Prime polynomial values of linear functions in short intervals
aa r X i v : . [ m a t h . N T ] O c t Prime polynomial values of linearfunctions in short intervals
Efrat Bank ∗ Lior Bary-Soroker † September 11, 2018
In this paper we establish a function field analogue of a conjecture in num-ber theory which is a combination of several famous conjectures, includingthe Hardy-Littlewood prime tuple conjecture, conjectures on the number ofprimes in arithmetic progressions and in short intervals, and the Goldbachconjecture. We prove an asymptotic formula for the number of simultaneousprime values of n linear functions, in the limit of a large finite field. Recently, several function field analogues of problems in analytic number theory weresolved in the limit of a large finite field, e.g. the Bateman-Horn conjecture [9]; theGoldbach conjecture [5]; the Chowla conjecture [8]; problems on variance of the numberof primes in short intervals and in arithmetic progressions [11] and on covariance ofalmost primes [13].Let us describe in more detail two classical problems in number theory and theirresolutions in the function field case. The two problems we describe relate to the workof this paper. Let be the prime characteristic function, i.e., ( h ) = ( , h is prime0 , otherwise . (1)The first problem is of counting primes in short intervals. By the Prime Number The-orem, it is conjectured that if I is an interval of length x ǫ , ǫ >
0, around large number ∗ School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel,[email protected] † School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel,[email protected] , then X h ∈ I ( h ) ∼ Z I dt log t ∼ x ǫ log x . (2)Let F q [ t ] be the ring of polynomials over the finite field F q with q elements. By abuseof notation, denote by the analogue of (1), i.e., the characteristic function of primepolynomials (which are by definition monic irreducibles), and let k f k = q deg f , for f ∈ F q [ t ] (where k k = 0). Rosenzweig and the authors [2] prove the following analogue of(2): Let f ∈ F q [ t ] be monic of degree k , k ≤ ǫ <
1, and I = I ( f , ǫ ) = { f ∈ F q [ t ] : k f − f k ≤ k f k ǫ } ; then X f ∈ I ( f ) = Ik (1 + O k ( q − / )) , (3)where the implied constant depends only on k and not on f or q . To compare between(2) and (3), we replace x ǫ with I , and log x with k .The second problem is the Hardy-Littlewood prime tuple conjecture, which assertsthat X
Let
B > and > ǫ > be fixed real numbers. Then the asymptoticformula X f ∈ I ( f ,ǫ ) ( L ( f )) · · · ( L n ( f )) = I ( f , ǫ ) Q ni =1 deg( L i ( f )) (1 + O B ( q − / )) holds uniformly for all odd prime powers q , ≤ n ≤ B , distinct primitive linear functions L ( X ) , . . . , L n ( X ) defined over F q [ t ] each of height at most B , and monic f ∈ F q [ t ] ofdegree in the interval B ≥ deg f ≥ ǫ . Since n is fixed and the L i ’s are all linear, S ( L , . . . , L n ) = 1 + O ( q − ) (see [12, 1.3]).Hence, Theorem 1.1 is indeed the analogue of (6) over F q [ t ] in the limit q → ∞ . If n = 1and b = 1, Theorem 1.1 reduces to [2, Corollary 2.4]; and if n = 1 and ǫ > − B , itreduces to [2, Corollary 2.6].We not only count primes but also deal with other factorization types, see Theorem 3.1.The latter may be used to get independence of other arithmetic functions, e.g. the k -thdivisor function d k ( f ) = { ( f , . . . , f k ) : f = f · · · f k } , in parallel to [1].The main innovation of the paper is the calculation of the Galois groups of certainpolynomials, see §
2. The derivation of the main result from the Galois group calculationis then done in § The goal of this section is to calculate the Galois group of the product of n linear func-tions evaluated at a generic polynomial. For the rest of the section we fix an algebraicallyclosed field F of characteristic not equal
2. 3ecall that the discriminant of a monic separable polynomial F ( t ) is defined by theresultant of F and F ′ : disc( F ) = ± Res( F , F ′ ) = ± ν Y j =1 F ( τ j ) , (7)where F ′ = c Q νj =1 ( t − τ j ). Proposition 2.1.
Let L i ( X ) = f i ( t ) + g i ( t ) · X , where i = 1 , be distinct primitivelinear functions over F [ t ] . Let h = P mj =0 A j t j , where A j are variables. Assume that deg( f i ) > deg( g i ) + m , i = 1 , and that m ≥ . Denote by d i = disc t ( L i ( h )) thediscriminant of L i ( h ( t )) regarded as a polynomial in t . Then d , d are non-squares andare relatively prime in the ring F ( A , . . . , A m )[ A ] . Before proving the proposition we will prove three auxiliary results.
Lemma 2.2.
Let L ( X ) = f ( t ) + g ( t ) · X be a primitive linear function and let h ( A , t ) = P mj =0 A j t j be a polynomial with variable coefficients. Denote Ψ( t ) = f ( t ) g ( t ) + P mj =1 A j t j .Assume that ( L ( h ))( α ) = 0 for some α in an algebraic closure Ω of F ( A ) . Then ( L ( h )) ′ ( α ) = 0 ⇔ Ψ ′ ( α ) = 0 Proof.
Write ( L ( h ))( t ) = f ( t ) + g ( t ) · m X j =0 A j t j = ( A + Ψ( t )) · g ( t )Since f and g are relatively prime, they do not have a common zero. Therefore, if( L ( h ))( α ) = 0 then g ( α ) = 0, which means that A + Ψ( α ) = 0. Now,( L ( h )) ′ ( α ) = g ′ ( α )( A + Ψ( α )) + g ( α )Ψ ′ ( α ) = g ( α )Ψ ′ ( α ) . Hence, ( L ( h )) ′ ( α ) = 0 ⇔ Ψ ′ ( α ) = 0. Lemma 2.3.
Let L i ( X ) = f i ( t ) + g i ( t ) · X , i = 1 , be distinct primitive linear functions.Let h ( A , t ) = P mj =0 A j t j , with A , . . . , A m variables and let Ψ i ( t ) = f i ( t ) g i ( t ) + P mj =1 A j t j . If ρ , ρ in an algebraic closure of F ( A ) solve the linear system ( L ( h )) ′ ( ρ ) = 0( L ( h )) ′ ( ρ ) = 0Ψ ( ρ ) = Ψ ( ρ ) = − A (8) then they solve the linear system Ψ ′ ( ρ ) = 0Ψ ′ ( ρ ) = 0Ψ ( ρ ) = Ψ ( ρ ) . (9)4 roof. Note that since A +Ψ i ( ρ i ) = 0, it follows that ( L i ( h ))( ρ i ) = 0. Using Lemma 2.2,Ψ ′ i ( ρ i ) = 0 for i = 1 , Lemma 2.4.
Let ≤ m be an integer and let L = f + g X, L = f + g X be distinctprimitive linear functions over F [ t ] . Assume that deg( f i ) > deg( g i )+ m, and let h ( A , t ) = P mj =0 A j t j be a polynomial with variable coefficients and Ψ i ( t ) = f i ( t ) g i ( t ) + P mj =1 A j t j . Thenthe linear system (9) has no solution for ρ = ρ in an algebraic closure Ω of F ( A ) . The proof is in the spirit of the proof of [2, Lemma 3.5] and uses the tools of Carmon-Rudnick in [8].
Proof.
For short we write ρ = ( ρ , ρ ). Let − ϕ i ( t ) = (cid:18) ψ i ( t ) + m X j =3 A j t j (cid:19) ′ = ψ i ( t ) ′ + m X j =3 jA j t j − , where i = 1 , ψ i = f i g i . Let c ( ρ ) = ψ ( ρ ) − ψ ( ρ ) + m X j =3 ( ρ j − ρ j ) A j . Then if m ≥
3, Ψ ′ i ( t ) = 2 A t + A − ϕ i ( t ) c ( ρ ) = Ψ ( ρ ) − Ψ ( ρ ) − (( ρ − ρ ) A + ( ρ − ρ ) A ) . (10)The system of equations (9) defines an algebraic set T ⊆ A × A m in the variables ρ , ρ , A , . . . , A m . It takes the matrix form M ( ρ ) · (cid:0) A A (cid:1) = B ( ρ ) = (cid:16) ϕ ( ρ ) ϕ ( ρ ) c ( ρ ) (cid:17) , (11)where M ( ρ ) = (cid:18) ρ ρ ρ − ρ ρ − ρ (cid:19) .Let α : T → A and β : T → A m be the projection maps. We note that since Ψ = Ψ ,there are only finitely many solution of (9) with ρ = ρ .For every ρ ∈ U = { ρ | ρ = ρ , ϕ i ( ρ i ) = ∞ , i = 1 , } , the rank of M ( ρ ) is 2. Thus,the dimension of the fiber α − ( ρ ) is at most m − ρ ∈ U . Moreover, for a given ρ ∈ U , (11) is solvable if and only if rank( M | B ) = 2 if and only if d ( ρ ) = det( M | B ) = 0.This means that the solution space (restricting to ρ ∈ U ) lies in d ( ρ ) = 0.It suffices to prove that d ( ρ ) is a nonzero rational function in the variables ρ = ( ρ , ρ ).Indeed, this implies that dim( α ( T )) ≤ dim { d ( ρ ) = 0 } = 1, so dim T ≤ m − < m .Thus, β ( T ) does not contain the generic point of A m , which is A = ( A , . . . , A m ), andhence (9) has no solution with ρ ∈ Ω . 5 straightforward calculation gives d ( ρ ) = ( ρ − ρ )(2 c ( ρ ) + ( ρ − ρ )( ϕ ( ρ ) + ϕ ( ρ ))) . By (10), if m ≥
3, then the coefficient of A in 2 c ( ρ ) + ( ρ − ρ )( ϕ ( ρ ) + ϕ ( ρ )) is2( ρ − ρ ) − ρ + ρ )( ρ − ρ ) , which is nonzero in any characteristic and we are done.Assume m = 2. Then c ( ρ ) = ψ ( ρ ) − ψ ( ρ ) and ϕ i ( ρ i ) = − ψ ′ i ( ρ i ). So, d ( ρ ) = ( ρ − ρ )(2( ψ ( ρ ) − ψ ( ρ )) − ( ρ − ρ )( ψ ′ ( ρ ) + ψ ′ ( ρ ))) . Assume that d = 0 as a polynomial in ρ . Then also0 = 2( ψ ( ρ ) − ψ ( ρ )) − ( ρ − ρ )( ψ ′ ( ρ ) + ψ ′ ( ρ )) . (12)Let us solve (12) with ψ i rational function in ρ i , i = 1 ,
2. Choose α such that ψ ( α ) = 0.By replacing ρ by ρ + α , we may assume that α = 0. Substituting 0 for ρ gives riseto the differential equation 0 = 2 ψ ( ρ ) − ρ ψ ′ ( ρ ) − ρ ψ ′ (0) (13)As an element of the field of formal Laurent series, ψ solving (13) must have the form: ψ ( ρ ) = ρ ψ ′ (0) + ∞ X i = N a ip +2 ρ ip +21 , N ∈ Z . (14)Plug (13) and (14) in (12) to get0 = ρ ψ ′ (0) − ψ ( ρ ) − ρ ψ ′ ( ρ ) + ρ ψ ′ (0) + ∞ X i = N a ip +2 ρ ip +11 ! + ρ ψ ′ ( ρ ) . Substituting 0 for ρ we get0 = − ψ ( ρ ) + ρ ψ ′ (0) + ρ ψ ′ ( ρ ) , (15)which is almost identical to (14); therefore, ψ ( ρ ) = ρ ψ ′ (0) + ∞ X i = N c ip +2 ρ ip +22 . (16)Here, without loss of generality, we assume the series for ψ and ψ start at the same N , as we allow the coefficients to be zero. Plug (14) and (16) in the original equation(12) to get, 0 = 2 ∞ X i = N a ip +2 ρ ρ ip +11 − ∞ X i = N c ip +2 ρ ρ ip +12 ! (17)6y comparing the coefficients of ρ ρ ip +12 and ρ ρ ip +11 , one gets that a ip +2 = c ip +2 = 0 forall i = 0 and a = c . This means that ψ ( ρ ) = ρ ψ ′ (0) + a ρ ψ ( ρ ) = ρ ψ ′ (0) + a ρ in contradiction to the assumption that ψ i = f i g i where deg( f i ) > f i , g i are relativelyprime. Therefore, d ( ρ ) is not the zero polynomial, as needed to conclude the proof. Proof of Proposition 2.1.
By [2, Proposition 3.6], Gal( L i ( h ) , F ( A )) is the full symmetricgroup. Hence, d i is not a square in F ( A ) for each i = 1 , F ( A , . . . , A m )[ A ]. If d , d are not relatively prime in F ( A , . . . , A m )[ A ], then they have a common root (aspolynomials in A ). Now, d = disc t L ( h ( t )) = ± ν Y j =1 ( L ( h ))( τ j )= ± ν Y j =1 g ( τ j )( A + Ψ ( τ j ))where ( L ( h )) ′ ( t ) = c · Q νj =1 ( t − τ j ). A root ρ of d must therefore satisfy: ( ( L ( h )) ′ ( ρ ) = 0Ψ ( ρ ) = − A (18)(note that if g ( τ j ) = 0 then ( L ( h ))( τ j ) = 0). A root ρ of d satisfies the analoguesequations. Thus, the condition that d and d have a common root translates intothe linear system (8). By Lemma 2.3, the solutions for this system is a subset of thesolutions of the linear system (9), which is an empty set by Lemma 2.4 whenever ρ = ρ .If ρ = ρ , then Ψ ( ρ ) = Ψ ( ρ ) = − A hence f ( ρ ) g ( ρ ) − f ( ρ ) g ( ρ ) = 0 . So ρ is algebraic over F in contradiction to Ψ ( ρ ) = − A . Therefore, d · d is indeednot a square in F ( A ). Thus, d and d are relatively prime in F ( A , . . . , A m )[ A ]. Proposition 2.5.
Let L , · · · , L n be distinct primitive linear functions and f ∈ F [ t ] amonic polynomial of degree k . Let f = f + P mj =0 A j t j where ≤ m < k . Then, Gal n Y i =1 L i ( f ) , F ( A ) ! = n Y i =1 Gal( L i ( f ) , F ( A )) = S k × · · · × S k n , where k i = deg( L i ( f )) . roof. Let f = f + P mj =0 A j t j . Then, L i ( f ) = ˜ L i ( h )where ˜ L i = ˜ f i + g i · X , h = P mj =0 A j t j , and ˜ f i = f i + g i f . Since m < k , it follows thatdeg( ˜ f i ) > deg( g i ) + m for each i . Since m ≥
2, [2, Proposition 3.6] givesGal( L i ( f ) , F ( A )) = Gal( ˜ L i ( h ) , F ( A )) ∼ = S k i (19)By Proposition 2.1, the discriminants d i = disc t ( ˜ L i ( h )) = disc t ( L i ( f )) are non-squaresand pairwise relatively prime in F ( A , . . . , A m )[ A ]. So, d , . . . , d n are square indepen-dent (in the sense that any product is non-square). Together with (19), the discussionbefore [3, Lemma 3.4] gives thatGal n Y i =1 L i ( f ) , F ( A ) ! = n Y i =1 Gal( L i ( f ) , F ( A )) = S k × · · · × S k n , as needed. In this section we shall prove a generalization of Theorem 1.1.We follow the notation of [1]. The cycle structure of a permutation σ of k letters is thepartition λ ( σ ) = ( λ , . . . , λ k ) of k if in the decomposition of σ as a product of disjointcycles, there are λ j cycles of length j .For each partition λ ⊢ k , the probability that a random permutation on k letters hascycle structure λ is given by Cauchy’s formula: p ( λ ) = { σ ∈ S k : λ ( σ ) = λ } S k = k Y j =1 j λ j · λ j ! . (20)For f ∈ F q [ t ] of positive degree k , we say its cycle structure is λ ( f ) = ( λ , . . . , λ k ) ifin the prime decomposition f = Q j P j (we allow repetition), we have { i : deg( P i ) = j } = λ j .For a partition λ ⊢ k , we let λ be the characteristic function of f ∈ M n of cyclestructure λ : λ ( f ) = ( , λ ( f ) = λ , otherwise . (21) Theorem 3.1.
Let
B > and > ǫ > be fixed real numbers. Then the asymptoticformula X f ∈ I ( f ,ǫ ) λ ( L ( f )) · · · λ n ( L n ( f )) = p ( λ ) · · · p ( λ n ) I ( f , ǫ ) (cid:16) O B (cid:16) q − (cid:17)(cid:17) olds uniformly for all odd prime powers q , ≤ n ≤ B , distinct primitive linear functions L ( X ) , . . . , L n ( X ) defined over F q [ t ] each of height at most B , monic f ∈ F q [ t ] of degreein the interval B ≥ deg f ≥ ǫ , and partitions λ , · · · , λ n of deg( L ( f )) , . . . , deg( L n ( f )) ,respectively. Theorem 1.1 is the special case of Theorem 3.1 when taking all the λ i ’s to be thepartition into one part. (Since then = λ i and p ( λ i ) = L i ( f )) .)The proof of Theorem 3.1 is in the same spirit as proofs of other results in the literatureonce one has the Galois group calculation (Proposition 2.5). In fact, it is nearly identicalto the proof of [1, Theorem 1.4]. For the reader’s convenience we bring here the full proof. Theorem 3.2 ([1, Theorem 3.1]) . Let A = ( A , . . . , A m ) be an ( m + 1) -tuple of variablesover F q , let F ( t ) ∈ F q [ A ][ t ] be monic and separable in t , let L be a splitting field of F over K = F q ( A ) , and let G = Gal( F , K ) = Gal( L/K ) . Assume that F q is algebraically closedin L . Then there exists a constant c = c ( m, tot . deg( F )) such that for every conjugacyclass C ⊆ G we have (cid:12)(cid:12)(cid:12)(cid:12) { a ∈ F m +1 q : Fr a = C } − | C || G | q m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ cq m +1 / . Here Fr a denotes the Frobenius conjugacy class (cid:16) S/Rφ (cid:17) in G associated to the homo-morphism φ : R → F q given by A a ∈ F m +1 q , where R = F q [ A , disc F − ] and S is theintegral closure of R in the splitting field of F . See [1, Appendix A] for more details andfor a proof. Proof of Theorem 3.1.
Let ǫ > f be a monic polynomial of degree k where B ≥ k ≥ ǫ . Set m = ⌊ ǫk ⌋ and let f = f + P mj =0 A j t j . Note that 2 ≤ m < k . Define F i ( A , t ) = L i ( f ) and F = F · · · F n . Let L be the splitting field of F over K = F q ( A )and let F be an algebraic closure of F q . Let G = Gal( F , K ) = Gal( L/K ).By Proposition 2.5, S k × · · · × S k n ∼ = Gal( F L/ F K ) ∼ = Gal( L/L ∩ ( F K )) ≤ G, where k i = deg( L i ( f )). On the other hand, the factorization F = F · · · F n implies that G ≤ S k × · · · × S k n . So G = S k × · · · × S k n , (22)and L ∩ ( F K ) = K . It follows in particular that L ∩ F = K ∩ F = F q . Hence, we mayapply Theorem 3.2 with the conjugacy class C = { ( σ , . . . , σ n ) ∈ G : λ σ i = λ i } to get that (cid:12)(cid:12) { a ∈ F m +1 q : Fr a = C } − | C | / | G | · q m +1 (cid:12)(cid:12) ≤ c ( B ) q m +1 / .
9e note that | C | / | G | = p ( λ ) · · · p ( λ n ) and { a ∈ F m +1 q : disc t ( F )( a ) = 0 } = O B ( q m ).Also for a ∈ F m +1 q with disc t ( F ( a )) = 0 we have Fr a = C if and only if λ F i ( a ,t ) = λ i forall i = 1 , . . . , n (see the proof of [1, Theorem 3.1] where this is shown explicitly). Now, X f ∈ I ( f ,ǫ ) λ ( L ( f )) · · · λ n ( L n ( f ))= { a ∈ F m +1 q : λ F i ( a ,t ) = λ i for all i } = { a ∈ F m +1 q : disc t ( F )( a ) = 0 , λ F i ( a ,t ) = λ i for all i } + O B ( q m )= { a ∈ F m +1 q : Fr a = C } + O B ( q m )= | C | / | G | q m +1 + O B ( q m +1 / )= p ( λ ) · · · p ( λ n ) q m +1 (1 + O B ( q − / )) . This finishes the proof, since I ( f , ǫ ) = q m +1 . In this section we provide heuristic for (5). Fix 1 > ǫ >
0. A classical conjecture aboutprimes in short intervals of the form [ x, x + x ǫ ] asserts that X x ≤ h ≤ x + x ǫ ( h ) ∼ Z x + x ǫ x dt log t ∼ x ǫ log( x ) , x → ∞ . Another classical conjecture, on the number of primes in arithmetic progressions, saysthat if 0 < a < b < x − δ , then X 1, then the number of h ≡ a (mod b ) in [ x, x + x ǫ ] is at most 1. Therefore, tocombine these two conjectures together, we must at least demand that 1 < ǫ + δ . Thenit is natural to expect that: X x ≤ h ≤ x + uh ≡ a (mod b ) ( h ) ∼ φ ( b ) · u log( x ) , x → ∞ . (23)for 0 < a < b < x − δ and x ǫ ≤ u ≤ x where ǫ > L ( X ) = a + bX : X x ≤ h ≤ x + x ǫ ( L ( h )) = X x ≤ h ≤ x + x ǫ ( a + bh )= X a + y ≤ ˜ h ≤ a + y + b − ǫ y ǫ ˜ h ≡ a (mod b ) (˜ h ) ∼ φ ( b ) · b − ǫ y ǫ log( y ) ∼ φ ( b ) · bx ǫ log( L ( x )) , (24)where y = bx and b δ < x and 0 < a < b . (Note that when b δ < x , then b < y − δ ′ forsome δ ′ > 0, which implies that y ǫ ≤ b − ǫ y ǫ ≤ y ; therefore the prior to the last step isjustified.)Next we deal with the case where b < a > | b | . By division with remainder, thereare unique 0 < a ′ < | b | and 0 < r such that a = a ′ + | b | r . Thus a + bh = a ′ + | b | ( r − h ).So, putting h ′ = r − h and L ′ ( X ) = a ′ + | b | X , we get X x ≤ h ≤ x + x ǫ ( L ( h )) = X r − x − x ǫ ≤ h ′ ≤ r − x ( L ′ ( h ′ ))= X y ≤ h ′ ≤ y + x ǫ ( L ′ ( h ′ )) ∼ φ ( | b | ) · | b | x ǫ log( L ′ ( y )) ∼ φ ( | b | ) · | b | x ǫ log( L ( x )) , where y = r − x − x ǫ and | b | δ < a and | b | x α < a < | b | x β for 1 < α < β . (Note thatwhen these conditions hold, y ǫ ′ < x ǫ < y for ǫ ′ < ǫβ and | b | δ < y since y ∼ a | b | ; thereforethe prior to the step is justified.) References [1] Julio. C Andrade, Lior Bary-Soroker, and Zeev Rudnick. Shifted convolution andthe titchmarsh divisor problem over F q [ t ], arXiv:1407.2076, 2014.[2] Efrat Bank, Lior Bary-Soroker, and Lior Rosenzweig. Prime polynomials in shortintervals and in arithmetic progressions, Duke , in print.[3] Lior Bary-Soroker. Irreducible values of polynomials. Adv. Math. , 229(2):854–874,2012.[4] Lior Bary-Soroker. Hardy-Littlewood tuple conjecture over large finite fields. Int.Math. Res. Not. IMRN , (2):568–575, 2014.115] Andreas O. Bender. Decompositions into sums of two irreducibles in F q [ t ]. C. R.Math. Acad. Sci. Paris , 346(17-18):931–934, 2008.[6] Andreas O. Bender and Paul Pollack. On quantitative analogues of the Goldbachand twin prime conjectures over F q [ t ], arXiv:0912.1702, 2009.[7] Dan Carmon. The autocorrelation of the Mobius function and Chowla’s conjecturefor the rational function field in characteristic 2. preprint, 2014.[8] Dan Carmon and Ze´ev Rudnick. The autocorrelation of the M¨obius function andChowla’s conjecture for the rational function field. Q. J. Math. , 65(1):53–61, 2014.[9] Alexei Entin. On the Bateman-Horn conjecture for polynomials over large finitefields. arXiv:1409.0846, 2014.[10] Andrew Granville. Unexpected irregularities in the distribution of prime num-bers. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zrich,1994), 388–399, Birkhuser, Basel, 1995.[11] Jonathan P. Keating and Ze´ev Rudnick. The variance of the number of primepolynomials in short intervals and in residue classes. Int. Math. Res. Not. IMRN ,(1):259–288, 2014.[12] Paul Pollack. Simultaneous prime specializations of polynomials over finite fields.