On prime values of binary quadratic forms with a thin variable

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form $F$ and binary linear form $G$, there exist infinitely many $\ell, m\in\mathbb{Z}$ such that both $F(\ell, m)$ and $G(\ell, m)$ are primes as long as there are no local obstructions.


Introduction
One of the most difficult problems in number theory concerns finding primes among interesting subsets of the natural numbers. A particular example of such a problem is finding primes among values of a given polynomial. Several famous conjectures belong to this line of investigation, including the Bateman-Horn conjecture.
In the case of polynomials of a single variable it is unknown whether a given polynomial represents infinitely many primes, except for linear polynomials by the seminal work of Dirichlet. For polynomials in two variables we have some non-linear examples, including quadratic norm forms, all suitable quadratic polynomials by work of Iwaniec [8], binary cubic forms by work of Heath-Brown [11] and Heath-Brown and Moroz [13], and the polynomial x 2 + y 4 due to Friedlander and Iwaniec [4]. One obvious approach is to deduce the analogous results for single variable polynomials from their two-variable counterparts by restricting one variable. Currently we do not know how to do this, but in some cases we can restrict one of the variables to a sparse subset of the integers. This gives rise to an interesting family of problems.
One particular example that has been considered is the case of the quadratic form x 2 + y 2 , where y is restricted to a sparse subset of the integers, including the case of y being prime. This was worked out in great detail by Fouvry and Iwaniec [3]. Lam [14] [15] and Pandey [17] studied similar problems for principal forms of certain negative discriminants. In all these cases the number of admissible y up to size X cannot be less than O δ (X 1−δ ) for any positive δ. Friedlander and Iwaniec [4] were able to break this barrier in the special case of restricting y to the set of squares. Heath-Brown and Li [12] later refined their methods to restrict y to the set of prime squares.
In this paper we further generalize the work of Fouvry and Iwaniec [3] by considering arbitrary primitive positive definite binary quadratic forms. It is worth mentioning that Friedlander and Iwaniec [7] provided a simplified proof of [3] if y is restricted to the set of primes.This was followed by Lam [15] and Pandey [17] in their own works but we decided to follow the original argument.
Let F (x, y) ∈ Z[x, y] be a positive definite and primitive quadratic form (i.e. the greatest common divisor of its coefficients is equal to 1). For d ∈ N, we set We shall prove the following theorem.
Theorem 1.1. Let F (x, y) = αx 2 + βxy + γy 2 ∈ Z[x, y] be a primitive positive definite quadratic form and X be a positive real number. Let λ(ℓ) be a sequence of complex numbers supported on the natural numbers which satisfy the bound |λ(ℓ)| ≤ C log A ℓ for all ℓ ∈ N and some fixed A, C > 0. Suppose q ∈ N and q (log X) N for some N > 0. Then for any B > 0 and a ∈ Z with gcd(a, q) = 1 we have and P F = p≤C F p with C F depending only on F .
Note that H F,q is positive if ρ(2) = 2. For example the primitive positive definite binary quadratic form 2x 2 + xy + y 2 of discriminant −7 cannot represent infinitely many prime values with x a prime. On the other hand it exhibits infinitely many prime values with y prime. The flexibility provided by λ(ℓ) and F (ℓ, m) ≡ a (mod q) have applications in proving Vinogradov's three primes theorem with special types of primes; see [9] for details.
The purpose of introducing P F in our expression is to remove some small prime factors that avoid the use of Dirichlet composition law (see Section 2). In practice this dependence on P F can be removed via Möbius inversion. For example, a particularly attractive consequence of Theorem 1.1 is the following: Corollary 1.2. Let F be a positive definite binary quadratic form and G be a binary linear form. Assume that for every prime p there are x, y ∈ Z such that p ∤ F (x, y)G(x, y). Then there exist infinitely many ℓ, m ∈ Z such that both F (ℓ, m) and G(ℓ, m) are primes.
It is also possible to impose the conditions F (ℓ, m) ≡ a (mod q) and G(ℓ, m) ≡ b (mod q): be a primitive positive definite quadratic form of discriminant −∆. Suppose q ∈ N and q (log X) N for some N > 0. Then for any A > 0 and a, b ∈ Z with gcd(ab, q) = 1 we have where One way to phrase Corollary 1.2 is that given the complete norm form F (x, y) and restricting the first variable x to primes, the form still represents infinitely many primes. A natural extension of this question is to ask given an arbitrary primitive complete norm form N in n variables and restricting a subset of the variables to a special set S, does N still represent infinitely many primes? In this formulation Heath-Brown [11] and Heath-Brown and Moroz [13] can be viewed as restricting one variable in a complete cubic norm form to be equal to zero. More recently, Maynard [16] showed that complete norm forms still represent infinitely many primes even with as many as a quarter of the variables are set to zero.
More generally, one expects a polynomial G with exactly r factors over Q should take values which have exactly r prime factors if there are no local obstructions. Indeed this is included in Schinzel's hypothesis. Our corollary 1.2 is a step towards confirming this conjecture for binary cubic forms, following the theorem of Heath-Brown and Moroz [13], by confirming this for the case when F has one linear factor and negative discriminant.
be a binary cubic form with negative discriminant, that is reducible over Q. Assume that for every prime p there are x, y ∈ Z such that p ∤ H(x, y). Then there exist infinitely many integers x, y such that H(x, y) has exactly two prime factors.
In particular, there are infinitely many integers with exactly two prime factors that are sums of two cubes, see also work of Pandey [17].
To deduce Theorem 1.1 from the work of Fouvry and Iwaniec [3] we must overcome two difficulties. The first is that the proof of a key lemma which is critical in Fouvry and Iwaniec [3] fails for a general binary quadratic form. In particular they obtained an optimal spacing result of roots modulo d of the congruence ν 2 + 1 ≡ 0 modulo d. Fortunately an analogous result was developed by Balog, Blomer, Dartyge and Tenenbaum [1]. We will then mimic the argument from [6] to finish up the proof in Section 4 as the original argument in [3] is not sufficient for our case. The second issue is that in general the arithmetic over a ring of integers O K with K a quadratic number field is not analogous to the arithmetic over Z[i] when O K has a non-trivial class group. To overcome this issue we require several applications of the Dirichlet composition law. We will develop the necessary tools in Section 2 and then employ it in Section 5.
Notation: We write τ (n) for the divisor function of a natural number n. ♭ denotes a sum over positive squarefree integers.

Dirichlet composition
Let F (x, y) = αx 2 + βxy + γy 2 be a primitive positive definite quadratic form of discriminant −∆. For a sequence of complex numbers λ(ℓ), ℓ ∈ N and N ∈ N we define a sequence (2.1) As F (x, y) is assumed to be positive definite, this is a finite sum. In [3], a key component of the bilinear sum estimates is the identity where ℓ = ℜ(wz), when F (x, y) = x 2 + y 2 and gcd(m, n) = 1. This is based on the classical identity (a 2 + b 2 )(c 2 + d 2 ) = (ac − bd) 2 + (bc + ad) 2 and the fact that there is one binary quadratic form of discriminant −4 up to (proper) equivalence. We now extend this identity to the case when the class number is not equal to one. To generalize (2.2), we will use the Dirichlet composition law.
Definition 2.1 (Dirichlet composition). Let f (x, y) = ax 2 + bxy + cy 2 and F (x, y) = αx 2 + βxy + γy 2 be primitive positive definite forms of discriminant −∆ < 0 which satisfy gcd(a, α, (b + β)/2) = 1. Then the Dirichlet composition of f (x, y) and F (x, y) is the form where B is any integer such that See [2] for a good reference in Dirichlet composition. Note that (b + β)/2 ∈ Z since b 2 ≡ β 2 ≡ −∆ (mod 4). This composition makes the equivalence class of binary quadratic of discriminant ∆ into an abelian group. The term composition is justified by the following identity: and It is convenient to have explicit coefficients in the composition for our purposes.
To establish an analogue of (2.2), we need to study the solutions of when gcd(m, n) = 1. One can show that m can be represented by a binary quadratic form of the same discriminant, say f (x, y); and by composing with F (x, y) we obtain a form that represents n. But to work out the composition explicitly, the condition gcd(a, α, (b + β)/2) = 1 is needed. This motivates us to construct a set of binary quadratic forms, S F (t), in which this condition is always satisfied.
For any F (x, y) = αx 2 + βxy + γy 2 and any t ∈ Z, define S F (t) to be a set of binary quadratic form of discriminant −∆ such that (1) every primitive binary quadratic form of discriminant −∆ is properly equivalent to exactly one element in S F (t); (2) the principal form is contained in S F (t); and If S F (t) is the set of primitive reduced forms of discriminant −∆, then (1) and (2) are satisfied. Since each of them represent infinitely primes, if necessary, we can transform the form so that the coefficient of x 2 is one of these primes and thus it is clear that (3) can be satisfied. Now we put S F = S F (α) and define We assume P F is large enough so that Q F |P F . We also pick an integer B with the following properties: So B only depends on F and the choice of S F . Proposition 2.2. Let ∆ be a positive integer and F (x, y) = αx 2 + βxy + γy 2 be a primitive binary quadratic form of discriminant −∆. Let m, n be positive integers such that gcd(mn, P F ) = 1. If mn = F (X, Y ) for some integers X, Y with gcd(X, Y ) = 1, then there exists a unique binary quadratic form f (x, y) = ax 2 + bxy + cy 2 ∈ S F and integers u, v, w, z such that gcd(u, v) = gcd(w, z) = 1 and au 2 + buv + cv 2 = m, and if ∆ > 4 then there is exactly one more tuple, namely (−u, −v, −w, −z), that satisfies the properties. If ∆ = 3, 4 we have 6 or 4 solutions, respectively.

Solving (2.4) and (2.5) gives
These equations also imply that gcd(u, v)| gcd(X, Y ) and gcd(z, The and it can easily be checked that for any Now suppose ∆ > 4 and there is another tuple (u 0 , v 0 , w 0 , z 0 ) that satisfies the requirement. It is straightforward to verify that (2.10) Further, it is easy to see that uv 0 − u 0 v ≡ 0 (mod m). It thus follows that If ∆ = 4, we can take S F = {x 2 + y 2 } and B = β (note that β must be even). Then (2.10) becomes and this has 4 pairs of solutions (u 0 , v 0 ). The case for ∆ = 3 is similar.
On the other hand, if we have f (u, v) = m and f * (w, z) = n, by Dirichlet composition they can produce X, Y ∈ Z such that F (X, Y ) = mn via However even if gcd(u, v) = gcd(w, z) = 1, it does not guarantee gcd(X, Y ) = 1. We are not too far away because by (2.4) and (2.5), we have Similarly gcd(X, Y )|n. Hence if gcd(m, n) = 1, we have gcd(X, Y ) = 1. Furthermore, if gcd(mn, P F ) = 1, we also deduce that gcd(X, γ) = 1 since γ|P F . From Proposition 2.2 and the discussion above, we conclude that Here we set λ(ℓ) = 0 if ℓ < 0. If ∆ = 3 or 4, the constant 1 2 before the summation should be 1 6 and 1 4 respectively. The condition gcd(mn, P F ) = 1 also implies and we will need this later in Section 5.

Setting up a sieve problem
After the algebraic preparations we now present the general framework of sieving with which we aim to find prime values in the sequence F (ℓ, m) with ℓ restricted to a thin sequence. In order to prove Theorem 1.1 it suffices to consider the sum where Λ(N) is the von Mangoldt function and a N is defined as in (2.1) and χ is a Dirichlet character modulo q. The character χ is present to detect the congruence condition modulo q.
Then we have the analogue of Proposition 9 in [3].
Proposition 3.1. Let Y, Z ≥ 1 and X > Y Z. Then we have the identity and R(X; Y, Z, χ) is given by (3.6).
From Proposition 3.1 we see that Theorem 1.1 follows provided that acceptable estimates for δ(N; Y, Z), B(X; Y, Z, χ), R(X; Y, Z, χ), P (Z; χ) can be obtained. We will give appropriate bounds for all but B(X; Y, Z, χ) in this section.
When sieving for prime values of N we will need to study sums of the type then from gcd(ℓ, γm) = 1 we immediately have gcd(ℓ, d) = 1 as well. Hence we expect that A d (X; χ) is approximated by and M d (X; χ) = 0 otherwise. With this we set For a parameter D we define the complete remainder term R(X, D; χ) as As F (x, y) is assumed to be positive definite, there exists a positive constant C 1 only depending on F , such that F (ℓ, m) ≤ X implies that |ℓ|, |m| ≤ C 1 √ X. Now put (3.6) As in [3] we have the bound Proof of Proposition 3.1. Our goal is to derive Proposition 3.1 from [3, Proposition 9]. To do so we must check that that condition (7.16) in [3] holds with the function ρ(bc), with c a fixed integer. Let −D(l) denote the discriminant of the quadratic polynomial F (x, l), and let −d = −d(l) be the unique fundamental discriminant such that Q( −D(l)) = Q( −d(l)). Let ρ ′ d (n) be the multiplicative function Consider the Dirichlet series Then D(s) differs from the series by a holomorphic factor. Here It is then apparent that .

ON PRIME VALUES OF BINARY QUADRATIC FORMS WITH A THIN VARIABLE 11
Plainly, G(s) converges and is holomorphic for ℜ(s) > 1/2. We then obtain the bound for some positive number c d by standard estimates of the zero-free region of the Dirichlet L-function L(s, χ −d ) and the Selberge-Delange method. The desired conclusion then follows from partial summation.
The terms B(X; Y, Z, χ), R(X; Y, Z, χ) in Proposition 3.1 will be controlled by the following lemmas: Lemma 3.3. Suppose q ∈ N and q (log X) N for some N > 0. Let θ 1 , θ 2 be two real numbers such that 1/2 < θ 1 < 1 and 0 < θ 2 < 1 − θ 1 . Then for Y = X θ 1 and Z = X θ 2 and any B > 0, We follow a similar strategy to [3] in showing that the term R(X; Y, Z; χ) can be bounded by our Type I estimate from Lemma 4.1 and the trivial estimate With all these ingredients we can prove our main theorem and corollaries.
Proof of Theorem 1.1. Define Y, Z as in Lemma 3.3. Together with Lemma 3.2 and Lemma 3.3 we have shown that The condition gcd(F (ℓ, m), P F ) = 1 on the left can be removed because of the presence of Λ (F (ℓ, m)). Hence by orthogonality of χ, it gives Finally, we treat the remaining terms in Proposition 3.1. We can use the trivial bound P (Z; χ) ≪ ε Z 1+ǫ for any ε > 0. The contribution of the terms with δ(N; Y, Z) is negligible as in (7.18) in [3].

Level of Absolute Distribution
In this section we shall obtain Type I estimates that are needed to prove Lemma 3.2 and the corollaries to Theorem 1.1. The most pressing issue is to control the quantity R(X; D, χ) given (3.5). To this end, we have the following lemma: To prove Lemma 4.1, it is convenient to remove the restrictive condition gcd(N, P F ) = 1. We let the scripted letters A, M, R to denote the analogous quantities A, M, R which appeared in the previous section, but without the condition gcd(N, P F ) = 1. We also set M d (X; χ) = 0 if gcd(d, q) > 1. We then have the following analogue to Lemma 4.1: Lemma 4.1 will be a simple consequence of Lemma 4.2. Furthermore Lemma 4.2 will be used to prove the corollaries from our main theorem. As in [3] we deduce Lemma 4.2 from a version where the A d (X; χ) are smoothed with an auxiliary weight function. Since we need to accommodate extra assumption gcd(ℓ, γm) = 1 it is convenient to adopt the approach from [6] instead.
Let √ X Y X be an additional parameter to be chosen later, and let w : R + → R be a smooth function with the following properties: For a, ℓ ≥ 1 we define the function When gcd(d, q) = 1, define as well as the smoother remainder term When gcd(d, q) > 1, they are both defined to be 0. We obtain the following lemma.

By Möbius inversion
To simplify our notation, let c = d/ gcd(a, d). where F a,ℓ (z) is defined in (4.2). Therefore  Define M d (X; w, χ) to be the summand when h = 0, i.e., the expression Note that for any integer c, we have We wish to sum R d (X; w, χ) dyadically and hence we define |R d (X; w, χ)|.
Substituting b = d/c = gcd(a, d), each term in the above sum can be bounded by |W a (c, ν)|.

By dyadic division,
where H is a power of 2, H) is defined similarly for those h < 0. We only present the argument for V + a (C, H) below for simplicity. For a reduced residue class t (mod q), we define Then (4.10) The symbol * means we are summing over reduced residue classes only. Next, we need to employ the Proposition 3 from [1]. Proposition 4.4. Let F (x, y) = αx 2 + βxy + γy 2 ∈ Z[x, y] be an arbitrary quadratic form whose discriminant is not a perfect square. For any sequence α n of complex numbers, positive real numbers D, N, we have Notice that where Applying this inequality and Cauchy-Schwarz inequality on (4.10), we deduce that log 2A+3 (HX).
Hence we obtain To develop a similar bound for large values of H, we apply integration by parts twice in (4.7) as in [6], followed with the large sieve type estimate. We arrive at (4.12) When H aC √ XY −1 , we use (4.11) to deduce that and if H > aC √ XY −1 , we use (4.12) to deduce that The same estimates hold for V − a (C, H) as well. Therefore by (4.9) and by (4.8) Proof of Lemma 4.2. To complete the proof of Lemma 4.2, it suffices to show that the error we made when we replace A d (X; χ) with A d (X; w, χ) is negligible as well, i.e. both |A d (X; χ) − A d (X; w, χ)| and |M d (X; χ) − M d (X; w, χ)| are small. Note that Here we have used the estimate a N ≪ C τ (N)(log X) A . Similarly, Summing over d and choosing Y = D 1/4 X 3/4 , we have

Proof of Lemma 4.1. Follows from Lemma 4.2 and Mobius inversion.
Finally, we give proofs for the corollaries.
Proof of Corollaries 1.2 and 1.3. If λ is supported on primes, then starting from (3.8) again, the right hand side becomes Therefore it is also equal to The contribution when gcd(ℓ, e) > 1 is negligible. Hence by Lemma 4.2 The contribution when gcd(ℓ, γm) > 1 is also negligible; therefore by orthogonality, Corollary 1.3 follows by taking λ(ℓ) = Λ(ℓ) when ℓ ≡ b (mod q). For Corollary 1.2, let G(x, y) = mx + ny with gcd(m, n) = 1. Then there exist integers s, t such that ms − nt = 1. By a change of variables u = −tx − sy, v = mx + ny we obtain F (x, y) = F (nu + sv, −mu − tv), which is a binary quadratic form in u and v. The result follows from Corollary 1.3 on the pair of forms F (nu + sv, −mu − tv) and v with q = 1.

Bilinear sums
In this section we shall estimate B(X; Y, Z; χ) given in (3.2) by proving Lemma 3.3. For reasons of exposition, we first work under the assumption that |λ(ℓ)| 1 for all ℓ ∈ N. As we save an arbitrary power of log X in our arguments, the general case can then be obtained by changing the parameter A. We proceed as in [3]. First put θ = (log X) −A and write where the error term O(θX(log X) 2 ) represents a trivial bound for the contribution of µ(b)χ(bd)a bd with bd ≤ 2θX or e −2θ X < bd ≤ X, which terms are not covered exactly. As in [3], we need to show that each short sum B(M, N) satisfies B(M, N) ≪ θ 2 X(log X) 2 . Define α(n) = µ(n)χ(n). When applying Proposition 2.3 to decompose a mn into solutions of in fact later in (5.6) we will decompose the solutions of f * (w, z) = n again using the same proposition. We construct S f * in the same way we construct S F by taking and let g(x, y) = dx 2 + exy + f y 2 ∈ S f * . We pick an integer B such that (1) B ≡ b (mod 2a) for all ax 2 + bxy + cy 2 ∈ S F ; (2) B ≡ e (mod 2d) for all dx 2 + exy + f y 2 ∈ S f * ; (3) B ≡ β (mod 2α); and (4) B 2 +∆ ≡ 0 (mod 4adα) for all ax 2 +bxy+cy 2 ∈ S F and dx 2 +exy+f y 2 ∈ S f * .
Such B always exist since the coefficients of x 2 of elements in S F or S f * are distinct primes. So B depends only on F and the choices of S F and S f * ; and hence depends only on F . In the definition of P F we take C F large enough so that By Proposition 2.3, we can bound B 1 (M, N) by α(f * (w, z))λ(Q F (u, v; w, z)) .
Since n is squarefree, by Proposition 2.2 we can decompose f * (w, z) = n as for some g ∈ S f * and we have the relations We then see that the inner sum of (5.5) becomes α(rg * (w 0 , z 0 ))λ(Q F (u, v; w, z)).
Estimating trivially we find that the terms with r ≥ θ −2 , where we take θ = (log x) −A for some large positive number A as in [3], contribute O θMN In the remaining terms we ignore the conditions r 2 |h(P, Q), gcd(h(P, Q), P F ) = 1 and obtain α(rg * (w 0 , z 0 ))λ(z 0 P − αw 0 Q) .
We then write where the asterisk in the sum means that the sum is over primitive pairs. By [3] it then suffices to give a bound of the shape for every c, r, M, N with c < θ −4 , r < θ −2 , M θ 4 Z, N > θ 3 Y, and θ 5 X < MN < X. Our assumptions in Lemma 3.3 guarantee that M, N satisfy N ε < M < N 1−ε for some small ε > 0. This assumption will be used in (5.18) and (5.22) and we will give a bound of the form (5.10) Let A = N/r, B = √ M and α(u, v) = α(rg * (u, v)). Then α(u, v) is supported in the annulus A 2 < g * (u, v) ≤ 4A 2 . By applying the Cauchy-Schwarz inequality, we obtain and Q(u, v; w, z) = vw − αuz. Here ψ(w, z) can be any non-negative function with ψ(w, z) ≥ 1 if B 2 ≤ h(w, z) ≤ 4B 2 . We do not need to be specific at this point; nevertheless it will be convenient to assume that ψ(w, z) takes the form Ψ(h(w, z)), where Our desired estimate for D(α) is A 3 B with a saving of an arbitrary power of log N. Since ℓ runs over all integers (without any restriction), after squaring we obtain D(α) = * (w,z)∈Z 2 gcd(w,α)=1 ψ(w, z) This equality follows because Q(u, v; w, z) is a bilinear form. Note that The orthogonality relation Q(u, v; w, z) = 0 in (5.12) is equivalent to for some rational integer c ∈ Z since gcd(w, αz) = 1. It thus follows that =D 0 (α) + 2D * (α), (5.13) say, where D 0 (α) denotes the contribution of c = 0 and D * (α) that of all |c| > 0. Thus and , t gcd(s, t) (α * α)(s, αt). Note that g * (s, αt) ≤ 2A (from the support of α) and cB/2 < g * (s, αt) < 3cB (from the support of ψ). Observe that these imply that c < 4AB −1 , otherwise D(α; b, c) is zero. Let Ξ be a parameter such that 1 ≤ Ξ ≤ 4AB −1 = C, (5.18) say. We will take Ξ to be a power of log N at the end and this explains why N needs to be larger than M, say If h(w, z) = Dw 2 + Ewz + F z 2 with D > 0, then Then we can define By (9.14) of [3], To account for the large d appearing in the above sum, we need to invoke Proposition 15 of [3]. Our final obstacle is to develop an estimate of D d (α) for small values of d. Here the modulus d is less than a power of log N, which is analogous to the