An Asymptotic for the Number of Solutions to Linear Equations in Prime Numbers from Specified Chebotarev Classes
aa r X i v : . [ m a t h . N T ] N ov An Asymptotic for the Number of Solutions toLinear Equations in Prime Numbers fromSpecified Chebotarev Classes
Daniel M. KaneOctober 31, 2018
Abstract
We extend results relating to Vinogradov’s three primes Theorem toprovide asymptotic estimates for the number of solutions to a given linearequation in three or more prime numbers under the additional constraintthat each of the primes involved satisfies specialized Chebotarev condi-tions. In particular, we show that such solutions can be expected to existunless a solution would violate some local constraint.
In 1937 Vinogradov proved that any sufficiently large odd number could bewritten as the sum of three primes. In addition, he managed to provide anasymptotic for the number of ways to do so, proving (as stated in Iwaniec-Kowalski ([3]) Theorem 19.2)
Theorem 1 (Vinogradov) . For N a positive integer and A any real numberthen X n + n + n = N Λ( n )Λ( n )Λ( n ) = G ( N ) N + O ( N log − A ( N )) , (1) where G ( N ) = 12 Y p | N (1 − ( p − − ) Y p ∤ N (1 + ( p − − ) , Λ( n ) is the Von Mangoldt function, and the asymptotic constant in the O de-pends on A . It is easy to see that the contribution to the left hand side of Equation 1coming from one of the n i a power of prime is negligible, and thus this sideof the equation may be replaced by a sum over triples p , p , p of primes thatsum to N of log( p ) log( p ) log( p ). This implies that any sufficiently large odd1umber can be written as a sum of three primes since G ( N ) is bounded belowby a constant for N odd.It should also be noted that the main term, N G ( N ) can be written as C ∞ Y p C p , where C ∞ = N C p = ( (1 − ( p − − ) if p | N (1 + ( p − − ) else . When written this way, there is a reasonable heuristic explanation for Theo-rem 1. To begin with, the Prime Number Theorem says that the Von Mangoldtfunction, is approximated by the distribution assigning 1 to each positive inte-ger. The term C ∞ provides an approximation to the number of solutions basedon this heuristic. The C p can be thought of as corrections to this heuristic. Theycan be thought of as local contributions coming from congruential informationabout the primes p i . In particular, C p can easily be seen to be equal to p n ( n , n , n ) ∈ (( Z /p Z ) ∗ ) : n + n + n ≡ N (mod p ) o ( p − . This can be thought of as a correction factor coming from the fact that no prime(except for p ) is a multiple of p . In particular, C p is equal to the ratio of theprobability that three randomly chosen elements of ( Z /p Z ) ∗ sum to N modulo p to the probability that three randomly chosen elements of ( Z /p Z ) + sum to N modulo p .A number of generalizations of Vinogradov’s Theorem have since been proven.Some, such as Zhan in [11] and [12], deal with restrictions on the relative sizesof the primes involved. In particular, [11] shows that one of the primes can betaken to be as small as N / ǫ . It is shown in [12] that the primes involvedcan all be taken to be relatively close to each other.The problem was generalized to number fields by Tuljaganova in [10] andlater by Noda in [8], who ask which elements of the ring of integers can bewritten as a sum of generators of principle prime ideals.Several more papers deal with problems where the primes involved are re-quired to be taken from specified subsets of the set of all prime numbers. In [4]Li and Pan show that for any three sets of prime numbers with sufficient densitythat any sufficiently large odd N can be written as a sum of primes, one fromeach set. In particular, they show that for any three sets of primes with relativedensities δ , δ and δ within the set of all primes so that δ + δ + δ > p i are required to lie inspecified Chebotarev classes. In particular, after fixing Galois extensions K i / Q and conjugacy classes C i of Gal( K i / Q ), we find an asymptotic for the sum of Q ki =1 log( p i ) over primes p , . . . , p k ≤ X so that [ K i / Q , p i ] = C i for each i and P ki =1 a i p i = N . Note that the results on writing N as a sum of primes fromarithmetic progressions, will follow as a special case of this when K i is abelianover Q (although our bounds are probably worse). In particular we prove: Theorem 2.
Let k ≥ be an integer. Let K i / Q be finite Galois extensions (1 ≤ i ≤ k ) and G i = Gal ( K i / Q ) . Let a , . . . , a k be non-zero integers withno common divisor. Let C i be a conjugacy class of G i for each i . Let K ai bethe maximal abelian extension of Q contained in K i , and let D i be its discrim-inant. Let D be the least common multiple of the D i . Let H i be the subgroupof ( Z /D Z ) ∗ corresponding to K ai via global class field theory. Let H i be thecoset of H i corresponding to the projection of an element of C i to Gal ( K ai / Q ) .Additionally let N be an integer and let A and X be positive numbers, then X p i ≤ X [ K i / Q ,p i ]= C i P i a i p i = N k Y i =1 log( p i ) = k Y i =1 | C i || G i | ! C ∞ C D Y p ∤ D C p + O (cid:16) X k − log − A ( X ) (cid:17) , (2) where the sum of the right hand side is over sets of prime numbers p , . . . , p k ≤ X so that P ki =1 a i p i = N , and so that the Artin symbol [ K i / Q , p i ] lands in theconjugacy class C i of G i for all ≤ i ≤ k . On the right hand side, C ∞ = 1 P ki =1 a i Z x i ∈ [0 ,X ] P i a i x i = N k X i =1 a i ∂∂x i ! dx ∧ dx ∧ . . . ∧ dx k ,C D = D { ( x i ) ∈ (( Z /D Z ) ∗ ) k : x i ∈ H i , P ki =1 a i x i ≡ N (mod D ) } Q ki =1 | H i | ! , and the second product is over primes p not dividing D of C p = p { ( x i ) ∈ (( Z /p Z ) ∗ ) k : P ki =1 a i x i ≡ N (mod p ) } ( p − k ! . The implied constant in the O term may depend on k, K i , C i , a i , and A , but noton X or N . Additionally, if k = 2 and K i , C i , a i , A, X are fixed, then Equation(2) holds for all but O ( X log − A ( X )) values of N . The introduction of Chebotarev classes leads to two main differences betweenour asymptotic and the classical one. For one, the Chebotarev Density Theorem3ells us that there are fewer primes in these Chebotarev classes than out of themand causes us to introduce a factor of Q ki =1 (cid:16) | C i || G i | (cid:17) . Secondly, Global Class FieldTheory tells us that the prime p i will necessarily lie in the subset H i of ( Z /D Z ) ∗ ,giving us the correction factor C D rather than Q p | D C p to account for requiredcongruence relations that these primes satisfy.It should be noted that the error term is o ( X k − log( X ) − ), whereas if N is bounded away from both the largest and smallest possible values that canbe taken by P i a i x i for x i ∈ [0 , X ], then C ∞ will be on the order of X k − .For K i , C i fixed, the first term on the right hand side is a constant. Althoughit depends on N , C D will be bounded away from both 0 and ∞ unless N cannot be written as a sum P a i x i with x i ∈ H i . Lastly, for p ∤ Dn Q i a i ,inclusion-exclusion tells us that C p = 1 + O ( p − ), and for p | N , p ∤ D Q i a i , C p = 1 + O ( p − ). This means that unless C p = 0 for some p , Q p C p is withina bounded multiple of Q p | N (1 + O ( p − )) = exp( O (log log log N )) . Therefore,unless C D = 0, C p = 0 for some p , or N is near the boundary of the availablerange, the main term on the right hand side of Equation (2) dominates the error. Our proof will closely mimic the proof in [3] of Theorem 1. We provide a briefoverview of the proof given in [3], discuss our generalization and provide anoutline for the rest of the paper.
On a very general level, the proof given in [3] depends on writingΛ = Λ ♯ + Λ ♭ . Here Λ ♯ is a nice approximation to the Von Mangoldt function obtained essen-tially by sieving out multiples of small primes and Λ ♭ is an error term. It isrelatively easy to deal with the sum X n + n + n = N Λ ♯ ( n )Λ ♯ ( n )Λ ♯ ( n ) , yielding the main term in Equation (1). This leaves additional terms, eachinvolving at least one Λ ♭ . These terms are dealt with by showing that Λ ♭ issmall in the sense that its generating function has small L ∞ norm.To prove this bound on Λ ♭ , Iwaniec and Kowalski make use of Theorem 13.10of [3], which states that for any A X m ≤ x µ ( m ) e πiαm ≪ x log − A ( x )( µ is the M¨obius function) with the implied constant depending only on A . Thisin turn is proved by considering separately the case where α is near a rationalnumber of small denominator and the case where it is not.4f α is close to a rational number, the sum can be bounded through the useof Dirichlet L -functions. In particular, one has bounds on P n ≤ x χ ( n ) µ ( n ) for χ a Dirichlet character ([3] (5.80)). To prove this, Iwaniec and Kowalski useboth Theorem 5.13 of [3], which gives bounds on the sums of coefficients of thelogarithmic derivative of an L -function, and some bounds on zero-free regionsand Siegel zeroes.If α is not well approximated by a rational number with small denominator,an appropriate bound is proved by rewriting the sum using some combinatorialidentities ([3] (13.39)) and using the quadratic form trick. (The actual boundobtained is given in [3] Theorem 13.9.) Our proof of Theorem 2 is similar in spirit to the proof of Theorem 1 givenin [3]. We differ in a few ways, some just in the way we choose to organizeour information and some from necessary complications due to the increasedgenerality. We provide below an outline of our proof and a comparison of ourtechniques to those used in [3].Instead of dealing directly with Λ , Λ ♯ and Λ ♭ as is done in [3], we instead dealdirectly with their generating functions. In Section 3.3, we define G , which isour equivalent of the generating function for Λ. As it turns out, G is somewhatdifficult to deal with directly, so we define a related function F , that is bettersuited for techniques involving Hecke L -functions. In Proposition 6 we provethat we can write G approximately as an appropriate sum of F ’s.In Section 3.4 we define G ♯ and G ♭ , which are analogues of the generatingfunctions for Λ ♯ and Λ ♭ . We also define analogous F ♯ and F ♭ . The sievingtechnique that we use to write G ♯ is not quite analogous to that used in [3].Essentially, we write our version of Λ ♯ as a product of local factors. This willproduce some sums over smooth numbers later in our analysis, so in Lemma 9we bound the number of smooth numbers, so that we may bound errors comingfrom sums over them.We next work on proving that F ♭ has small L ∞ norm (this is somewhatequivalent to [3] showing that generating functions of Λ ♭ or µ are small). Asin [3], we split into two cases based on whether or not we are near a rationalnumber.In Section 4.1, we deal with the approximation near rationals. First, inSection 4.1.1, we generalize some necessary results about L -functions and Siegelzeroes. In Section 4.1.2, we use these to produce an approximation of F , andin Section 4.1.3, we show that this also approximates F ♯ .In Section 4.2, we deal with showing that F ♭ is small away from rationals.It should be noted that while the rest of this paper generalizes the correspond-ing proof in [3] in a relatively straightforward way by use of standard results,something new is needed for this Section. The primary reason for this is thatwhile Vinogradov’s bound on exponential sums over prime numbers reduces thesum in question to exponential sums over arithmetic progressions, the anal-ogous argument in our case requires bounding sums of the form P e πiαN ( a ) a in a number field. To deal with this issue, we will make use ofresults about exponential sums of polynomials. Unfortunately, standard resultsof this type will not be strong enough when the leading term of the polynomialis approximated by a rational number with relatively small (polylogarithmic)denominator. Thus, we require a new bound of this type which is given byLemma 20 below. In the process of deriving this Lemma, we need some resultsabout when multiples of a number with poor rational approximation have agood rational approximation, which we prove in Section 4.2.2. In Section 4.2.3,we use this result to prove bounds on sums of the type described above, and inSection 4.2.4 use these to obtain the necessary control on F . In Section 4.2.1,we prove bounds for F ♯ , and thus on F ♭ .In Section 5, we use our bounds on F ♭ to prove bounds on G ♭ . Finally, inSection 6, we use this bound to prove Theorem 2. In Section 6.1, we introducethe appropriate product generating functions, and deal with the terms comingfrom G ♭ ’s. In Section 6.2, we produce the main term of our Theorem.Finally, in Section 7, we show an application of our Theorem to constructingelliptic curves whose discriminants split completely over specified number fields. In this Section, we introduce some of the basic terminology and results that willbe used throughout the rest of the paper. In Section 3.1, we briefly recall someasymptotic notation. In Section 3.2, we recall some of the basic facts from classfield theory that will be used later. In Section 3.3, we define the functions F and G along with some of the basic facts relating them. In Section 3.4, we define F ♯ and G ♯ along with some related terminology and again prove some basic facts.Finally, in Section 3.5, we prove a result on the distribution of smooth numbersthat will prove useful to us later. Throughout we use O ( X ) to denote a quantity whose absolute value is boundedabove by some constant times X . Let Ω( X ) denote a positive quantity that isbounded below by some constant times the absolute value of X . We use, Θ( X )will be used to denote a quantity which is both O ( X ) and Ω( X ). Throughoutthe paper the implied constants will potentially depend on the number fields K i , K, L, etc. in question, but upon nothing else unless otherwise stated. Specifying the Artin symbol of a prime will sometimes force congruence condi-tions on it coming from global class field theory. In this Section, we review someof the basic facts of this theory that will be needed later. A reader interestedin proofs of these results is encouraged to read Milne [7]. The input that werequire from class field theory can be summarized in the following theorem:6 heorem 3.
There is a one-to-one correspondence between Galois extensions K/ Q with abelian Galois group and pairs ( H, N ) where N is a positive integerand H is a subgroup of ( Z /N Z ) ∗ so that H is not periodic modulo M for any M strictly dividing N . Furthermore, if K is the extension corresponding to this pair ( H, N ) , then K is ramified exactly at the primes dividing N (in fact N dividesthe discriminant of K ) and there exists an isomorphism ϕ : ( Z /N Z ) ∗ /H → Gal ( K/ Q ) so that for any rational prime p not dividing N , [ K/ Q , p ] = ϕ ( p (mod N )) . In particular, if χ : Gal ( K/ Q ) → C ∗ is a character, then χ ([ K/ Q , p ]) = ψ ( p ) for some Dirichlet character ψ .More generally, if K/L is any abelian extension of number fields, and χ : Gal ( K/L ) → C ∗ is a character, then there exists a Grossencharacter ψ so thatfor primes p relatively prime to the discriminant of K , we have χ ([ K/L, p ]) = ψ ( p ) . From this theorem, we obtain the following corollaries:
Corollary 4.
Let K/ Q be a Galois field extension. Let L ⊂ K be a subfield sothat Gal ( K/L ) is abelian. Let χ : Gal ( K/L ) → C ∗ be a character, correspondingas described in Theorem 3 to a Grossencharacter ψ on L . Then there exists aDirichlet character ρ so that ψ ( p ) = ρ ( N L/ Q ( p )) for all primes p if and only if χ can be extended to a character on Gal ( K/ Q ) ab .Proof. Let G = Gal( K/ Q ). First we claim that if a prime p of L has norm N L/ Q ( p ) = p n for some rational prime p , then [ K/L, p ] is conjugate to [ K/ Q , p ] n .This is because if the prime q of K sits over p , then it also sits over p . If theelement g ∈ G fixes q and acts via p -power Frobenius on O K / q , then g is in theconjugacy class of [ K/ Q , p ]. On the other hand, g n is the unique element of G that fixes q and acts on the residue field by p n power Frobenius. Since p hasresidue field F p n , this means that g n is in the conjugacy class of [ K/L, p ].Suppose that χ extends to a character of G ab , and thus to G . Letting K ab be the maximal abelian subextension of K over Q , χ gives a characterof Gal( K ab / Q ). Thus by Theorem 3, there is a Dirichlet character ψ so that ψ ( p ) = χ ([ K ab / Q , p ]) = χ ([ K/ Q , p ]). We claim that for primes p of L that χ ([ K/L, p ]) = ψ ( N L/ Q ( p )). This is because if N L/ Q ( p ) = p n then, by the above, χ ([ K/L, p ]) = χ ([ K/ Q , p ] n ) = χ ([ K/ Q , p ]) n = ψ ( p ) n = ψ ( p n ) = ψ ( N L/ Q ( p )) . Next assume that χ is a character on Gal( K/L ) and ψ is a Dirichlet characterso that χ ([ K/L, p ]) = ψ ( N L/ Q ( p )) for all p . Let M be the abelian extension of Q corresponding via the correspondence in Theorem 3 to the kernel of ψ . Let K ′ bethe compositum of M and K . Let G ′ = Gal( K ′ / Q ). Let H = Gal( K ′ /K ) ⊂ G ′ .7y Theorem 3, ψ corresponds to a character χ ′ on Gal( M/ Q ), and thus to acharacter on G ′ . If N L/ Q ( p ) = p n , we have that χ ([ K/L, p ]) = ψ ( p n ) = ψ ( p ) n = χ ′ ([ K ′ / Q , p ]) n = χ ′ ([ K ′ / Q , p ] n ) = χ ′ ([ K ′ /L, p ]) . By the Chebotarev Density Theorem, [ K ′ /L, p ] can take any possible value inGal( K ′ /L ). Thus, for all g ∈ Gal( K ′ /L ), we have that χ ( g/H ) = χ ′ ( g ). Thus χ ′ vanishes on H . On the other hand, χ ′ is necessarily injective on Gal( M/ Q ),and since this generates G ′ modulo G , this implies that H is trivial. Therefore,we have that χ ( g ) = χ ′ ( g ) for g ∈ Gal(
K/L ). Thus, χ ′ is an extension of χ to G and thus to G ab . Corollary 5.
Let L/ Q be a number field and χ a Dirichlet character on Q .Then χ ( N L/ Q ( p )) is trivial on primes p of L not dividing the discriminant of Q , if and only if the abelian extension, M , of Q corresponding to the kernel of χ is contained in L .Proof. Let K be the compositum of M and the Galois closure of L . By Theorem3, χ corresponds to some character ψ of Gal( M/ Q ) and thus of Gal( K/ Q ). Let p be a prime of L with N L/ Q ( p ) = p n . As in the proof of Corollary 4, we havethat χ ( N L/ Q ( p )) = χ ( p n ) = χ ( p ) n = ψ ([ K/ Q , p ]) n = ψ ([ K/ Q , p ] n ) = ψ ([ K/L, p ]) . By the Chebotarev Density Theorem, [
K/L, p ] can take on any value in Gal( K/L ),and thus χ vanishes on norms from L if and only if ψ vanishes on this set. On theother hand, by assumption, ψ is injective on Gal( M/ Q ) = Gal( K/ Q ) / Gal(
K/M ).Thus the kernel of ψ is exactly Gal( K/M ), and thus χ vanishes on norms from L if and only if Gal( K/L ) contains Gal(
K/M ), or equivalently if and only if M is contained in L . G and F We begin with a standard definition:
Definition 1.
Let e ( x ) denote the function e ( x ) = e πix . We now define G as the generating function for the set primes p ≤ X with[ K/ Q , p ] = C each weighted by log( p ) . Definition 2.
Suppose that K/ Q is a finite Galois extension with G = Gal ( K/ Q ) ,C a conjugacy class of G , and X a positive real number. We then define thegenerating function G K,C,X ( α ) = X p ≤ X [ K/ Q ,p ]= C log( p ) e ( αp ) . Where the sum is over primes, p , with p ≤ X and [ K/ Q , p ] = C .
8s it is a little awkward to deal with G directly, we would rather work witha related function defined in terms of characters. We first need one auxiliarydefinition: Definition 3.
Let L/ Q be a number field. Let Λ L be the Von Mangoldt functionon ideals of L , Λ L : { Ideals of L } → R defined by Λ L ( a ) = ( log( N ( p )) if a = p n otherwisewhich assigns log( N ( p )) to a power of a prime ideal p , and 0 to ideals that arenot powers of primes. We now define
Definition 4. If L/ Q is a number field, ξ a Grossencharacter of L , and X apositive number, define the function F L,ξ,X ( α ) = X N ( a ) ≤ X Λ L ( a ) ξ ( a ) e ( αN ( a )) . Where the sum above is over ideals a of L with norm at most X . Notice that the sum in the definition of F is determined up to O ( √ X ) bythe terms coming from primes a of prime norm.For both F and G , we will often suppress some of the subscripts when theyare clear from context. We now demonstrate the relationship between F and G . One may expect them to be related since we can write the characteristicfunction of a conjugacy class of G as a linear combination of characters inducedfrom cyclic subgroups. The generating functions for these cyclic subgroups willturn out to give copies of F . Proposition 6.
Let K and C be as above. Pick a c ∈ C . Let L ⊆ K be thefixed field of c . Then we have that G K,C,X ( α ) = | C || G | X χ χ ( c ) F L,χ,X ( α ) ! + O ( √ X ) , (3) where the sum is over characters χ of the subgroup h c i ⊂ G , which, by Theorem3, can be thought of as characters of L .Proof. We begin by considering the sum on the right hand side of Equation (3).It is equal to X χ χ ( c ) F L,χ,X ( α ) = X χ X N ( a ) ≤ X Λ L ( a ) χ ( c ) χ ( a ) e ( αN ( a ))= X N ( a ) ≤ X Λ L ( a ) e ( αN ( a )) X χ χ ( c ) χ ([ K/L, a ])= ord( c ) X N ( a ) ≤ X [ K/L, a ]= c Λ L ( a ) e ( αN ( a )) .
9p to an error of O ( √ X ), we can ignore the contributions from elements whosenorms are powers of primes, because there are O ( √ X/ log( X )) higher powersof primes with norm at most X . Therefore the above equalsord( c ) X N ( p ) ≤ X [ K/L, p ]= cN ( p ) is prime log( N ( p )) e ( αN ( p )) + O ( √ X ) . We need to determine now which primes p ∈ Z are the norm of an ideal p of L with [ K/L, p ] = c , and for such p , how many such p lie over it. Eachsuch p must have only one prime q of K over it and it must be the case that[ K/ Q , q ] = c . Hence the p we wish to find are exactly those that have a prime q lying over them with [ K/ Q , q ] = c . These are exactly the primes p so that[ K/ Q , p ] = C . Hence the term e ( αn ) appears in the above sum if and only if n is a prime p with [ K/ Q , p ] = C . We next need to compute the coefficient of thisterm. The coefficient will be ord( c ) log( p ) times the number of primes p of L over p with [ K/ Q , p ] = c . These primes are in 1-1 correspondence with primes q of K over p with [ K/ Q , q ] = c . Now for such p , there will be | G | ord( c ) primes of K over it, and | G || C | ord( c ) of them will have the correct Artin symbol. Hence thecoefficient of e ( αp ) for such p will be exactly | G || C | log( p ). Therefore the sum onthe right hand side of Equation (3) is | G || C | X p ≤ X [ K/ Q ,p ]= C log( p ) e ( αp ) + O ( √ X ) . Multiplying by | C || G | completes the proof of the Proposition. Here we define some simpler functions meant to approximate F and G . In orderto do so we will need a number of auxiliary definitions: Definition 5.
For p a prime let Λ p ( n ) = ( if p | n − p − else . Λ p can be thought of as a local approximation to the Von Mangoldt function,based only on the residue of n modulo p . Putting these functions together weget Definition 6.
Let z be a positive real. Define a function Λ z by Λ z ( n ) = Y p Let C ( z ) = Y p Definition 8. Let K/ Q be a Galois extension and C ⊂ G = Gal ( K/ Q ) aconjugacy class of the Galois group. Let the image of C in G ab correspond viaTheorem 3 to a coset H of some subgroup of ( Z /D K Z ) ∗ for D K the discriminantof K . We define Λ K,C to be the arithmetic function: Λ K,C ( n ) = ( φ ( D K ) | H | if n ∈ H otherwise . This accounts for the congruence conditions implied by n being a prime withArtin symbol C . Definition 9. Let L be a number field. Let L ′ be its maximal abelian subexten-sion. By Theorem 3, this corresponds to a subgroup H L of ( Z /D L Z ) ∗ for D L the discriminant of L . Let Λ L/ Q ( n ) = ( φ ( D L ) | H L | if n ∈ H L otherwise . L/ Q accounts for the congruence conditions that are implied by being anorm from L down to Q .We are now prepared to define our approximations F ♯ and G ♯ to F and G . Definition 10. For K/ Q Galois, C a conjugacy class in Gal ( K/ Q ) , and z and X positive real numbers, we define the generating function G ♯K,C,X,z ( α ) = | C || G | X n ≤ X Λ K,C ( n )Λ z ( n ) e ( αn ) . We also let G ♭K,C,X,z ( α ) = G K,C,X ( α ) − G ♯K,C,X,z ( α ) . Definition 11. For L/ Q a number field, ξ a Grossencharacter of L , X and z positive numbers, we define the function F ♯L,ξ,X,z ( α ) = (P n ≤ X Λ L/ Q ( n )Λ z ( n ) χ ( n ) e ( αn ) if ξ = χ ◦ N L/ Q for some character χ otherwise . We note that although there may be several Dirichlet characters χ so that ξ = N L/ Q ◦ χ , that the product of χ ( n ) with Λ L/ Q ( n ) is independent of the choice ofsuch a χ by Corollary 5. We also let F ♭L,ξ,X,z ( α ) = F L,ξ,X ( α ) − F ♯L,ξ,X,z ( α ) . Again for these functions we will often suppress some of the subscripts.We claim that F ♯ and G ♯ are good approximations of F and G , and inparticular we will prove that: Theorem 7. Let K/ Q be a finite Galois extension, and let C be a conjugacyclass of Gal ( K/ Q ) . Let A be a positive integer and B a sufficiently large multipleof A . Then if X is a positive number, z = log B ( X ) , and α any real number,then (cid:12)(cid:12)(cid:12) G ♭K,C,X,z ( α ) (cid:12)(cid:12)(cid:12) = O (cid:16) X log − A ( X ) (cid:17) , (4) where the implied constant depends on K, C, A , and B , but not on X or α . Theorem 8. Given L/ Q a number field, and ξ a Grossencharacter of L , let A be a positive number and B a sufficiently large multiple of A . Then if X is apositive number, z = log B ( X ) , and α any real number, then (cid:12)(cid:12)(cid:12) F ♭L,ξ,X,z ( α ) (cid:12)(cid:12)(cid:12) = O (cid:16) X log − A ( X ) (cid:17) , (5) where the implied constant depends on L, ξ, A, and B , but not on X or α . The proofs of these Theorems will be the bulk of Sections 4 and 5.12 .5 Smooth Numbers We also need some results on the distribution of smooth numbers. We beginwith a definition: Definition 12. Let S ( z, Y ) be the number of n ≤ Y so that n | P ( z ) . In otherwords the number of n ≤ Y so that n is squarefree and has no prime factorsbigger than z . We will need the following bound on S ( z, Y ): Lemma 9. If z ≤ log B ( X ) and Y ≤ X , then S ( z, Y ) ≪ Y − / (2 B ) exp (cid:16) O ( p log( X )) (cid:17) . Proof. Notice that Z Yy =0 S ( z, y ) dy = 12 πi Z i ∞ − i ∞ ( s ( s + 1)) − Y p Apply Lemma 9 with X = Y . 13 Approximation of F In this Section, we will prove Theorem 8.In order to prove Theorem 8, we will split into cases based upon whether α is well approximated by a rational number of small denominator. If it is (thesmooth case), we proceed to use the theory of L -functions to approximate F . If α is not well approximated (the rough case), we generalize results on exponentialsums over primes to show that | F | is small. In either case, F ♯ is not difficult toapproximate. We note that the use of the word “smooth” here has nothing todo with the concept of smooth numbers discussed in the previous section, andis merely an unfortunate coincidence of terminology.We note that by Dirichlet’s approximation Theorem, we can always find apair ( a, q ) with a and q relatively prime and q < M = Θ( X log − B ( X )) with (cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12) ≤ qM . We consider the smooth case to be the one where q ≤ z =log B ( X ). As we will often be concerned with whether a real number is wellapproximated by a rational number of given denominator, we make the followingdefinition: Definition 13. We say that a real number α has a rational approximation withdenominator q if there exists an integer a relatively prime to q so that (cid:12)(cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12)(cid:12) < q . α Smooth In this Section, we will prove the following Proposition: Proposition 11. Let L be a number field, and ξ a Grossencharacter. If z =log B ( X ) , Y ≤ X and α = aq with a and q relatively prime and q ≤ z , then forsome constant c > (depending only on L , ξ and B ), | F ♭L,ξ,Y,z ( α ) | = O (cid:16) X exp (cid:16) − c p log( X ) (cid:17)(cid:17) . We note that this result can easily be extended to all smooth α . In particularwe have: Corollary 12. Let L and ξ be as above. Let A be a constant, and B a sufficientlylarge multiple of A . Let z = log B ( X ) . Suppose that α = aq + θ with a and q relatively prime, q ≤ z and | θ | ≤ qM (for M = X log − B ( X ) ). Then | F ♭L,ξ,X,z ( α ) | = O ( X log − A ( X )) . Proof (Given Proposition 11). Noting that if F ♭L,ξ,X ( α ) = P n ≤ X a n e ( αn ), then14y Abel summation and Proposition 11, F ♭L,ξ,X ( α ) = X n ≤ X a n e (cid:18) naq (cid:19) e ( nθ )= (1 − e ( θ )) X Y ≤ X F ♭L,ξ,Y (cid:18) aq (cid:19) e ( Y θ ) + F ♭L,ξ,X (cid:18) aq (cid:19) e (( X + 1) θ )= O (cid:16) X − log B ( X ) (cid:17) X Y ≤ X O (cid:16) X log − A − B ( X ) (cid:17) + O (cid:16) X log − A − B ( X ) (cid:17) = O (cid:16) X log − A ( X ) (cid:17) . In order to prove Proposition 11, we will need to separately approximate F and F ♯ . For the former, we will also need to review some basic facts aboutHecke L -functions. L -functions Fix a number field L and a Grossencharacter ξ . We consider Hecke L -functionsof the form L ( ξχ, s ) where χ is a Dirichlet character of modulus q ≤ z = log B ( X )thought of as a Grossencharacter via χ ( a ) = χ ( N L/ Q ( a )). We let d be the degreeof L over Q , and let D L be the discriminant. We let m be the modulus of thecharacter ξ , and q the modulus of χ . We note that ξχ has modulus at most q m . Therefore by [3], in the paragraph above Theorem 5.35, L ( ξχ ) has analyticconductor q ≤ d | d L | N ( m ) q d , and by Theorem 5.35 of [3], for some constant c depending only on L , L ( ξχ, s ) has no zero in the region σ > − cd log( | d L | N ( m ) q d ( | t | + 3))except for possibly one Siegel zero. Note also that L ( ξχ, s ) has a simple pole at s = 1 if ξ = ¯ χ , and otherwise is holomorphic. Noting that − L ′ ( ξχ, s ) L ( ξχ, s ) = X a Λ L ( a ) ξχ ( a ) N ( a ) − s , and that the n − s coefficient of the above is at most d log( n ), we may applyTheorem 5.13 of [3] and obtain for a suitable constant c > X N ( a ) ≤ Y Λ L ( a ) ξ ( a ) χ ( a ) = (6) rY − Y β β + O (cid:18) Y exp (cid:18) − c log Y √ log Y + 3 log( q d ) + O (1) (cid:19) (log( Y q d ) + O (1)) (cid:19) , Y β β should be taken with β the Siegel zero if it exists; r = 0unless ξχ = 1, in which case, r = 1; and the implied constants may depend on L , and ξ but not on χ or Y .In order to make use of Equation 6, we will need to prove bounds on the sizeof Siegel zeroes. In particular we show that: Lemma 13. For all L and ξ , and all ǫ > , there exists a c ( ǫ ) > so that forevery Dirichlet character χ of modulus q and every Siegel zero β of L ( ξχ, s ) , β > − c ( ǫ ) q ǫ . Proof. We follow the proof of Theorem 5.28 part 2 from [3], and note the placeswhere we differ. We note that Theorem 5.35 states that we only need by con-cerned with the case when ξχ is totally real. We then consider two such χ having Siegel zeros. We use, L ( s ) = ζ L ( s ) L ( ξχ , s ) L ( ξχ , s ) L ( ξ χ χ ), whichhas conductor O ( q q ) d instead of the analogous one from [3]. This gives us aconvexity bound on the integral term of O (( q q ) d x − β ), instead of the onelisted. Again assuming that β > / 4, we take x > c ( q q ) d . We noticethat we still have (5.64) for σ > − /d + ǫ (for any ǫ > 0) by noting that | P N ( a ) ≤ x ξχ ( a ) | = O ( x − /d + max( x, q )). Therefore, Equation (5.75) of [3]becomes L ( ξχ , ≫ (1 − β )( q q ) − d (1 − β ) (log( q q )) − . The rest of the argument from [3] carries over more or less directly. F We prove Proposition 14. With L, ξ, χ, Y, r as above, X ≥ Y and z = log B ( X ) , F L,ξχ,Y (0) = rY + O (cid:16) X exp( − c p log( X )) (cid:17) . (7) Where again c depends on L, ξ but not χ, X, Y .Proof. Applying Lemma 13 with ǫ = 1 − / (2 B ) to Equation 6, we get that X N ( a ) ≤ Y Λ L ( a ) ξ ( a ) χ ( a ) = rY − Y β β + O (cid:16) X exp( − c p log( X )) (cid:17) = rY + O (cid:16) Y exp( − c ( ǫ ) p log( Y ) (cid:17) + O (cid:16) X exp( − c p log( X )) (cid:17) = rY + O (cid:16) X exp( − c p log( X )) (cid:17) . .1.3 Approximation of F ♯ Proposition 15. With L, ξ, χ, X, Y, r as above, z = log B ( X ) , F ♯L,ξχ,Y (0) = rY + O (cid:16) X exp( − c p log( X )) (cid:17) . Proof. If ξχ is not of the form χ ′ ◦ N L/ Q , then F ♯ = 0 and we are done. Otherwiselet ξχ be as above with χ ′ a character of modulus q ′ . We have that F ♯L,χ ′ ,Y (0) = X n ≤ Y Λ L/ Q ( n )Λ z ( n ) χ ′ ( n )= C ( z ) X n ≤ Y X d | ( P ( z,q ′ D L ) ,n ) µ ( d )Λ L/ Q ( n ) χ ′ ( n )= C ( z ) X d | P ( z,q ′ D L ) X n = dm ≤ Y µ ( d )Λ L/ Q ( n ) χ ′ ( n )= C ( z ) X d | P ( z,q ′ D L ) µ ( d ) χ ′ ( d ) X m ≤ Y/d Λ L/ Q ( dm ) χ ′ ( m ) . Consider for a moment the inner sum over m . It is periodic with period q ′ D L .Note that the sum over a period is 0 unless χ ′ is trivial on H L , in which casethe average value is χ ′ ( d ) φ ( q ′ D L ) q ′ D L . Since r = 1 if χ ′ vanishes on H L and r = 0otherwise, we have that: F ♯L,χ ′ ,Y (0) = C ( z ) (cid:18) φ ( q ′ D L ) q ′ D L (cid:19) X d | P ( z,qD L ) d ≤ Y (cid:18) rµ ( d ) Yd + O ( q ′ D L ) (cid:19) . The sum of error term here is at most O ( C ( z ) qS ( z, Y )) which by Lemma 9 is O (cid:16) Y − / (2 B ) log ( z ) q exp( O ( p log( X ))) (cid:17) . The remaining term is rY C ( z ) φ ( q ′ D L ) q ′ D L X d | P ( z,q ′ D L ) d ≤ Y µ ( d ) d . The error introduced by extending the sum to all d | P ( z, q ′ D L ) is at most O (cid:18) Y C ( z ) Z ∞ Y S ( z, y ) y − dy (cid:19) . By Lemma 9 this is O (cid:16) Y − / (2 B ) log( z ) exp( O ( p log( X ))) (cid:17) . rY C ( z ) φ ( q ′ D L ) q ′ D L X d | P ( z,q ′ D L ) µ ( d ) d = rY C ( z ) (cid:18) φ ( q ′ D L ) q ′ D L (cid:19) (cid:18) φ ( P ( z, q ′ D L )) P ( z, q ′ D L ) (cid:19) = rY C ( z ) (cid:18) φ ( P ( z )) P ( z ) (cid:19) = rY. Hence F ♯L,ξχ,Y,z (0) = rY + O (cid:16) Y − / (2 B ) log ( z ) q exp( O ( p log( X ))) (cid:17) = rY + O (cid:16) X exp( − c p log( X )) (cid:17) . Proof. Combining Propositions 14 and 15 we obtain that F ♭L,ξχ,Y,z (0) = O (cid:16) X exp( − c p log( X )) (cid:17) . Our Proposition follows immediately after noting that F ♭L,ξ,X,z (cid:18) aq (cid:19) = 1 φ ( q ) X χ mod q G ( ¯ χ, a/q ) F ♭L,ξχ,X,z (0) . Where G ( ¯ χ, a/q ) is the Gauss sum G ( ¯ χ, a/q ) = X x (mod q ) ¯ χ ( x ) e ( ax/q ) . α Rough In this Section, we will show that | F ♭ ( α ) | is small for α not well approximated bya rational of small denominator. We will do this by showing that both | F ( α ) | and | F ♯ ( α ) | are small. The proof of the latter will resemble the proof of Proposition15. The proof of the former will require some machinery including some Lemmasabout rational approximations and exponential sums of polynomials. F ♯ Proposition 16. Fix L a number field, and ξ a Grossencharacter. Fix B andlet z = log B ( X ) . Let α be a real number. If α has a rational approximationwith denominator q , then | F ♯L,ξ,z ( α ) | = O (cid:16) X log( X ) log( z ) q − + q log( q ) log( z ) + X − / (4 B ) exp( O ( p log( X )) (cid:17) , where the implied constant may depend on L and ξ but nothing else. roof. We note that the result is trivial unless ξ = N L/ Q ( χ ) for some Dirichletcharacter χ of modulus Q . Hence we may assume that F ♯L,ξ,z ( α ) = X n ≤ X Λ L/ Q ( n )Λ z ( n ) χ ( n ) e ( αn ) . Let D L be the discriminant of L . We note that F ♯L,ξ,z ( α ) = X n ≤ X Λ L/ Q ( n )Λ z ( n ) χ ( n ) e ( αn )= C ( z ) X n ≤ X X d | ( n,P ( z,QD L )) µ ( d )Λ L/ Q ( n ) χ ( n ) e ( αn )= C ( z ) X d | P ( z,QD L ) µ ( d ) χ ( d ) X md = n ≤ X Λ L/ Q ( dm ) χ ( m ) e ( αdm )= O C ( z ) X d | P ( z,QD L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ X/d Λ L/ Q ( dm ) χ ( m ) e ( αdm ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In order to analyze the last sum, we split it up based on the residue class of m modulo QD L . Each new sum is a geometric series with ratio of terms e ( αQD L d ).Hence we can bound this sum as min (cid:16) Xd , QD L || dQD L α || (cid:17) , where || x || is the distancefrom x to the nearest integer. Therefore we have that | F ♯χ,z ( α ) | = O C ( z ) X d ≤ X − / (4 B ) min (cid:18) Xd , QD L || dQD L α || (cid:19) + C ( z ) X / B S ( z, X ) . We bound the sum in the first term by looking at what happens as d rangesover an interval of length q QD L . We get that dQD L α = x + kQα for x the value at the beginning of the interval and k an integer at most q QD L .Notice that kQD L α is within q of kQD L aq , which must be distinct for differentvalues of k . Hence none of the fractional parts of dQD L α can be within q of each other. Hence the sum over this range of d is at most Xd + QD L / (3 q ) + QD L / q ) + . . . = O (cid:0) Xd + 3 qQD L log( q ) (cid:1) . Furthermore the Xd term does not showup in the first such interval, since when d = 0, dQD L α is an integer. We have3 QD L X − / (4 B ) /q + 1 of these intervals. Therefore, the first term is at most C ( z ) times O (cid:18) X ( q/ (3 QD L )) + X q/ (3 QD L )) + . . . + 9 Q D L log( q ) X − / (4 B ) + 3 qQD L log( q ) (cid:19) = O (cid:16) X log( X ) q − + log( q ) X − / (4 B ) + q log( q ) (cid:17) . The other term is bounded by Lemma 9 as O (cid:16) log( z ) X − / (4 B ) exp (cid:16) O ( p log( X )) (cid:17)(cid:17) . | F ♯L,ξ,z ( α ) | = O (cid:16) X log( X ) log( z ) q − + q log( q ) log( z ) + X − / (4 B ) exp( O ( p log( X )) (cid:17) . In the coming Sections, we will need some results on rational approximation ofnumbers. In particular, we will need to know how often multiples of a given α have a good rational approximation. We have the following Lemmas. Lemma 17. Let X, Y, A be positive integers. Let α be a real number with ratio-nal approximation of denominator q . Suppose that for some B , that XY B − >q > B . Then for all but O (cid:0) Y (cid:0) A / B − / + A B − + log( AY ) A X − (cid:1)(cid:1) of theintegers n with ≤ n ≤ Y , nα has a rational approximation with denominator q ′ for any XA − > q ′ > A .Proof. By Dirichlet’s approximation theorem, nα always has a rational approx-imation aq ′ with q ′ < XA − and (cid:12)(cid:12)(cid:12)(cid:12) nα − aq ′ (cid:12)(cid:12)(cid:12)(cid:12) < q ′ XA − . Therefore, nα lacks an appropriate rational approximation only when the abovehas a solution for some q ′ ≤ A . If such is the case then, dividing by n , we findthat α is within ( q ′ ) − n − X − A of some rational number of denominator d sothat d | nq ′ . Note that this error is at most max( n, d ) − X − A .Given such a rational approximation to α with denominator d , we claim thatit contributes to at most Y A d − bad n ’s. This is because there are at most A values of q ′ , and for each value of q ′ , we still need that n is a multiple of d ( d,q ′ ) ≥ dA − . Hence for each q ′ , there are at most Y Ad − bad n . Since thereare at most A values of q ′ , we have at most Y A d − bad n .Next, we pick an integer n . We will now consider only Y ≥ n ≥ n sothat αn has no suitable rational approximation. We do this by analyzing thedenominators d for which some rational number of denominator d approximates α to within X − A (max( d, n )) − . Suppose that we have some d = q which doesthis. α is within q − of a number with denominator q , and within X − n − A ofone with denominator d . These two rational numbers differ by at least ( dq ) − and therefore, ( dq ) − ≤ q − + X − An − . Hence, either dq − or X − An − dq is at least . Hence, either d ≥ q , or d ≥ Xn Aq ≥ n B AY . Therefore, the smallest such d is at least the minimum of q and n B AY .20ext, suppose that we have two different such denominators, say d and d ′ .The fractions they represent are separated by at least ( dd ′ ) − and yet are bothclose to α . Therefore, ( dd ′ ) − ≤ X − A ( d − + d ′− ) . Therefore, we have that max( d, d ′ ) ≥ X A . Hence, there is at most one suchdenominator less than X A .Next, we wish to bound the number of such denominators d in a dyadic inter-val [ K, K ]. We note that the corresponding fractions are all within X − AK − of α , and that any two are separated from each other by at least (2 K ) − . There-fore, the number of such d is at most 1 + 8 KX − A .To summarize we potentially have the following d each giving at most Y A d − bad n ’s. • One d at least min (cid:0) q , n B AY (cid:1) . • For each diadic interval [ K, K ] with K ≥ X A at most 10 KX − A such d ’sNotice that there are log(2 AY ) such diadic intervals, and that each contributesat most 10 Y A X − bad n ’s. We also potentially have n bad n ’s from thenumbers less than n . Hence the number of n for which there is no suitablerational approximation of nα is at most O (cid:0) n + Y A B − + Y A B − n − + log( AY ) Y A X − (cid:1) . Substituting n = Y A / B − / yields our result.We will also need the following related Lemma: Lemma 18. Let X, A, C be positive integers. Let α be a real number withrational approximation of denominator q . Suppose that for some B > A , that XB − > q > B . Then there exists a set S of natural numbers so that • elements of S are of size at least Ω( BA − ) . • The sum of the reciprocals of the elements of S is O ( A B − + X − A C ) . • for all positive integers n ≤ C , either n is a multiple of some elementof S or nα has a rational approximation with some denominator q ′ with XA − n − > q ′ > A .Proof. We use the same basic techniques as the proof of Lemma 17.We begin by letting S be the set of all integers of the form dD for someintegers d, D with A ≥ D , D | d , d ≤ AC and (cid:12)(cid:12)(cid:12) α − ad (cid:12)(cid:12)(cid:12) ≤ XA − , for some integer a relatively prime to d .21e begin by verifying the third claim for this set S . Note that nα always hasa rational approximation aq ′ accurate to within q ′ XA − n − with q ′ < XA − n − .This means that we have an appropriate rational approximation of nα unlessthis q ′ is less than A . If this happens, it is the case that (cid:12)(cid:12)(cid:12)(cid:12) α − anq ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ q ′ XA − ≤ XA − . Letting d = nq ′ / gcd( a, nq ′ ) ≤ AC and D = q ′ / gcd( a, q ′ ), we see that n is amultiple of dD , which is in S since (cid:12)(cid:12)(cid:12)(cid:12) α − a/ gcd( a, nq ′ ) d (cid:12)(cid:12)(cid:12)(cid:12) ≤ XA − . To verify the first property, we note that if we have integers a and d , with d not a multiple of q , so that (cid:12)(cid:12)(cid:12) α − ad (cid:12)(cid:12)(cid:12) ≤ XA − , then α is within q − of a rational number of denominator q and within X − A of one of denominator d . Hence,( dq ) − ≤ q − + X − A. Therefore, d ≥ min (cid:18) q , X A (cid:19) ≥ q . Therefore, every element of S is of the form dD with d ≥ q ≥ B and D ≤ A .Thus every such element is Ω( BA − ).Finally, we verify the second property. To each element dD of S , we mayassociate the rational number ad so that (cid:12)(cid:12)(cid:12) α − ad (cid:12)(cid:12)(cid:12) ≤ XA − . The sum of the reciprocals of elements of S associated to this fraction is at most P D ≤ A ( d/D ) − = O ( A d − ). Given two such approximations, ad and a ′ d ′ , theymust differ by at most XA − , and thus( dd ′ ) − ≤ XA − . Therefore the second largest such d is at least p XA − / . Next we consider the number of such approximations with d lying in a diadicinterval [ K, K ]. All of these approximations are within X − A of α and are sep-arated from each other by at least K . Therefore, taking K > p XA − / 2, thenumber of such approximations is O ( X − AK ), so the contribution they make22o the sum of the reciprocals of the elements of S is at most O ( X − A K ). Sum-ming this over K a power of 2 of size at most AC , yields a total contribution of O ( X − A C ). Thus, the sum of the reciprocals of elements of S correspondingto all of the appropriate rational approximations except for the one of mini-mal denominator is at most O ( X − A C ). The contribution coming from theapproximation with minimal denominator consists of the sum of reciprocals of O ( A ) terms each of size Ω( B/A ), and is thus O ( A ( B/A ) − ) = O ( A B − ). Com-bining this with the contribution from the other rational approximations yieldsour result.We will be using Lemma 18 to bound the number of ideals of L so that N ( a ) α has a good rational approximation. In order to do this we will also needthe following: Lemma 19. Fix L be a number field. Let n be a positive integer, and let X and ǫ be positive real numbers. Then for any ǫ > , we have that: X n | N ( a ) N ( a ) Pick a positive integer X . Let [ X ] = { , , . . . , X } . Let P be apolynomial with leading term cx k for some integer c = 0 . Let α be a real numberwith a rational approximation of denominator q . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ [ X ] e ( αP ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ | c | X (cid:18) q + 1 X + qX k (cid:19) − k , where the implied constant depends on k , but not on the coefficients of P . Note that the 10 − k in the exponent is not optimal and was picked for con-venience. Proof. We proceed by induction on k . We take as a base case k = 1. Thenwe have that P is a linear function with linear term c . α is within q − of arational number of denominator q . Therefore cα is within cq − of a numberof denominator between qc − and q . If c ≥ q/ 2, there is nothing to prove.Otherwise, cα cannot be within q − − cq − = O ( q − ) of an integer. Thereforethe sum is at most O (min( X, q )), which clearly satisfies the desired inequality.We now perform the induction step. We assume our inequality holds forpolynomials of smaller degree. Squaring the left hand side of our inequality, wefind that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ [ X ] e ( αP ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X a,b ∈ [ X ] e ( α ( P ( a ) − P ( b ))) / . n = a − b , we note that P ( n + b ) − P ( b ) is a polynomial in b of degree k − nckx k − .Letting [ X n ] = { , , . . . , X } ∩ { − n, − n, . . . , X − n } , we are left with at most X n ∈ [ − X,X ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X b ∈ [ X n ] e ( α ( P ( b + n ) − P ( b ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / = X n ∈ [ − X,X ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X b ∈ [ X n ] e (cid:18) ( nα ) (cid:18) P ( b + n ) − P ( b ) n (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / . Let B = min( q, X k /q ). We consider separately the terms in the above sumwhere nα has no rational approximation with denominator between B / and X k − B − / . By Lemma 17 with parameters A = B / , B = B, Y = X, X = X k − , the number of such n is at most O ( X ( B − / + log( X ) B / X − k )). Eachof those terms contributes O ( X ) to the sum and hence together they contributeat most O ( X ( B − / + log( X ) B / X (1 − k ) / )) . Which is within the required bounds.For the other terms, the inductive hypothesis tells us that the sum for fixed n is at most O | c | ( X − n ) (cid:18) B − / + 1 X − | n | + X k − B − / ( X − | n | ) k − (cid:19) − k − . Summing over n and taking a square root gives an appropriate bound.We apply this Lemma to get a bound on exponential sums of norms of idealsof a number field. In particular we show that: Lemma 21. Fix L a number field of degree d , and ξ a Grossencharacter ofmodulus m . Then given a positive number X and a real number α which has arational approximation of denominator q , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N ( a ) ≤ X ξ ( a ) e ( αN ( a )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O X (cid:18) q + 1 X /d + qX (cid:19) − d / ! . (8) Where the implied constant depends only on L and ξ .Proof. We begin by dividing the sum in question into pieces based on the classof a modulo multiplication by elements of O L congruent to 1 modulo m . It iswell known that there are only finitely many such classes, and thus it sufficesto show that for any such class, the sum over a in that class is bounded by theright hand side of Equation (8). We will henceforth proceed to bound the sumover a in one such class. 25ick a representative ideal a of the class in question. Let L m denote theset of elements of L congruent to 1 modulo m . Every ideal in our class can bewritten in the form b a for some b ∈ L m ∩ a − (where a − is the appropriatefractional ideal). Furthermore, this representation is unique up to multiplying b by an element of O ∗ L ∩ L m . We have that ξ ( b a ) = ξ ( b ) ξ ( a ). Since ξ hasmodulus m and since b ∈ L m , ξ ( b ) = ψ ( b ) for ψ some continuous character ψ :( L ⊗ R ) ∗ → C ∗ , with ψ ( O ∗ L ∩ L m ) = 1. Additionally, we have that N L/ Q ( b a ) = | N L/ Q ( b ) | N L/ Q ( a ).We simplify the sum in question by considering the geometry of the set of b ’sin question. In particular, we note that T := L m ∩ a − is a translate of a latticein L ⊗ R . Furthermore N L/ Q ( b ) is easily seen to be a degree d polynomial withrational coefficients on this lattice. The sum that we wish to take is over all b inthis lattice with norm at most X/N L/ Q ( a ) in some fundamental domain of theaction of O ∗ L ∩ L m . We can obtain such a region by letting D be a fundamentaldomain of O ∗ L ∩ L m within the set of unit norm elements of ( L ⊗ R ) ∗ . Wecan then take our sum to be over all b in R := D · (0 , ( X/N L/ Q ( a )) /d ]. ByDirichlet’s Unit Theorem, D can be taken to be a bounded region with finitevolume and surface area and finitely many connected components within theunit norm elements of ( L ⊗ R ) ∗ (which by taking logarithms is isomorphic to atorus times some number of copies of R , giving us notions of volume and surfacearea). For such D , it is easy to see that R produces a region in L ⊗ R withvolume Θ( X ) and surface area O ( X − /d ) (where the implied constant dependson our choice of D ) and finitely many connected components. Summarizing theabove, we find that the expression that we need to bound is ξ ( a ) X b ∈ T ∩ R ψ ( b ) e ( α | P ( b ) | ) , where P is some polynomial of degree d on T with rational coefficients. It shouldbe noted that P ( b ) will have constant sign on connected components of R (since P extends to a non-zero, continuous function on R ). Thus, by restricting oursum to a single connected component of R , we may ignore the absolute value of P taken above.In order to reduce the above sum to something that can be handled withLemma 20, we need to reduce to the case where we are summing e ( αp ( x )) for p some polynomial in one variable with integer leading term. In order to do this,we pick some non-zero vector v under which T is translation invariant. Thenfor any b ∈ T , P ( b + nv ) is a degree- d polynomial in n whose rational, leadingcoefficient does not depend on b . Perhaps replacing v by a positive multiple ofitself, we may assume that this leading coefficient is a non-zero integer. Fix aninteger Y = Θ( X / (2 d ) ). Define a line in T to be a subset of T of the form { b + v, b + 2 v, . . . , b + Y v } for some b ∈ T . Each element of T is contained inexactly Y lines, and thus the sum in question can be written as ξ ( a ) Y X lines N X b ∈ N ∩ R ψ ( b ) e ( αP ( b )) . 26e break the above sum into cases based upon whether or not N is containedin R . If N is neither contained in R nor disjoint from R , then it must be definedby some b ∈ T so that b + xv ∈ ∂R for some x ∈ [0 , Y ]. This means that b must lie within distance O ( Y ) of the boundary of R . Extending a fundamentaldomain of T around each such b , we find that their union is contained in a set ofvolume O ( X − /d Y ), and thus there are at most O ( X − /d Y ) such lines. Eachsuch line contributes O (1) to the above sum, thus the total contribution to theabove sum coming from lines N not contained in R is O ( X − /d Y ), which iswithin the desired bounds.Next consider the contribution coming from a particular line N contained in R . We note that each of these points corresponds to a b + v ∈ T ∩ R , and thusthat there are at most O ( X ) of these lines. Let N = { b + v, . . . , b + Y v } . Thesum in question over this line reduces to Y X n =1 ψ ( b + nv ) e ( αP ( b + nv )) = ψ ( b + v ) Y X n =1 ψ (1 + ( n − v ( b + v ) − ) e ( αp b ( n )) , where p b ( n ) is a degree d polynomial in n with coefficients dependent on b , butwhose leading term is integral and does not depend on b . Since ψ is continuous(and thus smooth) we may write ψ (1 + nvb − ) = 1 + O ( Y | ( b + v ) − | ), where | ( b + v ) − | is the maximum of the absolute value of ( b + v ) − at any of theinfinite places (where for complex places, we use the standard absolute valuerather than its square). Thus the absolute value of the sum in question is O (min( Y | ( b + v ) − | , Y )) + Y X n =1 e ( αp b ( n )) . By Lemma 20, the latter term above is O Y (cid:18) q + 1 Y + qY d (cid:19) − − d ! . Summing this latter term over all lines contained in R , gives a contribution toour final sum of size O X (cid:18) q + 1 Y + qY d (cid:19) − − d ! , which is of the appropriate size.We have left to bound the sum over lines N contained in R of O (min( Y | ( b + v ) − | , Y )). This is at most the sum over elements a ∈ T ∩ R of O (min( Y | a − | , Y )).This in turn is Y Z Y − { a ∈ T ∩ R : | a − | > s } ds. (9)We note that an element a has | a − | > s if and only if a has absolute valueat most s − at some infinite place. Furthermore, by construction, if a is in R ,27t must have absolute value at most O ( X /d ) at each real place. Let M νs bethe set of a ∈ R ∩ T so that | a | ν ≤ s − for some particular infinite place ν .Hence, { a ∈ T ∩ R : | a − | > s } ≤ P | M νs | . Pick a fundamental domain for T . Let ˜ M νs be the union of translates of this fundamental domain by elementsof M νs . It is clear that | M νs | = O (Vol( ˜ M νs )). On the other hand, for a ∈ ˜ M νs , | a | ν ≤ s − + O (1) and | a | µ = O ( X /d ) for other infinite places µ . Thus,Vol( ˜ M νs ) = O ( X − /d (1 + s − )) . Hence for any s we have that { a ∈ T ∩ R : | a − | > s } = O (min( X, X − /d s − )) . Thus the quantity in Equation (9) is at most Y Z Y − O (min( X, X − /d s − )) ds = O ( Y X − /d log( X /d /Y ))= O ( Y X − /d log( X )) . Thus the total contribution from these terms to our final sum is O ( Y X − /d log( X )) , which is within our desired bounds.Applying Abel summation and Lemma 21 yields the following Corollary. Corollary 22. Fix L a number field of degree d , and ξ a Grossencharacter ofmodulus m . Then given a positive number X and a real number α which has arational approximation of denominator q , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N ( a ) ≤ X log( N ( a )) ξ ( a ) e ( αN ( a )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O X log( X ) (cid:18) q + 1 X /d + qX (cid:19) − d / ! . F We are finally ready to prove our bound on F . Proposition 23. Fix a number field L of degree d and a Grossencharacter ξ .Let X ≥ be a real number. Let α be a real number with a rational approxima-tion of denominator q where XB − > q > B for some B > . Then F L,ξ,X ( α ) is O (cid:16) X (cid:16) log ( X ) B − − d / + log ( X ) X − − d / + log ( X ) X − − d / d + log d / ( X ) B − / (cid:17)(cid:17) . Where the asymptotic constant may depend on L and ξ , but not on X, q, B or α . Note that a bound for F L,ξ,X ( α ) is already known for the case when L/ Q is abelian. In [1], bounds are established for exponential sums over primes inan arithmetic progression. By Theorem 3, this is equivalent to proving boundson F (or more precisely, G ) when L is abelian over Q . Proposition 23 can bethought of as a generalization of this result.28 roof. Our proof is along the same lines as Theorem 13.6 of [3]. We first notethat the suitable generalization of Equation (13.39) of [3] still applies. Letting y = z = X / ( y and z are variables used in (13.39) of [3]), we find that F L,ξ,X ( α ) equals X N ( ab ) ≤ XN ( a ) Proof. We note that α can always be approximated by aq for some relativelyprime integers a, q with q ≤ X log − B ( X ) so that (cid:12)(cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12)(cid:12) ≤ qX log − B ( X ) . We split into cases based upon weather q ≤ z = log B ( X ).If q ≤ z our result follows from Corollary 12.If q ≥ z , our result follows from Propositions 16 and 23.31 Approximation of G In this Section, we prove Theorem 7. Proof. Recall Proposition 6 which states that G K,C,X ( α ) = | C || G | X χ χ ( c ) F L,χ,X ( α ) ! + O ( √ X ) . Where c is some element of C , and L is the fixed field of h c i ⊂ Gal( K/ Q ).Applying Theorem 8, this is within O (cid:16) X log − A ( X ) (cid:17) of | C || G | X χ χ ( c ) F ♯L,χ,X,z ( α ) ! = | C || G | X χ X n ≤ X χ ( c )Λ L/ Q ( n )Λ z ( n ) χ ( n ) e ( αn ) = | C || G | X n ≤ X Λ z ( n ) e ( αn ) Λ L/ Q ( n ) X χ χ ( c ) χ ( n ) ! . Note that in the above, χ is summed over characters of Gal( K/L ) and that χ ( n )is taken to be 0 unless χ can be extended to a character of Gal( K/ Q ) ab . Wewish the evaluate the inner sum over χ for some n ∈ H L .Let the kernel of the map h c i → Gal( K/ Q ) ab be generated by c k for some k | ord( c ). Then χ ( n ) is 0 unless χ ( c k ) = 1. Therefore we can consider the sumas being over characters χ of h c i /c k . Taking K a to be the maximal abeliansubextension of K over Q , this sum is then k if [ K a / Q , n ] = c and 0 otherwise.Hence the sum over χ is non-zero if and only if n ∈ H C . The index of H C in H L is [ H L : H K ], which is in turn the size of the image of h c i in Gal( K/ Q ) ab ,or |h c i / h c k i| = k . Hence Λ L ( n ) P χ χ ( c ) χ ( n ) = Λ K,C ( n ). Therefore G K,C,X ( α )is within O (cid:16) X log − A ( X ) (cid:17) of | C || G | X n ≤ X Λ K,C ( n )Λ z ( n ) e ( αn ) = G ♯K,C,X,z ( α ) . We now have all the tools necessary to prove Theorem 2. Our basic strategywill be as follows. We first define a generating function H for the number ofways to write n as P i a i p i for p i primes satisfying the appropriate conditions.It is easy to write H in terms of the function G . First, we will show that if H isreplaced by H ♯ by replacing these G ’s by G ♯ ’s, this will introduce only a smallchange (in an appropriate norm). Dealing with H ♯ will prove noticeably simplerthan dealing with H directly. We will essentially be able to approximate thecoefficients of H ♯ using sieving techniques. Finally we combine these results toprove the Theorem. 32 .1 Generating Functions We begin with some basic definitions. Definition 14. Let K i , C i , a i , X be as in the statement of Theorem 2. Then wedefine S K i ,C i ,a i ,X ( N ) := X p i ≤ X [ K i / Q ,p i ]= C i P i a i p i = N k Y i =1 log( p i ) . (i.e. the left hand side of Equation (2) ). We define the generating function H K i ,C i ,a i ,X ( α ) := X N S K i ,C i ,a i ,X ( N ) e ( N α ) . Notice that this is everywhere convergent since there are only finitely many non-zero terms. We know from basic facts about generating functions that H K i ,C i ,a i ,X ( α ) = k Y i =1 G K i ,C i ,X ( a i α ) . (10)We would like to approximate the G ’s by corresponding G ♯ ’s. Hence we define Definition 15. H ♯K i ,C i ,a i ,z,X ( α ) := k Y i =1 G ♯K i ,C i ,z,X ( a i α ) .H ♭K i ,C i ,a i ,z,X ( α ) := H K i ,C i ,a i ,X ( α ) − H ♯K i ,C i ,a i ,z,X ( α ) . We now prove that this is a reasonable approximation. Lemma 24. Let A be a constant, and z = log B ( X ) for B a sufficiently largemultiple of A . If k ≥ , (cid:12)(cid:12)(cid:12) H ♭K i ,C i ,a i ,z,X (cid:12)(cid:12)(cid:12) = O ( X k − log − A ( X )) . If k = 2 , (cid:12)(cid:12)(cid:12) H ♭K i ,C i ,a i ,z,X (cid:12)(cid:12)(cid:12) = O ( X / log − A ( X )) . In the above we are taking the L or L norm respectively of H ♭K ,C i ,a i ,X as afunction on [0 , , and the asymptotic constants in the big- O terms are allowedto depend on K i , a i , A and B , but not on X or N . roof. Our basic technique is to write each of the G ’s in Equation 10 as G ♯ + G ♭ and to expand out the resulting product. We are left with a copy of H ♯ anda number of terms which are each a product of k G ♯ or G ♭ ’s, where each suchterm has at least one G ♭ . We need several facts about various norms of the G ♯ and G ♭ ’s. We recall that the squared L norm of a generating function is thesum of the squares of it’s coefficients. • By Theorem 7, the L ∞ -norm of G ♭ is O (cid:16) X log − A − k ( X ) (cid:17) . • The L ∞ norm of G ♯ is clearly O ( X log log( X )). • | G ♯ | = O ( X log log ( X )). • | G | = O ( X log( X )). • Combining the last two statements, we find that | G ♭ | = O ( X log( X )).For k ≥ 3, we note that by Cauchy-Schwartz, the L norm of a product of k functions is at most the products of the L norms of two of them times theproducts of the L ∞ norms of the rest. Using this and ensuring that at least oneof the terms we take the L ∞ norm of is a G ♭ , we obtain our bound on | H ♭ | .For k = 2, we note that the L norm of a product of two functions is atmost the L norm of one times the L ∞ norm of the other. Applying this to ourproduct, ensuring that we take the L ∞ norm of a G ♭ we get the desired boundon | H ♭ | . H ♯ Now that we have shown that H ♯ approximates H , it will be enough to computethe coefficients of H ♯ . Proposition 25. Let z = log B ( X ) for B some positive constant. Pick ǫ > some other constant The e ( N α ) coefficient of H ♯K i ,C i ,a i ,z,X ( α ) is given by theright hand side of Equation (2) , or k Y i =1 | C i || G i | ! C ∞ C D Y p ∤ D C p + O (cid:16) X k − ǫ − / (3 B ) (cid:17) , where the implied constant above depends potentially on k , K i , a i , B and ǫ , butnot on X or N .Proof. We note that the quantity of interest is equal to k Y i =1 | C i || G i | ! X n ,...,n k ≤ X P ki =1 a i n i = N k Y i =1 Λ K i ,C i ( n i ) ! k Y i =1 Λ z ( n i ) ! . (11)34his is k Y i =1 | C i || G i | ! C ( z ) k k Y i =1 φ ( D ) | H i | ! · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ( n , . . . , n k ) ∈ { , , . . . , X } k : n i (mod D ) ∈ H i , ( n i , P ( z )) = 1 , k X i =1 a i n i = N )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thus our problem reduces to computing the size of the set S given by: ( ( n , . . . , n k ) ∈ { , , . . . , X } k : n i (mod D ) ∈ H i , ( n i , P ( z )) = 1 , k X i =1 a i n i = N ) . Our main technique for dealing with this term will be based of sieving. Inparticular, we sieve based on which primes less than z divide any of the n i . For d | P ( z, D ) we define S d = ( ( n , . . . , n k ) ∈ { , , . . . , X } k : n i (mod D ) ∈ H i , d (cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 n i , k X i =1 a i n i = N ) . It follows easily that | S | = X d | P ( z,D ) µ ( d ) | S d | . Thus, it suffices to estimate the sizes of the S d .We note that S d is the set of tuples ( n , . . . , n k ) with n i ≤ X , and P ki =1 a i n i = N so that the vector ( n , . . . , n k ) (mod dD ) lies in some restricted set of con-gruence classes. In particular, let T D := ( ( n , . . . , n k ) ∈ ( Z /D Z ) k : k X i =1 a i n i ≡ N (mod D ) , n i ∈ H i ) , and T p := ( ( n , . . . , n k ) ∈ ( Z /D Z ) k : k X i =1 a i n i ≡ N (mod p ) , p (cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 n i ) . Then the elements of S d are the tuples with n i ≤ X , P ki =1 a i n i = N , and( n , . . . , n k ) ∈ T D and ( n , . . . , n k ) ∈ T p for all p | d . To count the numberof such points, we first condition on their congruence classes modulo dD . Inparticular, by the Chinese Remainder Theorem, a ( n , . . . , n k ) ∈ S d can takeon only | T D | Q p | d | T p | different possible congruence classes modulo dD . Fixingsuch a class, c ∈ ( Z /Dd Z ) k with c (mod D ) ∈ T D and c (mod p ) ∈ T p for p | d ,the set of elements of S d congruent to c are simply those tuples with n i ≤ X ,35 ki =1 a i n i = N and ( n , . . . , n k ) ≡ c (mod dD ) . Therefore, we have that | S d | isthe sum over such c of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ( n , . . . , n k ) ∈ { , . . . , X } k : k X i =1 a i n i = N, ( n , . . . , n k ) ≡ c (mod dD ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (12)Notice that the set of k -tuples of integers n i with P ki =1 a i n i = N and( n , . . . , n k ) ≡ c (mod dD ) is an affine lattice within the space V of tuplesof real numbers x i so that P ki =1 a i x i = N . We induce a measure on V from thestandard measure on R n by putting the measure P ki =1 a i dx i on the quotient.We note that under this measure, C ∞ is the measure of R := [0 , X ] k ∩ V . Thelattice Z k has covolume 1 in R k . Since the a i are relatively prime, the imageof Z k under ( n , . . . , n k ) → P ki =1 a i n i is Z . Thus the projection of Z k to R k /V has covolume 1. Therefore Z k ∩ V has covolume 1 within V . Let L be the affinelattice Z k ∩ V . For c ∈ T d × Q p | d T p , let L c be the sublattice of L consistingof elements congruent to c modulo dD . The covolume of L c is ( dD ) k − timesthe covolume of L , and is thus ( dD ) k − . Notice that the set in Equation (12)is exactly L c ∩ R . We now try to estimate its size.Consider a fundamental domain M of L c . We can construct M so that ithas diameter O ( d ). Take the union of translates of M centered at the elementsof L c ∩ R . It is clear that this produces a set whose symmetric difference with R is contained within the set of points within distance O ( d ) of the boundary of R . It is thus, easy to see that this union has volume Vol( R ) + O ( dX k − + d k − ).On the other hand the volume of this region equals the covolume of L c timesthe number of points in L c ∩ R . Thus, | L c ∩ R | = C ∞ + O ( dX k − + d k − )( dD ) k − . Therefore, | S d | is the sum over T D × Q p | d T p of this quantity, or | S d | = | T D | · Y p | d | T p | ( C ∞ + O ( dX k − ))( dD ) − k . In order to obtain proper control on the error term above, we will want tobound the size of T p . For a tuple ( n , . . . , n k ) ∈ T p at least one of the n j mustbe zero modulo p . Fixing such an j , it must still be the case that P ki =1 a i n i ≡ N (mod p ). Unless a i = 0 for all i = j and N ≡ p ), there are only p k − such solutions. If p | N and p | a i for all i = j , we claim that our propositionholds trivially. In particular, we have that a j is not divisible by p (since the a i are relatively prime). Therefore if we have integers n i with P ki =1 a i n i ≡ N (mod p ), then n j must be divisible by p . Therefore S is empty, and C p is 0, andso both of sides of the equation in question are 0. Hence, we may assume thatthis is not the case and therefore assume that | T p | ≤ kp k − for all p . Thus the36rror term above is at most O Y p | d kp k − ( d − k X k − + 1) = O (cid:0) d ǫ ( X + d ) k − (cid:1) . While the above bound will prove sufficient for d ≪ X , we will need a differ-ent bound for larger values of d . We claim for any d that | S d | = O ( X k − d ǫ − ).This is because for ( n , . . . , n k ) ∈ S d , we must have some d i with d i | n i and d = Q ki =1 d. There are τ k ( d ) = d ǫ ways to pick the d i . For each way of pickingthe d i , the set of points in L with d i | n i for each i forms a lattice of covolume d . If any d i is bigger than X , there is nothing to prove. Otherwise, this latticehas a fundamental domain of diameter O ( X ) and thus extending translates ofthis fundamental domain around each point in the intersection of this latticewith R yields a figure of volume O ( X k − ). Thus, the number of such points is O ( X k − d − ). Thus, | S d | = O ( X k − d ǫ − ).To summarize, we have that for d ≤ X , we have that | S d | = | T D | · Y p | d | T p | C ∞ ( dD ) − k + O ( X k − ǫ ) . And for d ≥ X , we have that | S d | = O ( X k − d ǫ − ) = | T D | · Y p | d | T p | C ∞ ( dD ) − k + O ( X k − d ǫ − ) . Thus, | S | = | T D | D − k C ∞ X d | P ( z,D ) Y p | d −| T p | p k − + X d | P ( z,D ) ,d ≤ X O ( X k − ǫ ) + X d | P ( z,D ) ,d ≥ X O ( X k − d ǫ − ) . We begin by dealing with the error term above. By Corollary 10, it is at most O ( X k − ǫ S ( z, X )) + Z ∞ X O ( X k − ) S ( z, y ) y ǫ − dy = O ( X k − ǫ − / (3 B ) ) + Z ∞ X O ( X k − ) y ǫ − / (3 B ) − dy = O ( X k − ǫ − / (3 B ) ) . Therefore, this term can be safely ignored, and up to acceptable error wemay approximate | S | as | T D | D − k C ∞ X d | P ( z,D ) Y p | d −| T p | p k − = | T D | D k − C ∞ Y p | P ( z,D ) (cid:18) − | T p | p k − (cid:19) . k Y i =1 | C i || G i | ! C ∞ (cid:18) | T D | D k − (cid:19) Y p | D (1 − p − ) − k Y p | P ( z,D ) (cid:18) − | T p | p k − (cid:19) (1 − p − ) − k = k Y i =1 | C i || G i | ! C ∞ (cid:18) D | T D | φ ( D ) k (cid:19) Y p | P ( z,D ) p ( p k − − | T p | )( p − k = k Y i =1 | C i || G i | ! C ∞ C D Y p | P ( z,D ) C p . This completes our proof. We are finally able to prove Theorem 2 Proof. Let B be a sufficiently large multiple of A , and z = log B ( X ).For k ≥ S K i ,C i ,a i ,X ( N ) = Z H K i ,C i ,a i ,X ( α ) e ( − N α ) . By Lemma 24 this is Z H ♯K i ,C i ,a i ,z,X ( α ) e ( − N α )up to acceptable errors. This is the e ( N α ) coefficient of H ♯K i ,C i ,a i ,z,X ( α ), whichby Proposition 25 is as desired.For k = 2, we let T K i ,C i ,a i ,X ( N ) be the corresponding right hand side ofEquation 2. It will suffice to show that X | n |≤ P i | a i | X ( S K i ,C i ,a i ,X ( N ) − T K i ,C i ,a i ,X ( N )) = O ( X log − A ( X )) . If we define the generating function J K i ,C i ,a i ,X ( α ) = X | N |≤ P i | a i | X T K i ,C i ,a i ,X ( N ) e ( N α )we note that the above is equivalent to showing that | H K i ,C i ,a i ,X − J K i ,C i ,a i ,X | = O ( X / log − A ( X )) . But by Lemma 24, we have that | H K i ,C i ,a i ,X − H ♯K i ,C i ,a i ,z,X | = O ( X / log − A ( X )) , and by Proposition 25, we have | H ♯K i ,C i ,a i ,z,X − J K i ,C i ,a i ,X | = O ( X / log − A ( X )) . This completes the proof. 38 Application We present an application of Theorem 2 to the construction of elliptic curveswhose discriminants are divisible only by primes with certain splitting proper-ties. Theorem 26. Let K be a number field. Then there exists an elliptic curvedefined over Q so that all primes dividing its discriminant split completely over K .Proof. We begin by assuming that K is a normal extension of Q . We will choosean elliptic curve of the form: y = X + AX + B. Here we will let A = pq/ B = npq where n is a small integer and p, q areprimes that split over K . The discriminant is then − A + 27 B ) = − p q / − n p q = − p q ( p + 432 n q ) . Hence it suffices to find primes p, q, r that split completely over K with p +432 n q − r = 0. We do this by applying Theorem 2 with k = 3, K i = K , C i = { e } , and X large. As long as C D > C p > p , the main termwill dominate the error and we will be guaranteed solutions for sufficiently large X . If n = D , this will hold. This is because for C D to be non-zero we needto have solutions n + 0 n − n ≡ D ) with n i all in some particularsubgroup of ( Z /D Z ) ∗ . This can clearly be satisfied by n = n . For p = 2, C p is non-zero since there is a solution to n + 0 n − n ≡ n i divisible by 2 (take (1 , , p > 2, we need to show that thereare solutions to n + 432 D n − n ≡ p ) with none of the n i p . This can be done because after picking n , any number can be written as adifference of non-multiples of p . 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