One-level density and non-vanishing for cubic L-functions over the Eisenstein field
aa r X i v : . [ m a t h . N T ] F e b ONE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS OVER THE EISENSTEIN FIELD CHANTAL DAVID AND AHMET M. G ¨ULO ˘GLU
Abstract.
We study the one-level density for families of L-functions asso-ciated with cubic Dirichlet characters defined over the Eisenstein field. Weshow that the family of L -functions associated with the cubic residue symbols χ n with n square-free and congruent to 1 modulo 9 satisfies the Katz-Sarnakconjecture for all test functions whose Fourier transforms are supported in( − / , / trivial range ( − ,
1) for a family of cubic L-functions. This impliesthat a positive density of the L-functions associated with these characters donot vanish at the central point s = 1 /
2. A key ingredient in our proof is abound on an average of generalized cubic Gauss sums at prime arguments,whose proof is based on the work of Heath-Brown and Patterson [23, 22]. Introduction
Let F be a family of primitive Dirichlet characters χ defined over Q , or moregenerally over a number field K . From the work of Dirichlet and Hecke, we knowthat the L -functions L ( s, χ ) satisfy a functional equation relating the values of L ( s, χ ) to those of L (1 − s, χ ), and the distribution of the non-trivial zeros of L ( s, χ ) in the central critical strip is of particular interest.The one-level density for the family F measures the density of the low-lyingzeros (i.e. the zeros near s = 1 /
2) of the L-functions associated with the charactersin F . Following the work of Montgomery [36], and then Katz and Sarnak [29, 28],we believe that the statistics of the low-lying zeros of these L -functions match thoseof the eigenvalues of random matrices in a certain symmetry group associated withthe family F , usually symplectic, orthogonal, or unitary.Let φ be an even Schwartz test function. For a fixed character in F , the sum X ρ =1 / iγL ( ρ,χ )=0 φ (cid:16) γ log X π (cid:17) counts, with multiplicity, the zeros of L ( s, χ ) that are within O (1 / log X ) of thecentral critical point s = 1 /
2. To study the statistics of these zeros, one has toconsider the average over the family. In this paper, we consider smoothed averages.
Key words and phrases.
One-level density, low-lying zeros, non-vanishing, cubic Dirichlet char-acters, cubic Gauss sums, Hecke L-functions.Both authors are supported by T ¨UB˙ITAK Research Grant no. 119F413.
Let w : R → (0 , ∞ ) be an even Schwartz function, and D ( X ; φ, F ) = 1 A F ( X ) X χ ∈ F w (cid:18) N(cond( χ )) X (cid:19) X γL (1 / iγ,χ )=0 φ (cid:16) γ log X π (cid:17) A F ( X ) = X χ ∈ F w (cid:18) N(cond( χ )) X (cid:19) , where N(cond( χ )) is the norm of the conductor of the primitive character χ . Theone-level density is then defined aslim X →∞ D ( X ; φ, F ) . Conjecture 1.1 (Katz-Sarnak [28, 29]) . With the notation above, we have lim X →∞ D ( X ; φ, F ) = Z ∞−∞ φ ( x ) W G ( x ) dx, where W G ( x ) measures one-level density of eigenvalues near 1 of the classical com-pact group G = G ( F ) corresponding to the symmetry type of the family F . We refer the reader to [28, page 409] for the precise formulae of the densities W G ( x ) for the different symmetry groups G .The conjecture of Katz and Sarnak is still open, but evidence for the conjecturecan be obtained by proving that the conjecture holds for test functions φ whoseFourier transforms b φ have compact support.Assuming GRH, ¨Ozl¨uk and Snyder [38] showed that the one-level density for thefamily of quadratic characters satisfies the Katz-Sarnak conjecture with symplecticsymmetry for test functions φ with b φ supported in ( − , Q ( √− D ) [15],automorphic L-functions [26, 12, 27, 41, 42], elliptic curve L-functions [2, 3, 6, 24,35, 46], Hecke L-functions for characters of infinite order [45], symmetric powers ofGL(2) L-functions [13, 19], and a family of GL(4) and GL(6) L-functions [13].We study in this paper the one-level density for families of primitive cubic Dirich-let characters defined over the Eisenstein field Q ( ω ), where ω = e πi/ . Many newconceptual and technical difficulties appear when considering cubic (and not qua-dratic) characters, and the results in the literature are fewer and weaker. Meisner[33] and Cho and Park [5] showed that, under GRH, the one-level density for fam-ilies of cubic characters over Q satisfies the Katz-Sarnak conjecture with unitarysymmetry for test functions φ with b φ supported in ( − , − , b φ is a natural boundary for families attachedto Dirichlet characters, and our work provides the first example of a family of cubicL-functions in which the support is extended past this trivial range, assuming GRH.In the recent work [11], Drappeau, Pratt, and Radziwill computed the one-leveldensity over the family of all primitive Dirichlet characters, and proved the firstunconditional result extending the trivial range for this family. NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 3 An important tool to extend the support past the trivial range for quadraticcharacters is the use of Poisson summation. This is more intricate for cubic char-acters, as Poisson summation leads to averages of Gauss sums, and while quadraticGauss sums are given by a simple formula, their cubic analogues exhibit chaoticbehaviour; in order to understand them, we must invoke the deep work of Kubotaand Patterson. Furthermore, other features of the cubic families seem to conspireto make this strategy fail: for cubic characters over Q , the Gauss sums are notdefined over the ground field, and for cubic characters over Q ( ω ), there are “toomany characters”. For this reason, it had then become customary in the literatureto consider a thin subfamily of the cubic characters over Q ( ω ), as in [17, 31, 16, 10].Only recently, moments for the whole family of cubic characters when the base fieldcontains the cubic root of unity were considered in the work of David, Florea andLalin, who computed the first moment for the whole family over function fields [7].1.1. Statement of the results.
The main result of this paper is the followingtheorem, where we extend the support of the Fourier transform of the test functionfor the thin family F ′ . Theorem 1.2.
Let F ′ be the family of primitive cubic Dirichlet characters definedby (12) . Let φ be an even Schwartz function with b φ supported in ( − , ) . If GRHholds for L ( s, χ ) for each χ ∈ F ′ , then lim X →∞ D ( X ; φ, F ′ ) = Z ∞−∞ φ ( x ) W U ( x ) dx = b φ (0) , where W U ( x ) is the kernel measuring the one-level density for the eigenvalues ofunitary matrices. A folklore conjecture of Chowla predicts that L ( , χ ) = 0 for all L-functions L ( s, χ ) attached to Dirichlet characters. Over function fields, this is false, and ina recent paper, Li [32] showed that there are infinitively many quadratic DirichletL-functions such that L ( , χ ) = 0 in this case. It is believed that the number ofsuch characters should be of density zero among all quadratic characters, which isimplied by (the function field version of) the conjecture of Katz and Sarnak. It iswell-known that proving Conjecture 1.1 for test functions where the support of b φ is large enough, yields a positive proportion of non-vanishing for the correspondingset of L-functions, bringing evidence to Chowla’s conjecture. For the family ofcubic characters, one needs to extend the support beyond ( − ,
1) to get a positiveproportion. Hence, by Theorem 1.2, we can prove the following result for the thinsubfamily of cubic characters.
Corollary 1.3.
Let F ′ be the family of primitive cubic Dirichlet characters definedin (12) . If GRH holds for the corresponding L-functions, then L ( , χ ) = 0 for atleast / of the characters in F ′ . Our result is the first result showing a positive proportion of non-vanishing forany cubic family over number fields. Unconditionally, it is known that there areinfinitely many cubic Dirichlet characters χ such that L ( , χ ) = 0, over Q [1]and over Q ( ω ) [31]. Over function fields, a positive proportion of non-vanishingwas obtained by David, Florea and Lalin [8] for the family of cubic characterswhen F q does not contain a third root of unity (which is the equivalent of cubiccharacters over Q ), improving previous work [7, 14] exhibiting infinitively many CHANTAL DAVID AND AHMET M. G¨ULO ˘GLU cubic Dirichlet characters χ such that L ( , χ ) = 0 over function fields. The proofof [8] uses a completely different technique from this paper, based on the mollifiedmoments. It is interesting to compare the techniques and results, and we believethat the results of [8] could be obtained for cubic characters over Q , where theone-level density approach seems to fail. Of course, we would then need to assumeGRH (which is proven over function fields). We also speculate that the mollifiedmoments approach is more likely to succeed in obtaining a positive proportionof non-vanishing for the full family of primitive cubic characters over Q ( ω ) thanthe one-level density approach, as breaking the ( − , Q ( ω ), we include the followingresult, which supports the Katz-Sarnak conjecture for test functions whose Fouriertransform has support in the trivial range ( − , Theorem 1.4.
Let F be the family of primitive cubic Dirichlet characters definedin Section 3. Let φ be an even Schwartz function with b φ supported in ( − , .Assume GRH for L ( s, χ ) for each χ ∈ F . Then, lim X →∞ D ( X ; φ, F ) = Z ∞−∞ φ ( x ) W U ( x ) dx, where W U ( x ) is the kernel measuring the one-level density for eigenvalues of unitarymatrices. Finally, we state some unconditional results.
Theorem 1.5.
Let F be the family of primitive cubic Dirichlet characters over Q ( ω ) defined in Section 3. Let φ be an even Schwartz function with b φ supported in ( − , ) . Then, lim X →∞ D ( X ; φ, F ) = Z ∞−∞ φ ( x ) W U ( x ) dx = b φ (0) , where W U ( x ) is the kernel measuring the one-level density for eigenvalues of unitarymatrices. The same result holds for the subfamily F ′ defined in Section 4 with b φ supported in ( − , ) . We remark that since we consider smooth sums, the support of b φ for the family F ′ in Theorem 1.5 is slightly better than the one obtained by Gao and Zhao in[17], which requires supp( b φ ) ⊆ ( − , ) for the same family.1.2. Structure of the paper.
In Section 2, we collect the relevant facts aboutcubic characters, cubic families and cubic Gauss sums. In Section 3, we prove The-orem 1.4. In Section 4, we use Poisson summation to reduce the computation ofthe one-level density to averages of generalized cubic Gauss sums at prime argu-ments, and we prove Theorem 1.2, assuming the bounds for those averages given byTheorem 4.4. We prove Theorem 4.4 in Section 5 and Section 6 by generalising thework of Heath-Brown [22] and Heath-Brown and Patterson [23] on the distributionof cubic Gauss sums at prime arguments. The proof of Corollary 1.3 is given inSection 7.
NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 5 L -functions of cubic characters and cubic Gauss sums Cubic Dirichlet L -functions. Let K = Q ( ω ), ω = e πi/ . The ring ofintegers Z [ ω ] of K has class number one and six units (cid:8) ± , ± ω, ± ω (cid:9) . Each non-trivial principal ideal n co-prime to 3 has a unique generator n ≡ Z [ ω ] are given by the cubic residue symbols.For each prime π ∈ Z [ ω ] with π ≡ π ; the cubic residue symbol χ π ( a ) satisfying χ π ( a ) = (cid:16) aπ (cid:17) ≡ a (N( π ) − / mod π, and its conjugate χ π = χ π . In general, for n ∈ Z [ ω ] with n ≡ χ n is defined multiplicatively using the characters of primeconductor by χ n ( a ) = (cid:16) an (cid:17) = Y π | n χ π ( a ) v π ( n ) . Such a character χ n is primitive when it is a product of characters of distinctprime conductors, i.e. either χ π or χ π = χ π . Moreover, χ n is a cubic Heckecharacter of conductor n Z [ ω ] if χ n ( ω ) = 1. Since (cid:16) ωn (cid:17) = Y π | n ω v π ( n )( N ( π ) − / = ω P π | n v π ( n )( N ( π ) − / = ω ( N ( n ) − / , we conclude that a given Dirichlet character χ is a primitive cubic Hecke characterof conductor n n Z [ ω ], co-prime to 3, provided that χ = χ n , where(1) n = n n , where n , n are square-free and co-prime, and(2) N( n ) ≡ n ) ≡ N( n ) mod 9. Remark 2.1.
Given n ∈ Z [ ω ], co-prime to 3, write n = n n n , where n n is square-free, and n , n are co-prime. Then, the character χ n modulo n Z [ ω ] isinduced by the primitive character χ n χ n of conductor n n Z [ ω ] unless n is a cube;that is, n , n are units.We recall the cubic reciprocity for cubic characters. Lemma 2.2.
Let m, n ∈ Z [ ω ] , m, n ≡ ± . Then, (cid:16) mn (cid:17) = (cid:16) nm (cid:17) . Let χ be a primitive cubic Hecke character to some modulus m = m Z [ ω ], co-prime to 3. The completed Hecke L -series is then defined byΛ( s, χ ) = ( | d K | N ( m )) s/ (2 π ) − s Γ( s ) L ( s, χ ) , where d K = − K . Proposition 2.3 ([37, VII. Cor. 8.6]) . The completed L -series above is entire, pro-vided χ is primitive and m = Z [ ω ] . Futhermore, it satisfies the functional equation Λ( s, χ ) = W ( χ )(N m ) − / Λ(1 − s, χ ) where (1) W ( χ ) = X x mod m ( x, m )=1 χ ( x ) e (cid:0) Tr( x/m √− (cid:1) , where x varies over a system of representatives of ( Z [ ω ] / m ) × . CHANTAL DAVID AND AHMET M. G¨ULO ˘GLU
Explicit Formula and averaging over the family.
We now state the ex-plicit formula for cubic L-functions, which relate sums over the zeroes to sums overthe coefficients of the L -functions. Averaging over the family, we get the main termfor the one-level density, and our results are obtained by bounding the error term.We state but not prove the next two lemmas as their proofs are standard. Lemma 2.4.
Let χ be a primitive cubic Hecke character modulo c Z [ ω ] . Then,uniformly for s = σ + it satisfying | t | > and − σ , or that / < σ , (2) L ′ L ( s, χ ) = X | γ − t | s − ρ + O (cid:0) log (cid:0) N( c )(3 + | t | ) (cid:1)(cid:1) where the sum runs over the zeros ρ = β + iγ of Λ( s, χ ) counted with multiplicity. Lemma 2.5.
With the character as in the previous lemma, for any T > , thereis some T ∈ [ T, T + 1] such that (3) L ′ L ( σ ± iT , χ ) ≪ log (cid:0) N( c )(3 + T ) (cid:1) uniformly for − σ . Lemma 2.6 (Explicit Formula) . Let χ be a primitive cubic Hecke character modulo n Z [ ω ] with ( n,
3) = 1 , and let φ ( x ) be an even function of Schwartz class on R whoseFourier transform b φ ( y ) has compact support in ( − v, v ) . Then, X ρ φ (cid:16) ( ρ − /
2) log X πi (cid:17) = b φ (0) log N( n )log X + O (cid:16) X (cid:17) − X p X k (cid:0) χ ( p k ) + χ ( p k ) (cid:1) b φ (cid:18) k log N p log X (cid:19) log N p (N p ) k/ log X , where the implied constant depends only on φ .Proof. Note that G ( s ) := φ (cid:16)(cid:16) s − (cid:17) log X πi (cid:17) , is holomorphic in − Re( s ) G ( s ) = G (1 − s ) , s G ( s ) ≪ . Let T be a large real number and T ∈ [ T, T + 1] be as in Lemma 2.5. Let R bethe rectangle with vertices 2 − iT , iT , − iT , − − iT . By Cauchy’s ResidueTheorem we obtain X ρ G ( ρ ) = 12 πi Z R G ( s ) Λ ′ Λ ( s, χ ) ds, where the integral is taken counter-clockwise around R . By Lemma 2.5 and (4), thecontribution of the horizontal integrals is ≪ T − log ( T N( c )). Thus, taking limitas T → ∞ and using the functional equations for G ( s ) and Λ( s, χ ), we obtain X ρ G ( ρ ) = 12 πi Z σ =2 G ( s ) (cid:18) Λ ′ Λ ( s, χ ) + Λ ′ Λ ( s, χ ) (cid:19) ds = 12 πi Z σ =2 G ( s ) (cid:18) L ′ L ( s, χ ) + L ′ L ( s, χ ) + 2 Γ ′ Γ ( s ) + log 3N( c )4 π (cid:19) ds. NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 7 Using (4) again we can shift the contour to σ = 1 / X ρ G ( ρ ) = F (1) log 3N( c )4 π + 12 πi Z (1 / G ( s ) Γ ′ Γ ( s ) ds − X p H ( p ) , where H ( p ) = X n > (cid:0) χ ( p n ) + χ ( p n ) (cid:1) F (N( p n )) log N p and F ( y ) = 12 πi Z σ =1 / G ( s ) y − s ds = b φ (cid:16) log y log X (cid:17) / ( √ y log X ) . By the approximate formula (cf. [20, 8.363.3])Γ ′ Γ ( a + ib ) + Γ ′ Γ ( a − ib ) = 2 Γ ′ Γ ( a ) + O (cid:0) ( b/a ) (cid:1) we obtain 1 πi Z σ =1 / G ( s ) Γ ′ Γ ( s ) ds = 2log X Z ∞−∞ φ ( t ) Γ ′ Γ (cid:18)
12 + i πt log X (cid:19) dt = 2(Γ ′ / Γ)(1 / X b φ (0) + O (cid:0) (log X ) − (cid:1) . Finally, noting that X p X k> (cid:0) χ ( p k ) + χ ( p k ) (cid:1) b φ (cid:18) k log N p log X (cid:19) log N p (N p ) k/ ≪ X p p − / log p ≪ (cid:3) Let F be one of the families that will be defined in the next two sections. Recallthat D ( X ; φ, F ) = 1 A F ( X ) X χ ∈ F w (cid:18) N(cond( χ )) X (cid:19) X γL (1 / iγ,χ )=0 φ (cid:16) γ log X π (cid:17) , where N(cond( χ )) is the norm of the conductor of the primitive character χ and A F ( X ) = X χ ∈ F w (cid:18) N(cond( χ )) X (cid:19) . Using the explicit formula, Lemma 2.6, we obtain D ( X ; φ, F ) = b φ (0) A F ( X ) log X X χ ∈ F w (cid:18) N(cond( χ )) X (cid:19) log N(cond( χ ))+ O (cid:16) X v/ + |E F ( X ) | A F ( X ) log X (cid:17) , (5)where(6) E F ( X ) = X χ ∈ F ∪{ } w (cid:18) N(cond( χ )) X (cid:19) X p ∤ X k χ ( p ) b φ (cid:18) k log N p log X (cid:19) log N p (N p ) k/ . CHANTAL DAVID AND AHMET M. G¨ULO ˘GLU
To establish Theorems 1.2 and 1.4, we need to find the largest v such that E F ( X ) = o ( A F ( X ) log X ) for each family in question. Thus, the rest of the paper will bedevoted to the estimate of the sum(7) S F ( y ) = X χ ∈ F ∪{ } w (cond( χ ) /X ) X N p y p ∤ χ ( p ) log N p . Cubic Gauss sums and Poisson Summation for cubic characters.
Wenow define the generalized Gauss sums associated with the cubic residue symbols χ n , where n ≡ ∈ Z [ ω ]. Notice that we do not suppose that n is square-free,and these characters are not necessarily primitive. Let(8) g ( r, n ) = X α mod n χ n ( α ) e (cid:0) Tr( rα/n ) (cid:1) . Then, for ( n, √−
3) = 1, W ( χ n ) = χ n ( √− g (1 , n ) , where W ( χ ) is the sign of the functional equation given by (1).The following two lemmas are classical results about cubic Gauss sums whichcan be found in [23], or easily checked, and we include them without proof. Lemma 2.7.
Let n, n , n ≡ and s, r be elements of Z [ ω ] .If ( s, n ) = 1 , g ( rs, n ) = χ n ( s ) g ( r, n ) . If ( n , n ) = 1 , g ( r, n n ) = χ n ( n ) g ( r, n ) g ( r, n ) = g ( rn , n ) g ( r, n ) . Lemma 2.8.
Let π ≡ be a prime, ( π, r ) = 1 , where r ≡ . Let k, j be integers with k > and j > .If k = j + 1 , g ( rπ j , π k ) = N( π j ) × − if | kg ( r, π ) if k ≡ g ( r, π ) if k ≡ . If k = j + 1 , g ( rπ j , π k ) = ( ϕ K ( π k ) if | k, k j otherwise. Our next lemma is obtained from the Poisson summation formula over Z , whichis essentially [22, Lemma 10]. Lemma 2.9.
Let χ be a primitive character of ( Z [ ω ] / f Z [ ω ]) × . Then, X n ∈ Z [ ω ] χ ( n ) w (N( n ) /Y ) = YW ( χ ) X n ∈ Z [ ω ] χ ( n ) b w ( p Y N( n ) / N( f )) , where W ( χ ) is given by (1) and for t ∈ R , t > , b w ( t ) = Z R Z R w (N( x + yω )) e (cid:0) Tr( t ( x + yω ) / √− (cid:1) dxdy, = Z R Z R w (N( x + yω )) e ( − ty ) dxdy. NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 9 We will often use the following lemma, which makes it easier to keep trackof the size of various parameters when optimizing an estimate of the form U ≪ AH a + BH − b , where A, B, a, b are positive constants and H can be chosen. Simplychoosing H to satisfy AH a = BH − b leads to U ≪ ( A b B a ) / ( a + b ) , which is the bestbound apart for the value of the other parameters in the implied constant. Thiscan be generalized to the following lemma. Lemma 2.10 ([21, Lemma 2.4]) . Suppose that L ( H ) = m X i =1 A i H a i + n X j =1 B j H − b j , where A i , B j , a i , b j are positive. Suppose that H H . Then, there is some H with H H H such that L ( H ) ≪ m X i =1 n X j =1 (cid:16) A b j i B a i j (cid:17) / ( a i + b j ) + m X i =1 A i H a i + n X j =1 B j H − b j , where the implied constant depends only on m and n . Proof of Theorem 1.4
Let F be the family of cubic residue symbols χ ab where a, b ≡ ∈ Z [ ω ]are square-free and co-prime, and ab ≡ χ ), we write F = n ab ∈ Z [ ω ] \ { } : a, b ≡ , ( a, b ) = 1 , ab ≡ o . Lemma 3.1.
Let φ be a Schwartz class test function whose Fourier transform issupported in ( − v, v ) . Then, for the family F defined above, A F ( X ) = 2 π w (1)9 h (9) √ Y p ∤ (cid:16) − p + 2N p (cid:17) X log X + O ( X ) , X ab ∈ F w (N( ab ) /X ) log N( ab ) = A F ( X ) log X + O ( X log X ) , and for any ε > , E F ( X ) ≪ ( X / v/ ε under GRH X / v + ε unconditionally,where E F ( X ) is defined in (6) , w ( s ) = R ∞ w ( x ) x s − ds is the Mellin transform of w and h (9) = | J (9) /P (9) | is the order of the ray class group modulo .Proof. We first write A F ( X ) + w (1 /X ) = 1 h (9) X ψ mod 9 X q ≡ q sqf w (N( q ) /X ) ψ ( q ) X b ≡ b | q ψ ( b ) , where ψ runs over the ray class characters modulo 9. By Mellin inversion, A F ( X ) + w (1 /X ) = 1 h (9) X ψ mod 9 πi Z Re( s )=2 w ( s ) X s G ψ ( s ) ds, with generating series G ψ ( s ) = X q ∈ Z [ ω ] q ≡ q square-free ψ ( q )N( q ) s X b ∈ Z [ ω ] b ≡ b | q ψ ( b )= Y π ≡ (cid:16) ψ ( π )N( π ) s (cid:0) ψ ( π ) (cid:1)(cid:17) = L ( s, ψ ) L ( s, ψ ) F ( s, ψ ) , (9)where L ( s, ψ ) = Y π ≡ (cid:16) − ψ ( π )N( π ) s (cid:17) − = X a ψ ( a )N a s is the Hecke L-function associated with the character ψ , and F ψ ( s ) = Y π ≡ (cid:16) − ψ ( π )N( π ) s (cid:17)(cid:16) − ψ ( π )N( π ) s (cid:17)(cid:16) ψ ( π )N( π ) s (cid:0) ψ ( π ) (cid:1)(cid:17) = Y π ≡ (cid:16) − ψ ( π ) + ψ ( π ) + ψ ( π )N( π ) s + ψ ( π ) + ψ ( π )N( π ) s (cid:17) Note that F ψ ( s ) converges absolutely for Re( s ) > / ψ modulo 9, whereas L ( s, ψ ) and L ( s, ψ ) both have analytic continuation to entire functions except fora simple pole at s = 1 when the characters are principal.Hence, moving the contour to Re( s ) = 1 / ε , we get(10) 12 πi Z Re( s )=2 w ( s ) X s G ψ ( s ) ds = X ψ = ψ ψ = ψ Res s =1 (cid:0) X s w ( s ) L ( s, ψ ) L ( s, ψ ) F ψ ( s ) (cid:1) + O (cid:16) X / ε (cid:17) , where, for the error term, we used the fact that w ( s ) ≪ | s | − n for all n >
1, with theclassical convexity bound L ( s, ψ ) , L ( s, ψ ) ≪ ψ | s | max(0 , ε − Re( s )) for 0 Re( s ) ε > ψ = ψ , there is a pole of order 2 at s = 1 with residue (cid:18)
49 (Res s =1 ζ K ( s )) F ψ (1) w (1) (cid:19) X log X + O ( X ) , and when ψ = ψ , ψ = ψ , then there is a simple pole and the contribution of theresidue is O ( X ). Substituting in (10), this proves the first result.For the second assertion, using Mellin inversion, we have X ab ∈ F ∪{ } w (N( ab ) /X ) log N( ab ) = − πih (9) X ψ mod 9 Z Re( s )=2 w ( s ) X s G ′ ψ ( s ) ds, where G ψ ( s ) was defined above in (9). Using integration by parts, for each character ψ , the above integral is − Z Re( s )=2 w ′ ( s ) X s G ψ ( s ) ds − log X Z Re( s )=2 w ( s ) X s G ψ ( s ) ds. Working as above, the main contribution from each integral comes from the doublepole of G ψ ( s ) at s = 1. For the first integral, we bound this contribution by NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 11 O ( X log X ). Summing the second integral over the characters gives A F ( X ) + w (1 /X ). This proves the second assertion.Finally, we prove the last assertion. Let S F ( y ) = X < N p y log N p X ab ∈ F ∪{ } χ ab ( p ) w (N( ab ) /X ) . Writing each prime ideal p as p = ( π ) with π ≡ S F ( y ) = 1 h (9) X ψ mod 9 X π ≡ π ) y log N( π ) X q ∈ Z [ ω ] q ≡ q sqf ( χ π ψ )( q ) w (N( q ) /X ) X b ∈ Z [ ω ] b ≡ b | q ( χ π ψ )( b )= 1 h (9) X ψ mod 9 X π ≡ π ) y log N ( π ) 12 πi Z (2) X s w ( s ) G ( χ π ψ ) ( s ) ds. We evaluate the integral working as above. Since ( ψχ π ) is non-trivial for everycharacter ψ mod 9, the generating series has no pole when Re( s ) > /
2. We movethe integral to Re( s ) = 1 / ε , and we use the bound L ( + ε + it, χ ) ≪ ( t ε N(cond( χ )) ε under GRH | t | / ε N(cond( χ )) / unconditionally,(11)which holds for any non-trivial character χ . This gives S F ( y ) ≪ ( X / ε y ε under GRH X / ε y / unconditionally.By partial integration E F ( X ) = X k Z ∞ b φ (cid:16) k log y log X (cid:17) y − k/ dS F ( y )= X k Z X v/k S F ( y ) y − k/ − (cid:16) k b φ (cid:16) k log y log X (cid:17) − b φ ′ (cid:16) k log y log X (cid:17) k log X (cid:17) dy ≪ ( X / v/ ε under GRH X / v + ε unconditionally.This establishes the third assertion. (cid:3) Using the lemma in (5) of Section 2.2, the proof of Theorem 1.5 for the family F follows, and assuming GRH, the proof of Theorem 1.4 follows.4. Proof of Theorem 1.2
Let F ′ be the family of primitive cubic Dirichlet characters determined by thecubic residue symbols χ n , where n = 1 is square-free and congruent to 1 modulo 9.Again, with a slight abuse of notation, we write(12) F ′ = { n ∈ Z [ ω ] : n = 1 , n ≡ } . Lemma 4.1.
Let w : R → (0 , ∞ ) be an even Schwartz function. Then, A F ′ = ζ − K (2) π w (1)4 √ h (9) X + O ( X / ε ) , X n ∈ F ′ w (N( n ) /X ) log N( n ) = A F ′ log X + O ( X ) , and for any ε > , E F ′ ( X ) ≪ ( X / v/ ε under GRH X / v/ ε unconditionally,where w ( s ) = R ∞ w ( x ) x s − ds is the Mellin transform of w , h (9) = | J (9) /P (9) | isthe order of the ray class group modulo , and ζ K ( s ) is the Dedekind zeta functionof K .Proof. As in the proof of Lemma 3.1, we start by writing A F ′ = 12 πih (9) X ψ mod 9 Z σ =2 X s w ( s ) G ψ ( s ) ds − w (1 /X ) , where(13) G ψ ( s ) = X a ∈ J (9) ψ ( a ) | µ ( a ) | (N a ) s = ( ζ − K (2 s ) ζ K ( s )(1 + 3 − s ) − if ψ = ψ F ( s ) L ( s, ψ ) if ψ = ψ , with ψ the principal character modulo 9, and F ( s ) = X a µ ( a ) ψ ( a )(N a ) s , L ( s, ψ ) = X a ψ ( a )(N a ) − s . Proceeding as before, shifting the contour to Re( s ) = 1 / ε and using the convexitybound we conclude that A F ′ = 1 h (9) Res s =1 ( X s w ( s ) G ψ ( s )) + O ( X / ε ) . Since G ψ ( s ) has a simple pole at s = 1, we haveRes s =1 ( X s w ( s ) L ψ ( s )) = X w (1) lim s → ( s − G ψ ( s ) = X w (1) ζ − K (2) π √ , where we used (cf. [37, Ch VII. Corollary 5.11])Res s =1 ζ K ( s ) = 2 πh K R | d K | / = π √ . This gives the first claim. The second identity follows as in the previous sectionalong the same lines using integration by parts. For the third identity, using thebound (11) and working as in the proof of Lemma 3.1 with the generating series(13), we get S F ′ ( y ) ≪ ( X / ε y ε under GRH X / ε y / unconditionally.The third identity follows by partial integration. (cid:3) Using Lemma 4.1 in equation (5) of Section 2.2, the proof of Theorem 1.5 forthe family F ′ follows. We also remark that using the bound for E F ′ ( X ) of thelemma, under GRH, gives the one-level density for the family F ′ only when thesupport of the Fourier transform is contained in ( − , NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 13 Using Lemma 4.1 in (5), we have that D ( X ; φ, F ′ ) = b φ (0) + O (cid:16) X (cid:17) + O (cid:16) X v/ + |E F ′ ( X ) | X log X (cid:17) , and we want to show that E F ′ ( X ) = o ( X log X ) when supp( b φ ) ⊆ ( − v, v ). For thefamily F ′ in (12), the sum defined in (7) can be written as S F ′ ( y ) = X < N p y log N p X n ∈ F ′ ∪{ } χ n ( p ) w (N( n ) /X )= X n ∈ Z [ ω ] n ≡ w (N( n ) /X ) X < N p y χ n ( p ) log N p X d ∈ Z [ ω ] d ≡ d | n µ K ( d )= X d ∈ Z [ ω ] d ≡ µ K ( d ) X n ∈ Z [ ω ] n ≡ d − mod 9 w (N( nd ) /X ) X < N p y χ nd ( p ) log N p . where µ K is Moebius function over K and we use the detector X d ∈ Z [ ω ] d ≡ d | n µ K ( d ) = ( n is square-free0 otherwise.We write S F ′ ( y ) = S ( y ) + S ( y ) where S ( y ) = X < N p y log N p X d ∈ Z [ ω ] d ≡ d ) D µ K ( d ) χ d ( p ) X n ∈ Z [ ω ] n ≡ d − mod 9 w (cid:16) N( nd ) X (cid:17) χ n ( p )(14) S ( y ) = X d ∈ Z [ ω ] d ≡ d ) >D µ K ( d ) X n ∈ Z [ ω ] n ≡ d − mod 9 w (N( nd ) /X ) X < N p y χ nd ( p ) log N p . (15)4.1. Estimate of S ( y ) .Lemma 4.2. Given n ≡ in Z [ ω ] with N( n ) ≡ , write n = n n n with n , n square-free and co-prime, and n i ≡ . Assuming GRH for L ( s, χ n ) , and that n is not a cube, the estimate X N p x χ n ( p ) log N p ≪ x / log ( x N( n )) holds for x > .Proof. We shall give only a sketch of the proof as it uses standard techniques.The estimate trivially holds for x < x >
3, and write L ( s, χ n ) = Y p ∤ n (cid:0) − χ n ( p )N p − s (cid:1) − = Y p | d (cid:0) − χ ( p )N p − s (cid:1) L ( s, χ ) , where χ = χ n n is the primitive character to modulus n n Z [ ω ] that induces χ n ,and where d denotes the product of primes dividing n , but not n n . For T > we find T ∈ [ T, T + 1] given by Lemma 2.5 for L ( s, χ ). Using Perron’s formulawith a = 1 + (2 log x ) − , X N p x χ n ( p ) log N p = 12 πi Z a + iT a − iT − L ′ L ( s, χ n ) x s s ds + O (cid:18) x / + x log xT + log x + x log xT (cid:19) . Next we shift the contour to b = 1 / x ) − and use (2) and (3) to estimatethe horizontal and vertical integrals to get X N p x χ n ( p ) log N p ≪ x log (N( n n )(3 + T )) T log x + (cid:16) x / log T + xT log x (cid:17) log N( d )+ x log xT + x / (cid:0) log x log(3N( n n )) + log T log (N( n n )(3 + T )) (cid:1) . Choosing T = x gives the claimed estimate. (cid:3) Lemma 4.3.
Let < D √ X . Then, (16) S ( y ) ≪ Xy / D log ( yX ) + y min n XD , X / log X o under GRH yXD unconditionallyProof. Trivially estimating the innermost sum over primes in (15) gives S ( y ) ≪ y X d ∈ Z [ ω ] d ≡ d ) >D µ K ( d ) X n ∈ Z [ ω ] n ≡ d − mod 9 w (N( nd ) /X ) . Since 1 D √ X , we have X d ∈ Z [ ω ] D< N( d ) √ X (cid:18) X n ∈ Z [ ω ]N( n ) X/ N( d ) + X n ∈ Z [ ω ]N( n ) >X/ N( d ) (cid:19) w (N( nd ) /X ) ≪ X/D, and X d ∈ Z [ ω ] d ≡ d ) > √ X X n ∈ Z [ ω ] w (N( nd ) /X ) ≪ X / X/D.
If we assume GRH, we can use Lemma 4.2 whenever nd is not a cube. Thecontribution of these terms to S ( y ) is ≪ y / X d ∈ Z [ ω ]N( d ) >D X n ∈ Z [ ω ] w (N( nd ) /X ) log ( y N( nd )) ≪ Xy / D log ( yX ) . NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 15 Note that nd is a cube if and only if n = da for some a ∈ Z [ ω ], since d issquare-free. Thus, the contribution of cubes to S ( t ) is ≪ y X d ∈ Z [ ω ] d ≡ d ) >D X a ∈ Z [ ω ] w (N( a d ) /X ) ≪ y min { X/D , X / log X } . Combining all the estimates, the result follows. (cid:3)
Estimate of S ( y ) . Writing each prime ideal p co-prime to 3 as p = π Z [ ω ]with π ≡ χ nd ( p ) = χ nd ( π ) = χ π ( nd ) = χ π ( n ) χ π ( d ) , where the first equality follows since nd ≡ S ( y ) = 1 h (9) X ψ mod 9 X π ≡ π ) y log N( π ) X d ≡ d ) D µ K ( d ) χ π ( d ) ψ ( d ) × X n ∈ Z [ ω ] w (cid:16) N( nd ) X (cid:17) χ π ( n ) ψ ( n ) , where ψ runs over the ray class characters modulo 9, and h (9) is the order of the rayclass group modulo 9. When ψ is the principal character modulo 9, the innermostsum over n is X n ∈ Z [ ω ] w (cid:16) N( nd ) X (cid:17) χ π ( n ) − X n ∈ Z [ ω ] w (cid:16) N((1 − ω ) nd ) X (cid:17) χ π ((1 − ω ) n ) . Hence, we can write S ( y ) = S ( y ) + S ( y ) − S ( y ), where S ( y ) = 1 h (9) X ψ mod 9 ψ = ψ X π ≡ π ) y log N( π ) × X d ≡ d ) D µ K ( d ) χ π ( d ) ψ ( d ) X n ∈ Z [ ω ] w (cid:16) N( nd ) X (cid:17) ( ψχ π )( n ) ,S ( y ) = 1 h (9) X π ≡ π ) y log N( π ) X d ≡ d ) D µ K ( d ) χ π ( d ) X n ∈ Z [ ω ] w (cid:16) N( nd ) X (cid:17) χ π ( n ) , and S ( y ) = 1 h (9) X π ≡ π ) y log N( π ) X d ≡ d ) D µ K ( d ) χ π ((1 − ω ) d ) X n ∈ Z [ ω ] w (cid:16) nd ) X (cid:17) χ π ( n ) . Note that since 9 is a prime power, each non-principal character ψ takes on thesame values as the primitive character that induces ψ . Hence, we can treat each ψ as primitive, and we denote its conductor by f ψ . Then, ψχ π is a primitive characterof modulus f Z [ ω ], where f = f χ π , since ( π,
9) = 1.
We now apply Poisson summation (Lemma 2.9) to the innermost sums over n ∈ Z [ ω ]. For S ( y ), this gives(17) X n ∈ Z [ ω ] w (N( nd ) /X )( ψχ π )( n ) = Xψ ( − χ π ( f ψ √− ψ ( π ) W ( ψ ) g (1 , π )N( d )N( f ψ π ) X n ∈ Z [ ω ] ψ ( n ) χ π ( n ) ˜ w (cid:18) X N( n )N( d )N( f ψ π ) (cid:19) , where ˜ w ( t ) = b w ( √ t ) , and we used the identities W ( χ π ψ ) = χ π ( − ψ ( − W ( χ π ψ ) = ψ ( − W ( χ π ψ ) , | W ( χ π ψ ) | = N ( π ) N ( f ψ ) , and (from the Chinese Remainder Theorem), W ( χ π ψ ) = χ π ( f ψ ) ψ ( π ) W ( χ π ) W ( ψ ) = χ π ( f ψ ) ψ ( π ) W ( ψ ) χ π ( √− g (1 , π )with g (1 , π ) defined by (8).We now insert (17) into S ( y ), then write each ˜ n ∈ Z [ ω ] as un (1 − ω ) k , where n ≡ u is a unit and k >
0, and use χ π ( d ) χ π ( n ) g (1 , π ) = χ π ( nd ) g (1 , π ) = ( g ( nd, π ) when ( nd, π ) = 10 otherwiseto get S ( y ) = Xh (9) X ψ mod 9 ψ =1 W ( ψ ) ψ ( − f ψ ) X d ≡ d ) D µ K ( d ) ψ ( d )N( d ) X u ∈ Z [ ω ] × X n ∈ Z [ ω ] n ≡ ψ ( un ) × X π ≡ π,nd )=1N( π ) y g ( nd, π )N( π ) ˜ w (cid:18) X N( n )N( d f ψ π ) (cid:19) χ π ( u √− f ψ ) ψ ( π ) log N( π ) . (18)We get similar and simpler (since there is no character ψ ) formulae for S ( y )and S ( y ), namely S ( y ) = Xh (9) X d ≡ d ) D µ K ( d )N( d ) X u ∈ Z [ ω ] × k > X n ∈ Z [ ω ] n ≡ × X π ≡ π,nd )=1N( π ) y g ( nd, π )N( π ) χ π ( u (1 − ω ) k √−
3) ˜ w (cid:18) X k N( n )N( d π ) (cid:19) log N( π ) , and S ( y ) = Xh (9) X d ≡ d ) D µ K ( d )N( d ) X u ∈ Z [ ω ] × k > X n ∈ Z [ ω ] n ≡ × X π ≡ π,nd )=1N( π ) y g ( nd, π )N( π ) χ π ( u (1 − ω ) k +1 √−
3) ˜ w (cid:18) X k − N( n )N( d π ) (cid:19) log N( π ) . NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 17 We first estimate the sum over primes. To this end, we define for any character λ on Z [ ω ] and r ≡ H ( Z, r, λ ) = X π ≡ π,r )=1N( π ) Z ˜ g λ ( r, π ) log N( π ) = X c ≡ c,r )=1N( c ) Z ˜ g λ ( r, c )Λ( c )˜ g λ ( r, c ) = g ( r, c ) λ ( c )N( c ) − / , (19)where the second equality in the first line follows from Lemma 2.8. Theorem 4.4.
Let λ be a character on Z [ ω ] and r ∈ Z [ ω ] with r ≡ . Thenfor any ε > , H ( Z, r, λ ) ≪ Z ε (cid:16) Z / N( r ) / ε + Z / N( r ) / ε + Z / N( r ) / ε + Z / N( r ) / ε (cid:17) , where the implied constant depends on the modulus of λ and ε . We will prove this result in Section 5 following the techniques of [22]. Theorem4.4 can be compared with Theorem 1 of [22] which corresponds to the case r = 1with a bound of Z / ε .We first show how Theorem 4.4 implies Theorem 1.2. Write the estimate inTheorem 4.4 as H ( Z, r, λ ) ≪ P j =1 Z ϑ j N( r ) θ j with each 0 < θ j < / < ϑ j < S ( y ) since the estimates for S ( y ) and S ( y ) follow similarly.By partial summation, we have for λ ( π ) = χ π ( u √− f ψ ) ψ ( π ) , which is a character on Z [ ω ] of bounded modulus, X π ≡ π,nd )=1N( π ) y g ( nd, π )N( π ) λ ( π ) log N( π ) ˜ w (cid:18) XN ( n ) N ( d f ψ ) N ( π ) (cid:19) = Z y ˜ w (cid:18) XN ( n ) N ( d f ψ ) Z (cid:19) Z − / dH ( Z, nd, λ )= ˜ w (cid:16) XN ( n ) N ( d f ψ ) y (cid:17) y / H ( y, nd, λ ) + Z y H ( Z, nd, λ ) (cid:18) ˜ w ′ (cid:16) XN ( n ) N ( d f ψ ) Z (cid:17) Z / + ˜ w (cid:16) XN ( n ) N ( d f ψ ) Z (cid:17) Z / (cid:19) dZ ≪ X j =1 N ( nd ) θ j min y ϑ j − / , (cid:18) XN ( n ) N ( d f ψ ) (cid:19) − y ϑ j +3 / ! , using Theorem 4.4. We also used the fact that ˜ w is bounded and ˜ w ′ ( x ) ≪ x − forthe first bound, and the fact that ˜ w ( x ) ≪ x − and ˜ w ′ ( x ) ≪ x − for the secondbound. Inserting this estimate in (18) yields S ( y ) ≪ X X j =1 X d ≡ d ) D N( d ) θ j − X n ∈ Z [ ω ]N( n ) yN ( d f ψ ) /X N( n ) θ j y ϑ j − / + X N( n ) >yN ( d f ψ ) /X N( n ) θ j (cid:18) N( d f ψ ) X N( n ) (cid:19) y ϑ j +3 / ! ≪ X X j =1 y ϑ j − / X d ≡ d ) D N( d ) θ j − (cid:18) y N( d ) X (cid:19) θ j + y ϑ j +3 / X d ≡ d ) D N( d ) θ j − (cid:18) N( d ) X (cid:19) X N( n ) >yN ( d ) /X N( n ) θ j − ! ≪ X j =1 y ϑ j + θ j +1 / D θ j X θ j . The estimates for S ( y ) and S ( y ) are similarly carried out and result in thesame bound, so we conclude that(20) S ( y ) ≪ X j =1 y ϑ j + θ j +1 / D θ j X θ j . Finding the maximal support.
Combining (16) (under GRH) and (20),and assuming that 1 D X / and y X v , we obtain S F ′ ( y ) ≪ X ε (cid:18) y / D / X / + y / D / X / + y / D / X / + y / D / X / ε (cid:19) + Xy / D log X + yX / log X. Using partial integration as we did at the end of the proof of Lemma 3.1, andreplacing y = X v , this gives E F ′ ( X ) ≪ X ε (cid:0) X v/ − / D / + X v/ − / D / + X v/ − / D / + X v/ − / D / (cid:1) + X log XD + X v/ / log X, and choosing D = X ε , we have that E F ′ ( X ) = o ( X ) for any v < / v comes from the term X v/ − / D / above,which in turns comes from the term Z / N ( r ) / ε of Theorem 4.4.This completes the proof of Theorem 1.2, assuming Theorem 4.4, which is provenin the next two sections. NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 19 Proof of Theorem 4.4
The proof of Theorem 4.4 is a slight generalization of the proof of Theorem 1 of[22], where the author proves the bound X c ≡ c ) X g ( c )N( c ) / Λ( c ) ≪ X / ε . Comparing the above and the statement of Theorem 4.4, and replacing g ( c ) = g (1 , c ) by g ( r, c ) λ ( c ) when ( r, c ) = 1, we need to keep the dependence on the shift r (the character λ has absolutely bounded conductor). Lemma 5.1 (Vaughan’s Identity [22, p. 101]) . Let r ∈ Z [ ω ] , and Σ j ( Z, r, u ) = X a,b,c Λ( a ) µ K ( b )˜ g λ ( r, abc ) , (0 j where ˜ g λ is defined in (19) and each sum runs over a, b, c ∈ Z [ ω ] which are square-free with a, b, c ≡ , Z < N ( abc ) Z , ( r, abc ) = 1 , and subject to theconditions N ( bc ) u, j = 0 N ( b ) u, j = 1 N ( ab ) u, j = 2 ′ N ( a ) , N ( b ) u < N ( ab ) j = 2 ′′ N ( b ) u < N ( a ) , N ( bc ) j = 3 N ( a ) < N ( bc ) u, j = 4 . Then, Σ ( Z, r, u ) = Σ ( Z, r, u ) − Σ ′ ( Z, r, u ) − Σ ′′ ( Z, r, u ) − Σ ( Z, r, u ) + Σ ( Z, r, u ) . Furthermore, Σ ( Z, r, u ) = H (2 Z, r, λ ) − H ( Z, r, λ ) , and if we suppose that u Z / , then Σ ( Z, r, u ) = 0 . When using Lemma 5.1, the sums Σ j are divided into the so-called Type I sums(Σ and Σ ′ ) and Type 2 (bilinear) sums (Σ ′′ and Σ ), and each type is boundedwith a different technique. For the Type II sums, the proof of [22] goes throughin the exact same way, replacing ˜ g (1 , e ) by ˜ g λ ( r, e ) with the obvious modifications,see Section 5.2. For the Type I sums, we have to use a general version of the workof Patterson for the distribution of the generalized Gauss sums ˜ g λ ( r, e ), and keepthe dependence on the parameter r , see Section 5.1.5.1. Type I Sums.
The so-called Type I sums of Vaughan’s formula are Σ ( Z, r, u )and Σ ′ ( Z, r, u ) and they are bounded using the work of Patterson on the distribu-tion of the generalized Gauss sums ˜ g λ ( r, c ). Lemma 5.2.
For u Z / , | Σ ( Z, r, u ) | Z ) X a ≡ a,r )=1N( a ) u | µ K ( a ) | sup Z z Z | F a ( z, r, λ ) || Σ ′ ( Z, r, u ) | u X a ≡ a,r )=1N( a ) u | µ K ( a ) | sup Z z Z | F a ( z, r, λ ) | , where (21) F a ( z, r, λ ) = X b ≡ r,b )=1 ,a | b N( b ) z ˜ g λ ( r, b ) . Proof.
We haveΣ ( Z, r, u ) = X N ( abc ) ∼ ZN ( b ) u ( abc,r )=1 Λ( a ) µ K ( b )˜ g λ ( r, abc ) = X N ( bd ) ∼ ZN ( b ) u ( bd,r )=1 µ K ( b )˜ g λ ( r, bd ) X a | d Λ( a )= X N ( b ) u ( b,r )=1 µ K ( b ) X N ( d ) ∼ Z ( d,r )=1 b | d ˜ g λ ( r, d ) log N ( d/b ) , and using partial integration for the inner sum gives the first assertion. As for thesecond, we haveΣ ′ ( Z, r, u ) = X N ( abc ) ∼ ZN ( ab ) u ( abc,r )=1 Λ( a ) µ K ( b )˜ g λ ( r, abc )= X N( d ) u ( d,r )=1 (cid:18) X ab = d µ K ( b )Λ( a ) (cid:19) X c ≡ c,r )=1 ,d | c N( c ) ∼ Z ˜ g λ ( r, c ) X N( d ) u ( d,r )=1 | µ K ( d ) || F (2 Z, a, r, λ ) − F ( Z, a, r, λ ) | log N ( d ) , where on the second line we used the fact that the sum in the parenthesis is sup-ported only on square-free d . This proves the second assertion. (cid:3) Type II (Bilinear) Sums.
In this section, we bound the sumsΣ ′′ ( Z, r, u ) = X N ( abc ) ∼ Z ( abc,r )=1N( a ) ,N ( b ) u< N( ab ) Λ( a ) µ K ( b )˜ g λ ( r, abc )Σ ( Z, r, u ) = X N ( abc ) ∼ Z ( r,abc )=1N( a ) , N( bc ) >uN ( b ) u Λ( a ) µ K ( b )˜ g λ ( r, abc ) . NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 21 Note that since ( abc, r ) = 1, g ( r, abc ) = χ ab ( c ) g ( r, ab ) g ( r, c ) whenever ( ab, c ) = 1and is zero otherwise, by Lemmas 2.7 and 2.8. Therefore, in all cases, we have˜ g λ ( r, abc ) = χ ab ( c )˜ g λ ( r, ab )˜ g λ ( r, c ) , and we can write Σ ′′ ( Z, r, u ) = X N( vw ) ∼ Z N( v ) ,N ( w ) >u A ( v ) B ( w ) χ v ( w )Σ ( Z, r, u ) = X N( vw ) ∼ Z N( v ) ,N ( w ) >u C ( v ) D ( w ) χ v ( w )where we put A ( v ) = X ab = v N( a ) , N( b ) u Λ( a ) µ K ( b )˜ g λ ( r, ab ) , B ( w ) = ˜ g λ ( r, w ) C ( v ) = Λ( v )˜ g λ ( r, v ) , D ( w ) = X bc = wN ( b ) u µ K ( b )˜ g λ ( r, bc )whenever ( r, vw ) = 1 and zero otherwise. Notice that we have used the fact that u Z / to write N ( w ) > u in the sum for Σ ′′ .Note also that A ( v ) , B ( w ) , C ( v ) , D ( w ) ε X ε , for all relevant v, w and that the functions are supported on square-free integersin Z [ ω ] which are congruent to 1 modulo 3 by the hypothesis on a, b, c . We cannow use directly the proof of [22, Lemma 2] with the new functions A, B, C, D asdefined above, which differ only by fact that ˜ g ( v ) = ˜ g (1 , v ) is replaced by ˜ g λ ( r, v ),which does not change the size. This uses the large sieve [22, Theorem 2] for cubiccharacters to catch the oscillation of the character χ v ( w ) in the above equationsfor Σ ′′ and Σ , and we get the following. Lemma 5.3.
For any ε > , and u Z / , we have Σ ′′ ( Z, r, u ) , Σ ( Z, r, u ) ≪ Z / ε (cid:0) Z / u − / + Z / (cid:1) , where the implied constant depends only on ε . In particular, choosing u = Z / ,this gives Σ ( Z, r, Z / ) , Σ ( Z, r, Z / ) ≪ Z / ε . Proof.
We remark that [22, Lemma 2] states only the second part of the abovelemma. For the first part, one uses the bound on the bottom of page 104 whichleads Σ ′′ ( Z, r, u ) , Σ ( Z, r, u ) ≪ Z / ε (cid:0) V / + W / + Z / (cid:1) together with (9) of page 103 which implies that V, W ≤ Z/u. (cid:3)
Putting things together.
Using Lemmas 5.1, 5.2 and 5.3, we have for 1 u Z / that H (2 Z, r, λ ) − H ( Z, r, λ ) ≪ log Z X a ≡ a,r )=1N( a ) u sup Z z Z | F a ( z, r, λ ) | + Z / ε (cid:16) Z / u − / + Z / (cid:17) . Assuming Proposition 6.2, which gives an upper bound for | F a ( z, r, λ ) | , we de-duce that H (2 Z, r, λ ) − H ( Z, r, λ ) ≪ Z / ε (cid:16) Z / u − / + Z / (cid:17) + Z / ε N( r ) − / ε N( r ∗ ) ε + u / Z / ε N( r ) / ε + u / Z / ε N( r ) + ε . Finally, using Lemma 2.10 with u ∈ [1 , Z / ] to balance the terms containing u proves Theorem 4.4, where the term Z / ε N ( r ) / ε which gives the final esti-mate corresponds to u = Z / N ( r ) − / .6. Estimate of F a ( z, r, λ )Recall from (21) that F a ( z, r, λ ) = X b ≡ a | b, ( b,r )=1N( b ) z ˜ g λ ( r, b ) , where ˜ g λ ( r, b ) was defined in (19). To evaluate F a ( z, r, λ ), we will use the non-normalized generating function defined by h a ( r, λ, s ) = X b ≡ a | b, ( b,r )=1 g λ ( r, b )N( b ) − s , where g λ ( r, b ) = λ ( b ) g ( r, b ) = ˜ g λ ( r, b )N( b ) / . The following lemma contains theanalytic information on the generating function h a ( r, λ, s ). Lemma 6.1.
Write r = r r r , where r , r are square-free and co-prime, and r i ≡ . Let a ≡ be square-free and ( a, r ) = 1 . Then, h a ( r, λ, s + 1 / can be meromorphically continued to the whole complex plane; it is entire for ℜ ( s ) > / except possibly for a simple pole at s = 5 / with residue Res s =5 / h a ( r, λ, s + ) ≪ N ( a ) − N( r ) − / log 2N( ar ) log 2N( r ∗ ) when r = 1 and λ = λ , and zero otherwise.Let ε > and σ = 1 + ε . For s = σ + it satisfying σ − / σ σ and | s − / | > / , (22) h a ( r, λ, s + ) ≪ N( r r ) ( σ − σ ) N ( a ) ( σ − σ ) − σ N ( ar r ∗ ) ε (1 + t ) σ − σ . Furthermore, (23) Z T − T | h a ( r, λ, σ + it ) | dt ≪ T N ( a ) − / − ε N ( r r r ∗ ) / ε . We assume Lemma 6.1 for now, and we prove the following proposition.
NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 23 Proposition 6.2.
Suppose a ∈ Z [ ω ] is square-free with ( a, r ) = 1 , and let λ be aDirichlet character on Z [ ω ] . Write r = r r r with r i ≡ , r , r square-freeand co-prime, and let r ∗ be the product of the primes dividing r but not r r .Then, for any ε > , F a ( z, r, λ ) ≪ z / N ( a ) − ε N( r ) − / ε N( r ∗ ) ε + z / ε N ( a ) − / N( r ) / ε + z / ε N ( a ) − / N( r ) / ε . where the first term appears only when r = 1 , and the implied constant dependson ε and the character λ .Proof of Proposition 6.2. It follows from Perron’s Formula (cf. [9, Ch.17 p.105])that for z − ∈ Z + and σ = 1 + ε , F a ( z, r, λ ) − πi Z σ + iTσ − iT h a ( r, λ, s + ) z s s ds ≪ X b ≡ a | b, ( b,r )=1 ( z/ N( b )) σ min (cid:0) , T − | log( z/ N( b )) | − (cid:1) ≪ ε T − z ε N ( a ) − . (24)Shifting the contour to Re s = + ε , we pick up the possible residue of h a ( r, λ, s )at s = 4 / πi Z σ + iTσ − iT h a ( r, λ, s + ) z s s ds = 6 z / s =5 / h a ( r, λ, )+ 12 πi (cid:18)Z σ − / − iTσ − iT + Z σ + iTσ − / iT + Z σ − / iTσ − / − iT (cid:19) h a ( r, λ, s + ) z s s ds. Using the convexity bound (22) of Lemma 6.1, we see that (cid:18) Z σ − / − iTσ − iT + Z σ + iTσ − / iT (cid:19) h a ( r, λ, s + ) z s s ds ≪ N ( ar r ∗ ) ε Z σ σ − / T σ − σ ) − N( r r ) ( σ − σ ) N ( a ) ( σ − σ ) − σ z σ dσ ≪ N ( ar r ∗ ) ε (cid:0) T − N ( a ) − − ε z ε + N( r r ) / N ( a ) − / − ε z / ε (cid:1) . (25)By the mean value estimate (23) of Lemma 6.1 and Cauchy-Schwarz inequality, weobtain Z T − T | h a ( r, λ, σ + it ) | dt ≪ T / N ( a ) − / − ε/ N ( r r r ∗ ) / ε , so that Z T − T (cid:12)(cid:12)(cid:12) h a ( r, λ, σ + it ) σ + it (cid:12)(cid:12)(cid:12) dt ≪ T / N ( a ) − / − ε/ N ( r r r ∗ ) / ε on integrating by parts. Thus,(26) Z σ − / iTσ − / − iT h a ( r, λ, s + ) z s s ds ≪ T / z / ε N ( a ) − / − ε/ N ( r r r ∗ ) / ε . Combining (24), (25) and (26) together with the bound on the residue fromLemma 6.1, we obtain F a ( z, r, λ ) ≪ N ( ar r ∗ ) ε (cid:0) T / z / ε N( r r ) / N ( a ) − / − ε + T − N ( a ) − − ε z ε + z / N ( a ) − N( r ) − / (cid:1) where the last term is needed only when r = 1.Using Lemma 2.10 with T ∈ [1 , z / ] to bound the first two terms inside theparentheses yields the desired estimate. (cid:3) Remark 6.3.
Using Lemma 2.10, we obtain automatically a result independent ofthe various parameters. It would have been equivalent to choosing T = z / N ( a ) − / N ( r r ) − / , which gives the bound F a ( z, r, λ ) ≪ N ( ar r ∗ ) ε (cid:16) z / ε N ( a ) − / N ( r r ) / + z / N ( a ) − N( r ) − / (cid:17) assuming that T > N ( a ) N ( r r ) z , which is true since we are taking N ( a ) z / in Section 5.3.The rest of the section is devoted to the proof of Lemma 6.1. We first stateintermediate results in lemmas 6.4, 6.5 and 6.6.Our first goal is to write h a ( r, λ, s ) in terms of the generating function(27) ψ ( r, λ, s ) = X b ≡ λ ( b ) g ( r, b )N( b ) − s , which appears in the work of Patterson; namely, in [40] when λ is trivial, and[39] for the general case including the ray class character λ , following the work ofKazhdan and Patterson in [30].We also define, for any prime π ≡ ψ π ( r, λ, s ) = X b ≡ b,π )=1 λ ( b ) g ( r, b )N( b ) − s . We now express h a ( r, λ, s ) in terms of the function ψ ( r, λ, s ) . Our lemma issimilar to [1, Lemma 3.6], or [7, Lemma 3.11] for the function field case, where theauthors of those papers are dealing with slightly different functions.
Lemma 6.4.
Suppose a ∈ Z [ ω ] is square-free with ( a, r ) = 1 . Write r = r r r with r i ≡ , r , r square-free and co-prime Let r ∗ be the product of the primesdividing r but not r r . Then, h a ( r, λ, s ) = g ( r, a ) λ ( a ) N ( a ) − s Y π | ar r (1 − λ ( π ) N( π ) − s ) − × X d ≡ d | r ∗ µ K ( d ) λ ( d ) g ( ar r , d ) N ( d ) s Y π | d (1 − λ ( π ) N( π ) − s ) − × X c ≡ c | dar µ K ( c )N( c ) − s λ ( c ) g ( dar r /c, c ) ψ ( dar r /c, λ, s ) . NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 25 Proof.
Recall that when ( r, b ) = 1, g ( r, b ) = 0 when b is not square-free by Lemmata2.7 and 2.8. Then, rewriting b in the sum h a ( r, λ, s ) as b = ab ′ with b ′ ≡ a , and using Lemma 2.7, this yields h a ( r, λ, s ) = X b ≡ a | b, ( b,r )=1 λ ( b ) g ( r, b ) N ( b ) − s = g ( r, a ) λ ( a ) N ( a ) − s X b ≡ b,ar )=1 λ ( b ) g ( ar, b ) N ( b ) − s . Since ( b, r ) = 1 in the above sum, g ( ar, b ) = χ b ( r ) g ( ar r , b ) = g ( ar r , b ). Usingalso ( a, r ) = 1, it follows that X b ≡ b,ar )=1 λ ( b ) g ( ar, b ) N ( b ) − s = X b ≡ b,ar r )=1 λ ( b ) g ( ar r , b ) N ( b ) − s X d ≡ d | ( b,r ∗ ) µ K ( d )= X d ≡ d | r ∗ µ K ( d ) λ ( d ) N ( d ) s X b ≡ bd,ar r )=1 λ ( b ) g ( ar r , bd ) N ( b ) − s = X d ≡ d | r ∗ µ K ( d ) λ ( d ) N ( d ) s X b ≡ b,adr r )=1 λ ( b ) g ( ar r , bd ) N ( b ) − s = X d ≡ d | r ∗ µ K ( d ) λ ( d ) g ( ar r , d ) N ( d ) s X b ≡ b,adr r )=1 λ ( b ) g ( adr r , b ) N ( b ) − s , using again lemmas 2.7 and 2.8. Since adr r is square-free (recall that d | r ∗ ), itfollows from [1, Lemma 3.6] with r replaced by adr that X b ≡ b,adr r )=1 λ ( b ) g ( adr r , b ) N ( b ) − s = Y π | r (1 − λ ( π ) N( π ) − s ) − X b ≡ b,adr )=1 λ ( b ) g ( adr r , b )N( b ) − s = Y π | r (1 − λ ( π ) N( π ) − s ) − Y π | adr (1 − λ ( π ) N( π ) − s ) − × X c ≡ c | adr µ K (( c ))N( c ) − s λ ( c ) g (cid:18) adr r c , c (cid:19) ψ (cid:18) adr r c , λ, s (cid:19) . Combining all of the above, we arrive at the desired result. (cid:3)
Next, we need to understand h a ( r, λ, s + ) in the strip 1 / ε σ ε . Lemma 6.5 (Lemma p. 200 [39]) . Let r ∈ Z [ ω ] . Then, ψ ( r, λ, s ) can be mero-morphically continued to the whole complex plane; it is entire for ℜ ( s ) > , exceptpossibly for a simple pole at s = 4 / with residue ρ ( r, λ ) (which can occur only when λ is principal). Write r = r r r , where r , r are square-free and co-prime, and r i ≡ . Then ρ ( r, λ ) = 0 if r = 1 , and ρ ( r, λ ) ≪ N ( r ) − / , when r = 1 .Let ε > , and σ = 3 / ε . If s = σ + it , σ − / < σ < σ , and | s − / | > / ,then ψ ( r, λ, s ) ≪ N ( r ) ( σ − σ ) (1 + t ) σ − σ , where both bounds above are dependent on the conductor of the character λ . The convexity bound of the above lemma can be used to bound the integrandsinvolved in estimating F a ( z, r, λ ), as was used in [23]. Again, we are adaptingthe further work of [22] to get better bounds, by replacing pointwise bounds on theintegrands by mean value bounds. Our starting point is the following lemma, whichcorresponds to equation (20) of [22] with the difference that we are considering thefunction ψ ( r, λ, s ) defined in (27), and Heath-Brown considers only the case where λ is trivial. The proof of the general case is identical, using the generalisations of[30, 39]. Lemma 6.6 ([22, Equation 20]) . Z T − T | ψ ( r, λ, ε + it ) | dt ≪ T N ( r ) / . We remark that using the convexity bound of Lemma 6.5 would lead to theweaker bound Z T − T | ψ ( r, λ, ε + it ) | dt ≪ Z T − T (cid:12)(cid:12)(cid:12) N ( r ) / t (cid:12)(cid:12)(cid:12) dt ≪ T N ( r ) / . Combining the previous three lemmas we arrive at the following result for thefunction h a ( r, λ, s ). Proof of Lemma 6.1.
By Lemma 6.4 h a ( r, λ, s + ) = g ( r, a ) λ ( a ) N ( a ) − s − / Q π | ar r (1 − λ ( π ) N( π ) / − s ) − × X d ≡ d | r ∗ µ K (( d )) λ ( d ) g ( ar r , d ) N ( d ) s +1 / Y π | d (1 − λ ( π ) N( π ) / − s ) − × X c ≡ c | dar µ K (( c ))N( c ) − s λ ( c ) g ( dar r /c, c ) ψ ( dar r /c, λ, s + 12 ) . Hence,Res s =5 / h a ( r, λ, s + ) = g ( r, a ) λ ( a ) N ( a ) − / Q π | ar (1 − λ ( π ) N( π ) − ) − × X d ≡ d | r ∗ µ K ( d ) λ ( d ) g ( ar , d ) N ( d ) / Y π | d (1 − λ ( π ) N( π ) − ) − × X c ≡ c | dar µ K ( c )N( c ) − / λ ( c ) g ( dar /c, c ) ρ ( dar /c, λ ) , NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 27 which givesRes s =5 / h a ( r, λ, s + ) ≪ N ( a ) − N( r ) − / X d ≡ d | r ∗ N ( d ) − X c ≡ c | dar N( c ) − ≪ N ( a ) − N( r ) − / log 2N( ar ) log 2N( r ∗ ) . Using again Lemma 6.4, we have for s = σ + it as in the hypotheses, h a ( r, λ, s + ) ≪ N( r r ) ( σ − σ ) N ( a ) ( σ − σ ) − σ (1 + t ) σ − σ × X d ≡ d | r ∗ N ( d ) ( σ − σ ) − σ X c ≡ c | dar N( c ) − σ − ( σ − σ ) ≪ N( r r ) ( σ − σ ) N ( a ) ( σ − σ ) − σ N ( ar r ∗ ) ε (1 + t ) σ − σ , which proves (22). We now proceed to prove (23). Again, by Lemma 6.4, h a ( r, λ, ε + it ) ≪ ε N ( a ) − / − ε X d ≡ d | r ∗ | µ K ( d ) | N ( d ) / ε × X c ≡ c | dar | µ K ( c ) | N( c ) − / − ε (cid:12)(cid:12) ψ ( dar r /c, λ, σ + it ) (cid:12)(cid:12) . Using Cauchy-Schwarz twice, we bound | h a ( r, λ, ε + it ) | by ≪ ε N ( a ) − − ε X d ≡ d | r ∗ | µ K ( d ) | N ( d ) ε X d ≡ d | r ∗ (cid:12)(cid:12)(cid:12)(cid:12) X c ≡ c | dar | µ K ( c ) | N( c ) / ε | ψ ( dar r /c, λ, σ + it ) | (cid:12)(cid:12)(cid:12)(cid:12) ≪ ε N ( a ) − − ε X d ≡ d | r ∗ | µ K ( d ) | N ( d ) ε × X d ≡ d | r ∗ (cid:18) X c ≡ c | dar | µ K ( c ) | N( c ) ε X c ≡ c | dar | ψ ( dar r /c, λ, σ + it ) | (cid:19) ≪ ε N ( a ) − − ε X d ≡ d | r ∗ X c ≡ c | dar | ψ ( dar r /c, λ, σ + it ) | . Using Lemma 6.6, this gives Z T − T | h a ( r, λ, σ + it ) | dt ≪ T N ( a ) − − ε N ( ar r ) / X d ≡ d | r ∗ N ( d ) / X c ≡ c | dar N ( c ) − / ≪ T N ( a ) − / − ε N ( r r r ∗ ) / ε , which proves (23). (cid:3) A positive proportion of non-vanishing
To prove Corollary 1.3, we choose φ ( x ) = φ v ( x ) = (cid:16) sin( πvx ) πvx (cid:17) . Then, b φ v ( t ) = v − | t | v if | t | ≤ v − v, v ).For m ∈ Z , m >
0, let p m ( X ) = 1 A F ′ ( X ) X χ ∈ F ′ w (cid:16) N(cond( χ )) X (cid:17) δ ( χ ; m ) δ ( χ ; m ) = ( s = L ( s, χ ) = m φ v (0) = 1 and φ v ( x ) ≥ x ∈ R and the zeros are counted withmultiplicity, we have (under GRH)1 A F ′ ( X ) X χ ∈ F ′ w (cid:16) N(cond( χ )) X (cid:17) ∞ X m =1 p m ( X ) A F ′ ( X ) X χ ∈ F ′ w (cid:16) N(cond( χ )) X (cid:17) ∞ X m =1 mp m ( X ) A F ′ ( X ) X χ ∈ F ′ w (cid:16) N(cond( χ )) X (cid:17) X ρ = + iγL ( ρ,χ )=0 φ v (cid:16) γ log X π (cid:17) = D ( X ; φ v , F ′ ) . Since P m > p m ( X ) = 1, this yields p ( X ) = 1 A F ′ ( X ) X χ ∈ F ′ L ( ,χ ) =0 w (cid:16) N(cond( χ )) X (cid:17) > −D ( X ; φ v , F ′ ) > − b φ v (0)+ o X (1) , where the last inequality follows from Theorem 1.2. This proves a weighted versionof Corollary 1.3. We can easily re-state this as a counting version by choosing w asfollows. Assume X ∈ Z , and let w ( t ) = t (cid:0) − / (1 − X ( x − ) (cid:1) < x < /X t > /X. Then, w is smooth on [0 , ∞ ) and A F ′ ( X ) counts exactly the characters χ ∈ F ′ with N(cond( χ )) X . Hence we conclude that { χ ∈ A F ′ : N(cond( χ )) X, L (1 / , χ ) = 0 } { χ ∈ A F ′ : N(cond( χ )) X } = 1 A F ′ ( X ) X χ ∈ F ′ L ( ,χ ) =0 w (cid:16) N(cond( χ )) X (cid:17) > − b φ v (0) + o X (1) , NE-LEVEL DENSITY AND NON-VANISHING FOR CUBIC L -FUNCTIONS 29 and letting X → ∞ over the integers, we have { χ ∈ A F ′ : N(cond( χ )) X, L (1 / , χ ) = 0 } { χ ∈ A F ′ : N(cond( χ )) X } > − b φ v (0) + o X (1) → − . References
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Department of Mathematics and Statistics, Concordia University, 1455 de Maison-neuve West, Montr´eal, Qu´ebec, Canada H3G 1M8
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