A generalization of Bombieri-Vinogradov theorem and its application
aa r X i v : . [ m a t h . N T ] D ec A GENERALIZATION OF BOMBIERI-VINOGRADOV THEOREMAND ITS APPLICATION
PETER CHO-HO LAM
Abstract.
In this paper we establish a generalization of Bombieri-Vinogradovtheorem for primes represented by a fixed positive definite binary quadraticform. Then we apply this theorem to generalize a result of Vatwani on boundedgap between Gaussian primes. Introduction
Let π ( x ) be the number of primes less than x and π ( x ; q, a ) be the number ofprimes p less than x that satisfies p ≡ a (mod q ). It is widely believed that primesare equdistributed over arithmetic progressions; that is(1.1) π ( x ; q, a ) ∼ π ( x ) φ ( q ) as x → ∞ ,at least when q is not too close to x . For example, the celebrated Siegel-Walfisztheorem states that (1.1) is true when q ≤ (log x ) N where N is a fixed positiveconstant. Assuming GRH, one can prove that (1.1) holds for all q ≤ x / − ǫ forany ǫ >
0. We are very far from proving this unconditionally, but the Bombieri-Vinogradov theorem asserts that this is true on average: for any 0 < θ < / A >
0, we have X q
Mathematics Subject Classification.
Key words and phrases.
Binary Quadratic Forms, Gaussian Primes.
Vatwani mentioned that the same result should hold for all imaginary quadraticfield of class number one (e.g. for Q ( x, y ) = x + 163 y ) with minor changes inthe proof. This is because the corresponding ring of integers would still enjoyunique factorization and are therefore easier to work with. In this paper, we willprove a generalization of Bombieri-Vinogradov theorem that allows us to generalizeTheorem 1 to most primitive positive definite binary quadratic forms: Proposition 1.
Let Q ( x, y ) = ax + bxy + cy be a primitive positive definite binaryquadratic form with discriminant − ∆ such that ∤ ∆ and ∆ is not divisible by anyodd prime squares. Then for any A > and any θ < / , we have X q Theorem 2. Let Q ( x, y ) = ax + bxy + cy be a primitive positive definite binaryquadratic form with discriminant − ∆ such that ∤ ∆ and ∆ is not divisible by anyodd prime squares. Then there is a constant c ( Q ) > such that there are infinitelymany primes of the form p = Q ( m, n ) and p = Q ( m, n + h ) , with m, n, h ∈ Z ,such that < | h | ≤ c ( Q ) . In Section 2, we will establish Proposition 1 using the Bombieri-Vinogradov the-orem for number fields developed by Huxley [2]. The proof of Theorem 2 will beprovided in Section 3.In the following sections we will fix a binary quadratic form Q ( x, y ) = ax + bxy + cy and denote − ∆ < > 4. The proof for ∆ ≤ Bombieri-Vinogradov theorem for Q ( x, y )Let K be a number field with ring of integers O K . Let r be the number of realembeddings of K in C , h K be the class number of K and U be the group of unitsin O K . Eventually we will take K = Q ( − ∆), and since ∆ > U = {± } .To generalize the notion of arithmetic progression, we introduce the notion ofray class group. For an integral ideal q of O K , we consider the group of fractionalideals coprime to q and define an equivalence relation between them: we say that a ∼ b if there exists α, β ∈ O K that satisfies:(1) ( α ) , ( β ) coprime to q ,(2) α − β ∈ q ,(3) α/β ≻ 0, that is, σ ( α/β ) > σ of K , GENERALIZATION OF BOMBIERI-VINOGRADOV THEOREM AND ITS APPLICATION 3 such that ( α ) a = ( β ) b . The equivalence classes form a group, the ray class group (mod q ) and we denoteit by C q . Let h ( q ) denote the cardinality of C q . Then the value of h ( q ) is providedby the formula h ( q ) = 2 r φ ( q ) h K [ U : U q , ]where U q , = { α ∈ U : α ≡ q ) , α ≻ } and φ ( q ) is defined by φ ( q ) = N ( q ) Y p | q (cid:18) − N ( p ) (cid:19) . For the proof see [4]. Furthermore, the proof there also implies that φ ( q ) is thenumber of elements of ( O K / q ) ∗ . In this setting, Huxley [2] proved the followinggeneralization of Bombieri-Vinogradov theorem for number fields: Proposition 2 (Huxley) . Let χ P be the characteristic function for prime idealsand π ( x, K ) be the number of prime ideals with norm less than x . Then for any A > and any θ < / , we have X N ( q ) ≤ x θ h ( q ) φ ( q ) max C q ∈ C q max y ≤ x (cid:12)(cid:12)(cid:12)(cid:12) X N ( a ) ≤ y a ∈ C q χ P ( a ) − π ( y, K ) h ( q ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ A x (log x ) − A . Since we are only concerned with imaginary quadratic fields, we have r = 0 and α ≻ α ∈ O K . Since ∆ = 3 , 4, we also know that [ U : U q , ] = 2 unless 2 ∈ q .This only happens when N ( q ) = 2 , 4, and is therefore negligible.In Proposition 1, note that if Q ( x, y ) represents an integer k , then we can applya SL ( Z ) action on Q ( x, y ) so that Q (1 , 0) = k . Since Q ( x, y ) is known to repre-sent infinitely many primes, we assume Q (1 , 0) is a sufficiently large prime that iscoprime to all q in Proposition 1. We also assume ( Q ( m, n ) , q ) = 1 since we onlycount prime values of Q ( m, n ). The main idea of the proof of Proposition 1 is torelate Q ( m, n ) to certain integral ideals. This can be achieved by using aQ ( m, n ) = N (( α m + α n ))where α = a, α = − b + √ b − ac . From our assumptions, the ideal ( α m + α n ) is coprime to the ideal ( q ). Definethe ideal a = ( α , α ). Then we have N a = a and ( α m + α n ) ⊆ a . Thus we canwrite Q ( m, n ) = N ( s ) where as = ( α m + α n ) . On the other hand, if s is an integral ideal of O K such that as = ( α ) is principal, wemust have α ∈ ( α , α ) and hence as = ( α m + α n ) for some m, n ∈ Z . Thereforewe can work on the ideals ( α m + α n ) / a with m ≡ u, n ≡ v (mod q ) instead. Todetect this congruence condition, we resort to ray class group as we promised. PETER CHO-HO LAM Proposition 3. Let Q ( x, y ) = ax + bxy + cy be a primitive positive definite binaryquadratic form with discriminant − ∆ such that ∤ ∆ and ∆ is not divisible by anyodd prime squares. Let q ∈ N with ( a, q ) = 1 and u, v ∈ Z with ( Q ( u, v ) , q ) = 1 .Then for any m, n ∈ Z , m ≡ ± u, n ≡ ± v (mod q ) if and only if ( α m + α n ) ∼ ( α u + α v ) in C ( q ) .Proof. WLOG suppose ( m, n ) ≡ ( u, v ) (mod q ). Then( α m + α n ) − ( α u + α v ) = mq for some m ∈ O K . Therefore( α m + α n − mq )( α m + α n ) = ( α m + α n )( α u + α v ) . By taking α = α m + α n − mq and β = α m + α n , we obtain ( α )( α m + α n ) =( β )( α u + α v ) and hence ( α m + α n ) ∼ ( α u + α v ) in C ( q ) .On the other hand, if ( α m + α n ) ∼ ( α u + α v ), then there exists α, β coprimeto ( q ) such that α − β ∈ ( q ) and α ( α m + α n ) = ± β ( α u + α v ) . Since α − β = tq for some t ∈ O K , this gives tq ( α m + α n ) = − β (cid:18) α ( m ± u ) + α ( n ± v ) (cid:19) . As β is coprime to ( q ) we deduce that( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) α ( m ± u ) + α ( n ± v ) (cid:19) . This implies ( α ( m ± u ) + α ( n ± v )) /q ∈ O K , or alternatively2 a ( m ± u ) − b ( n ± v )2 q + n ± v q √− ∆ ∈ O K . If b is even, then ∆ is also even and 4 || ∆. From the imaginary part we have q | n ± v .From the real part it forces 2 q | a ( m ± u ), and hence q | m ± u as well. If b is odd,then ∆ is also odd and the imaginary part gives q | n ± v again. Note that − b ( n ± v )2 q , n ± v q are both integers or half-integers. Therefore 2 a ( m ± u ) / q ∈ Z and the result followsagain. (cid:3) The above proposition shows that the classes in the ray class group indeed gener-alize the notion of arithmetic progressions, but up to a unit. Moreover, the function ν ( q ) is related to the Euler-phi function for integral ideals: Proposition 4. For any q ∈ N , we have ν ( q ) = φ (( q )) . In particular, if p is a prime, then ν ( p ) = ( p − if p splits, p − if p is inert, p − p if p ramifies. GENERALIZATION OF BOMBIERI-VINOGRADOV THEOREM AND ITS APPLICATION 5 Proof. Since ( Q (1 , , q ) = 1, we have ( Q ( m, n ) , q ) = 1 if and only if α m + α n isa unit in ( O K / q ) ∗ with q = ( q ). The result then follows from the fact that φ ( q ) isthe number of elements in ( O K / q ) ∗ . (cid:3) Proof of Proposition 1. X Q ( m,n ) ≤ yu ≡ u (mod q ) v ≡ v (mod q ) χ P ( Q ( m, n )) = X Q ( m,n ) ≤ y ( α m + α n ) ∼ ( α u + α v ) χ P (cid:18) N ( α m + α n ) a (cid:19) = 2 X N s ≤ y s ∈ C ( q ) χ P ( s )where C ( q ) = h ( α u + α v ) ih a i − and h·i is the equivalence class that the idealbelongs to. On the other hand π ( y, K ) h (( q )) = 2 π ( y, K ) ν ( q ) h K . Therefore X N ( a ) ≤ y a ∈ C ( q ) χ P ( a ) − π ( y, K ) h (( q )) = 12 (cid:18) X Q ( m,n ) ≤ yu ≡ u (mod q ) v ≡ v (mod q ) χ P ( Q ( m, n )) − π ( x, Q ) ν ( q ) (cid:19) + 1 ν ( q ) (cid:18) π ( x, Q ) − π ( y, K ) h K (cid:19) . By Proposition 4 and the fact that 1 /ν ( p ) = 1 /p + O (1 /p ), we deduce that X d 1] be a symmetric smooth function that is supported on∆ k := { ( t , ..., t k ) ∈ [0 , ∞ ) k : t + ... + t k ≤ } . Then we can define our sieve weights λ ( d , ..., d k ) by λ ( d , ..., d k ) = µ ( d ) · · · µ ( d k ) F (cid:18) log d log R , ..., log d k log R (cid:19) . The value of R will be chosen to be a power of X . From the support of F we cansee that d d · · · d k ≤ R . Our goal here is to show that there exist ρ > k ≥ X . However, some difficulties mightarise if Q ( m + g j , n + h j ) are not coprime to each other. Fortunately these potentialcommon factors must be relatively small: Lemma 1. Let Q ( x, y ) = ax + bxy + cy . If Q ( m, n ) ≡ Q ( m + g, n + h ) ≡ p ) , then p | aQ ( g, h ) Q ( g, − h ) .Proof. If p ∤ n ( n + h ), then we have mn − ≡ ( m + g )( n + h ) − (mod p ) and hence mh ≡ ± ng (mod p ). Therefore Q ( mh, nh ) ≡ Q ( ± ng, nh ) ≡ n Q ( g, ± h ) (mod p ) . This implies p | Q ( g, h ) or p | Q ( g, − h ). On the other hand if p | n , we have p | am .Thus we either have p | a or p | m and0 ≡ Q ( m + g, n + h ) ≡ Q ( g, h ) (mod p ) . (cid:3) Therefore we can put a restriction m ≡ r , n ≡ r (mod W ) where W is the productof all primes less than log log log X and r , r are integers such that Q ( r + g j , r + h j ) W )for j = 1 , , ..., k . This guarantees that Q ( m + g j , n + h j ) are mutually coprime.Define S ( X, ρ ) := X Q ( m,n ) ∼ Xm ≡ r (mod W ) n ≡ r (mod W ) (cid:18) k X j =1 χ P ( Q ( m + g j , n + h j )) − ρ (cid:19) × (cid:18) X d ,d ,...,d k ∈ N d i | Q ( m + g i ,n + h i ) for each i λ ( d , d , ..., d k ) (cid:19) . By interchanging the order of summations, we obtain S ( X, ρ ) = X d ,...,e k ∈ N ( d i e i ,W )=1 for each i λ ( d , ..., d k ) λ ( e , ..., e k ) X a i ,b i (mod [ d i ,e i ]) Q ( a i ,b i ) ≡ d i ,e i ]) for each i X Q ( m,n ) ∼ Xm ≡ r (mod W ) n ≡ r (mod W ) m ≡ a i (mod [ d i ,e i ]) for each in ≡ b i (mod [ d i ,e i ]) for each i (cid:18) k X j =1 χ P ( Q ( m + g j , n + h j )) − ρ (cid:19) GENERALIZATION OF BOMBIERI-VINOGRADOV THEOREM AND ITS APPLICATION 7 Since the k + 1 integers W and [ d i , e i ] for i = 1 , , ..., k are mutually coprime, thecongruence conditions on m, n in the innermost sum can be treated as a singlecongruence restriction modulo W Q i [ d i , e i ]. Note that(3.2) X Q ( m,n ) ≤ Xm ≡ a (mod q ) n ≡ b (mod q ) q πX √− ∆ + O (1) . By Proposition 1 and (3.2), we deduce that S ( X, ρ ) = 1 ν ( W ) δ Q Xh ( − ∆) log X · (cid:18) k X ℓ =1 S ,ℓ (cid:19) − ρ · W πX √− ∆ · S + o ( R )where S := X d ,...,e k ∈ N ( d i ,e j )=1 for i = j λ ( d , ..., d k ) λ ( e , ..., e k ) Y i ρ ([ d i , e i ])[ d i , e i ] and S ,ℓ := X d ,...,e k ∈ N ( d i ,e j )=1 for i = jd ℓ = e ℓ =1 λ ( d , ..., d k ) λ ( e , ..., e k ) Y i ρ ([ d i , e i ]) ν ([ d i , e i ]) . These two types of sums also appear in many other works on bounded gap be-tween primes and from today’s perspective they can be evaluated by fairly standardFourier-analytic techniques. Here we simply quote the following proposition, whichis a special case of Lemma 2.3.1 in [7]. Proposition 5. Let g be a multiplicative function with (3.3) 1 g ( p ) = p + O ( p t ) with t < . Define S ( g ) = X d ,...,e k ∈ N ( d i ,e j )=1 for i = j λ ( d , ..., d k ) λ ( e , ..., e k ) Y i g ([ d i , e i ]) and S ,ℓ ( g ) = X d ,...,e k ∈ N ( d i ,e j )=1 for i = jd ℓ = e ℓ =1 λ ( d , ..., d k ) λ ( e , ..., e k ) Y i g ([ d i , e i ]) . Then we have S ( g ) = (1 + o (1)) (cid:18) Wφ ( W ) (cid:19) k I ( F )(log R ) k , and S ,ℓ ( g ) = (1 + o (1)) (cid:18) Wφ ( W ) (cid:19) k J ℓ ( F )(log R ) k − . Here I ( F ) := Z ∆ k (cid:18) ∂ k F ∂t ...∂t k (cid:19) dt dt ... dt k PETER CHO-HO LAM and J ℓ ( F ) := Z ∆ k − (cid:18) ∂ k − F ∂t ...∂t ℓ − ∂t ℓ +1 ...∂t k (cid:19) ( t , ..., t ℓ − , , t ℓ +1 , ..., t k ) dt dt ... dt ℓ − dt ℓ +1 ... dt k . Since g ( d ) = ρ ( d ) /d and g ( d ) = ρ ( d ) /ν ( d ) both satisfy (3.3), by Proposition 5we obtain S ( X, ρ ) = (1+ o (1)) (cid:18) Wφ ( W ) (cid:19) k X (log R ) k (cid:18) ν ( W ) (cid:18) θ − δ (cid:19) δ Q kJ ℓ ( F ) h ( − ∆) − ρW π √− ∆ I ( F ) (cid:19) . Finally since W ≥ ν ( W ), it suffices to find ρ > ρ < (cid:18) θ − δ (cid:19) δ Q √− ∆2 πh ( − ∆) kJ k ( F ) I ( F ) . By Theorem 23 of [5], for sufficiently large k , we havesup F kJ k ( F ) I ( F ) ≫ log k. Thus when k is large enough, the right side of (3.4) is strictly greater than 1.Therefore we can pick ρ > References [1] H. Halberstam, H.E. Richert, Sieve methods , London Mathematical Society Monographs, No.4. Academic Press, London-New York, 1974. xiv+364 pp.[2] M. N. Huxley, The large sieve inequality for algebraic number fields. III. Zero-density results. ,J. London Math. Soc. (2) Small gaps between primes , Ann. of Math. (2) 181 (2015), no. 1, 383413.[4] J. S. Milne, Class field theory Variants of the Selberg sieve, and bounded intervals containing manyprimes. , Res. Math. Sci. 1 (2014), Art. 12, 83 pp.[6] J. Thorner, Bounded gaps between primes in Chebotarev sets , Res. Math. Sci. 1 (2014), Art.4, 16 pp.[7] A. Vatwani, Higher rank sieves and applications , Ph. D. Thesis, 2016.[8] A. Vatwani, Bounded gaps between Gaussian primes , J. of Number Theory (2017), 449-473.