (cid:16) x log x sup √ x ≤ y ≤ x ( e − η K/ Q ( y ) / ) log( eD K ) + x / log x + log( eD K ) (cid:17) . From (8.1), one can see that η ( y ) is an increasing function of y and also η ( x / ) ≥ η ( x ).Hence, as log D K ≤ x / , we conclude that | H || C H | π C H ( x, K/K H ) = π ( x ) + O (cid:16) x log x e − η K/ Q ( x ) log( eD K ) + x / log x (cid:17) . Proposition 8.4 now follows by an application of Lemma 10.2 and absorbing the arisingsecondary error term via (10.2). (cid:3)
Proof of Proposition 8.2.
The arguments are nearly identical to Proposition 8.4 so we onlysketch the procedure and highlight the differences. First, we select the weight function f ( · ) = f x,ε ( · ) from Lemma 10.1 for any x ≥ ε = x − / + min { / , e − η K/ Q ( x ) / } + min { / , e − η Q ( x ) / } . Second, for any conjugacy class C , we exhibit a subgroup H satisfying conditions (i) and (ii)of Definition 8.3 but we will not assume condition (iii) holds. For any conjugacy class C ,take H to be the cyclic subgroup h g i generated by some g ∈ C so H ∩ C = ∅ . As H = h g i is abelian, the L -functions L ( s, χ, K H ) are Hecke L -functions and hence entire. Thus, both(i) and (ii) hold for this choice of H .Now, without any appeal to condition (iii), we may proceed with the same argumentsas Proposition 8.4 until we reach (10.3). Again, all the zeros in (10.3) are the non-trivialzeros ρ of ζ K ( s ) counted with multiplicity. Since ζ K ( s ) /ζ ( s ) is entire by the Aramata–Brauertheorem, we partition the zeros ρ of ζ K ( s ) according to whether they belong to ζ ( s ) or not.From these observations and (10.3), we have that | H || C H | e ψ C H ( x ; f )log x = F ( − log x ) + O (cid:16) X ζ K ( ρ ) /ζ ( ρ )=0 | F ( − ρ log x ) | + X ζ ( ρ )=0 | F ( − ρ log x ) | + x / log x (cid:17) . Lemma 10.1(iv) can be used to calculate the main term F ( − log x ). The sum over zerosof ζ K ( s ) /ζ ( s ) is estimated exactly the same as in Proposition 8.4 and, because of our newchoice for ε , the sum over zeros of ζ ( s ) is similarly handled. Overall, we obtain | H || C H | e ψ C H ( x ; f )log x = x log x + O (cid:16) x log x e − η K/ Q ( x ) / log( eD K ) + x log x e − η Q ( x ) / + x / log x (cid:17) . in contrast with (10.7). We mimic the arguments found in Proposition 8.4, which leads to (cid:12)(cid:12)(cid:12) π C ( x, K/ Q ) − | C || G | Li( x ) (cid:12)(cid:12)(cid:12) ≪ | C || G | x log x (cid:16) e − η K/ Q ( x ) log( eD K ) + e − η Q ( x ) (cid:17) + | C || G | x / log x . It only remains to replace Li( x ) with π ( x ). This can be achieved by invoking the above esti-mate with K = Q in which case C = G = { } . Observe η Q / Q ( x ) = log x by Definition 8.1,hence | π ( x ) − Li( x ) | ≪ x log x e − η Q ( x ) + x / log x . We combine this estimate with the prior display to finish the proof. (cid:3)
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