A Cubic analogue of the Friedlander-Iwaniec spin over primes

In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers by the Jacobi symbol, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $\mathbb{Z}[\zeta_{12}]=\mathbb{Z}[\zeta_3,i]$ by using the cubic residue character on the Eisenstein integers $\mathbb{Z}[\zeta_3]$. Our main theorem says that the cubic spin is equidistributed along prime ideals of $\mathbb{Z}[\zeta_{12}]$. The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $a^2+b^6$ on the Eisenstein integers.


Introduction
Friedlander and Iwaniec [1] famously showed that there are infinitely many prime numbers represented by a 2 + b 4 . Remarkable here is that numbers of this form are very sparse, that is, the number of such integers up to x is of order x 3/4 . Other similar results are Heath-Brown's proof that there are infinitely many primes of the form a 3 + 2b 3 [6], the generalization of Heath-Brown and Moroz of this to binary cubic forms [8], the extension of this by Maynard to general incomplete norm forms [11], and the result of Heath-Brown and Li [7] that the Friedlander-Iwaniec result holds also with b restricted to prime values.
If a prime p is of the form a 2 + b 4 , then p = ππ for some Gaussian prime π = b 2 + ia, so that the arithmetic in the work of Friedlander and Iwaniec really lies in Z [i]. For a Gaussian integer z = r + is with r odd, define the quadratic spin as where (s/r) 2 is the Jacobi symbol. One of the key ingredients in the proof of Friedlander and Iwaniec is to show that the spin is equidistributed along Gaussian primes [1, Theorem 2], which they obtained in the form p=r 2 +s 2 ≤x 2 ∤ r s r 2 ≪ x 1−1/77 . (1. 1) This has been generalized by Milovic to show equidistribution of (v/u) 2 over primes of the form p = u 2 − 2v 2 , which corresponds to the above with Z[ √ 2] in place of Z[i] [14,Theorem 2].
It is natural to ask if the argument can be extended to produce primes of the form a 2 +b 6 . Friedlander and Iwaniec have solved the ternary divisor problem for this sequence [3], and under the assumption of the existence of exceptional Dirichlet characters they have shown that there are infinitely many primes of this form [2].
At the moment there seems to be two major obstacles to solving the problem of primes of the form a 2 + b 6 . Firstly, the sequence of integers is now too sparse to replicate the steps in [1,. The second problem is structural. Recall that the proof of (1.1) relies on the law of quadratic reciprocity. With the sequence a 2 + b 6 we end up with cubic residues which unfortunately do not satisfy a suitable reciprocity law on Z.
The second obstacle can be overcome if we extend the whole set-up from Z to the Eisenstein integers Z[ζ 3 ], where the cubic residue character does satisfy a reciprocity law (see Lemma 4). Unfortunately the first issue remains and we are not able to detect primes of Z[ζ 3 ] of the form a 2 + b 6 with a, b ∈ Z[ζ 3 ] (see Section 9 for more details). However, we can still obtain the analogue of (1.1) in this situation and thus make partial progress on this problem. The Gaussian integers now correspond to the ring Z[ζ 12 ] = Z[ζ 3 , i] of integers of the twelfth cyclotomic extension, since the relative norm is N Q(ζ 12 )/Q(ζ 3 ) (r + is) = r 2 + s 2 for r, s ∈ Q(ζ 3 ).
We say that z ∈ Z[ζ 12 ] is primary if z ≡ ±1 (mod 3). For any (z, 3) = 1 there exists a unit µ such that µz is primary. For a primary number z = r + is ∈ Z[ζ 12 ] we define the cubic spin Similarly as in [1,Theorem 2], the exponent 1/143 is not the best that could be obtained and we have opted for simplicity in the proof over optimality.
The above cubic spin and the spin of Friedlander and Iwaniec [1] should not be confused with the spin of a prime ideal as defined by Friedlander, Iwaniec, Mazur, and Rubin [4].
Note that for a given prime p = (r + is) all of its Galois conjugates appear in the sum, which means that the sum is real. Indeed, for r + is primary we have σ∈Gal(Q(ζ 12 )/Q) [σ(r + is)] 3 3 . Note also that for π = r 2 + s 2 with r + is primary we have by cubic reciprocity (Lemma 4) since r + is being primary implies that r and π are primary in Z[ζ 3 ]. Thus, our main theorem implies that r is a cube modulo primes π asymptotically one third of the time (to prove this, expand 1 r≡t 3 (mod π) = (1 + [r/π] + [r/π] 2 )/3 and note that [p] 2 3 = [p] 3 ). Corollary 2. We have N Q(ζ 3 ) π≤x π=r 2 +s 2 , r≡±1 (mod 3), 3|s Remark 1. Since we prove that [a] 3 is independent of the choice of the primary generator z = r + is, this means by (1.2) that for primes π = r 2 + s 2 the property that r ≡ t 3 (mod π) is independent of the representation π = r 2 + s 2 where r + is is primary, so that the sum in the above corollary is well-defined. The corollary may be viewed as an approximation to the problem of primes of the form t 6 + s 2 on Z[ζ 3 ] -instead of r being a perfect cube, it is a cube modulo π = r 2 + s 2 .
For any integer n ≥ 1 define For rational primes we get the following corollary of Theorem 1 (the error term we get from the proof of Theorem 1 is actually O ε (x 1−1/142+ε ) so that the same error term holds for the corollary below).
Corollary 3. We have n≤x Λ(n)λ 3 (n) ≪ x 1−1/143 . Remark 2. The proof of Theorem 1 relies mainly on the law of cubic reciprocity. Thus, it seems plausible that the result can be generalized as follows. If an algebraic number field K contains a primitive mth root of unity, then we can define the mth power residue character on K which satisfies a reciprocity law (see [13,Chapter VIII,Theorem 5.11], for instance). Given a quadratic extension L/K we can then define a spin at elements of O L . It seems plausible that the argument could be generalized to obtain equidistribution of the spin along principal prime ideals of L. We hope to attack this question in a future article. Probably some assumptions are required here. At least in the simplest case of K = Q(ζ m ) with m odd prime and L = K[i] many parts of the argument seem to generalize nicely.
Remark 3. In [1, Section 23] Friedlander and Iwaniec write "We suspect, but have not examined thoroughly, that λ(n) are related to the Fourier coefficients of some kind of metaplectic Eisenstein series or a cusp form, by analogy to the Hecke eigenvalues (16.30) which generate a modular form of integral weight." Similarly, we expect that λ 3 (n) can be interpreted in terms of automorphic forms. Working out the details of this would be useful with a view towards the generalization outlined in the previous remark.
Remark 4. We suspect that our main theorem has some applications to elliptic curves over Z[ζ 3 ] but we do not have anything particularly interesting. For example, if π = r 2 + s 2 is a prime and r ≡ t 3 (mod π), then Corollary 2 provides us with a large family of elliptic curves E : Y 2 = X 3 + 3t 2 X ± 2s with bad reduction at some large prime π = π(E).

1.1.
A brief sketch of the Friedlander-Iwaniec argument. We present here a nonrigorous sketch of the proof of (1.1) which appears in [1,. Recall that the claim is that where [r+is] 2 = (s/r) 2 is the usual Jacobi symbol. The summation needs to be restricted to odd r but let us ignore this in the notation to simplify the presentation. Then by a sieve argument (essentially Vaughan's identity) the task is reduced to bounding Type I sums and Type II sums where α w and β z are arbitrary bounded coefficients and MN = x with sufficiently flexible ranges for M and N.
For w = u + iv with (u, v) = 1, let ω ≡ −vu −1 (mod u 2 + v 2 ), where u −1 denotes the multiplicative inverse, so that ω 2 ≡ −1 (mod u 2 + v 2 ). Similarly as in [1,Section 19], for any z = r + is we define Since ω 2 ≡ −1 (mod u 2 + v 2 ), this symbol is completely multiplicative in the upper variable and is therefore a quadratic character modulo u 2 + v 2 . We also have (z/w 1 ) 2 (z/w 2 ) 2 = (z/w 1 w 2 ) 2 provided that (w 1 , w 2 ) = 1. The general multiplicativity rule in the lower part is not much more complicated, but to simplify let us pretend that the symbol is completely multiplicative also in the lower varliable.
The key lemma is [1, Lemma 20.1], which morally states that the quadratic spin is twisted multiplicative in the sense that for some simple sign factor E. The proof of this relies on quadratic reciprocity multiple times. To simplify the presentation we pretend that this holds with E = 1. Then bounding the Type I and Type II sums is reduced to (absorbing factors into the coefficients α w and β w ) For the Type I sums (see [1, Section 22]) we fix w and write by making making the change of variables s → s + ωr, where I(r) denotes an interval of length ≪ √ N . The sum over s can be bounded using the Pólya-Vinogradov inequality for short character sums, which yields a non-trivial bound for the Type I sums (1.3) in the range M ≤ x 1/3−η for any η > 0.
To handle the Type II sums (see [1, Section 21]) we use Cauchy-Schwarz to morally get Since (z/w 1 w 2 ) 2 is a quadratic character modulo |w 1 w 2 | 2 , the sum over z is very small unless |w 1 w 2 | 2 is a perfect square (at least for N large compared to |w 1 w 2 | 2 ≍ M 2 ). The part where |w 1 w 2 | 2 is a perfect square is a very narrow subset of the variables, which gives a non-trivial bound for the Type II sums. This bound can be amplified by a suitable application of Hölder's inequality and by the reciprocity (z/w) 2 = (w/z) 2 . We get a non-trivial bound for the Type II sums (1.4) in the full range x η ≪ M, N ≪ x 1−η .

1.2.
Structure of the article. The proof of Theorem 1 follows the same lines as the proof of [1,Theorem 2]. In Section 2 we recall the law of cubic reciprocity and prove that the cubic spin [z] 3 satisfies a twisted multiplicativity relation (Lemma 5). This relation is the key ingredient in all of the arguments that follow. In Section 3 we recall basic facts about Q(ζ 12 ) and show that the definition of the spin [a] 3 = [z] 3 is independent of the choice of primary generator z of a (also for this we need Lemma 5).
In Section 4 we use Buchstab's identity to obtain a decomposition of the prime sum into sums of Type I and Type II. We could also use similar arguments as in [1] to this end. In Section 5 we explain how to choose unique primary generators for the ideals of a in a consistent manner.
After these steps the arugment is essentially same as in [1] with only minor modifications. In Sections 7 and 8 we compute the Type I and Type II sums, respectively, which by Section 4 completes the proof of Theorem 1. For this we need a version of the Poisson summation on Z[ζ 3 ], which is given in Section 6. The reason why the exponent in Theorem 1 is worse than that in [1,Theorem 2] is solely because in the Type I sums we essentially get a contribution from the error term in a lattice point counting problem on Z[ζ 3 ].
Lastly, in Section 9 we illustrate non-rigorously how the cupic spin arises from the problem of primes of the type α 2 + β 6 on Z[ζ 3 ]. The arguments follow the same lines as in [1]. We also explain why the density issue prevents us from completing the goal of detecting primes of this form.

1.3.
Notations. For functions f and g, we write f ≪ g or f = O(g) if there is a constant C such that |f |≤ C|g|. The notation f ≍ g means g ≪ f ≪ g. The constant may depend on some parameter, which is indicated in the subscript (e.g. ≪ ε ). We write f = o(g) if f /g → 0 for large values of the variable. For variables we write n ∼ N meaning N < n ≤ 2N.
For two functions f and g with g ≥ 0, it is convenient for us to denote f (N) ≺≺ g(N) if f (N) ≪ ε N ε g(N). A typical bound we use is τ k (n) ≺≺ 1, where τ k is the k-fold divisor function. For multivariable functions such as sums over two variables we write We say that an arithmetic function f is divisor bounded if |f (n)|≪ τ k (n) for some k.
For a statement E we denote by 1 E the characteristic function of that statement. For a set A we use 1 A to denote the characteristic function of A.
We let e(x) := e 2πix and e q (x) := e(x/q) for any integer q ≥ 1. We abbreviate modular arithmetic such as a ≡ b (mod c) by a ≡ b (c). For any (a, b) = 1 we let a −1 (b) denote the multiplicative inverse, so that aa −1 ≡ 1 (b).
We abbreviate the norm maps as follows. For any a = a (1) + a (2)

1.4.
Acknowledgements. I am grateful to my supervisor Kaisa Matomäki for helpful comments and support. I also thank Joni Teräväinen for comments on an early version of the manuscript. During the work the author was funded by UTUGS Graduate School. Part of the article was also completed while I was working on projects funded by the Academy of Finland (project no. 319180) and the Emil Aaltonen foundation.

Cubic reciprocity
In this section we recall basic properties of the Eisenstein integers Z[ζ 3 ] and the cubic residue character (cf. [10,Chapter 7], for instance). We also prove a twisted multiplicativity rule for the cubic spin [z] 3 (Lemma 5) by using the cubic reciprocity law. To simplify the notation we will abbreviate modular arithmetic such as Any rational prime p ≡ 1 (3) splits as p = ππ for a prime π ∈ Z[ζ 3 ]. Then for any a ∈ Z[ζ 3 ], π ∤ a, we have by Fermat's Little Theorem Since p ≡ 1 (3), we see that a (p−1)/3 ≡ ζ k 3 (π) for some k ∈ {0, 1, 2}, so that we may define the cubic residue character modulo π a π 3 := ζ k 3 .
For any non-zero λ ∈ Z[ζ 3 ] we have a unique factorization It is then clear that this is completely multiplicative in both variables. From here on we simplify notations by ignoring the subscript 3, that is, we write For any a, b ∈ Z[ζ 3 ] we write (a, b) = 1 if a and b are coprime. For any (a, b) = 1 we let ε(a, b) denote the cubic root of unity such that Note that for any a, b, c ∈ Z[ζ 3 ] with (a, bc) = 1 we have multiplicativity in the sense that ε(a, bc) = ε(a, b)ε(a, c) and ε(bc, a) = ε(b, a)ε(c, a).
By [10, Theorem 7.8] we have the following cubic reciprocity law (which can also be found in [13, Chapter VIII, Example 5.13]).
We will extend this definition to ideals of Z[ζ 12 ] in Section 3. We say that Analogously to the Dirichlet symbol defined in [1, Section 19], we define Since ω 2 ≡ −1 (u 2 + v 2 ), this is completely multiplicative in the upper variable, so that this is an extension of the character [r/( Similarly as [1, Lemma 20.1] follows from the quadratic reciprocity, the cubic reciprocity law implies that the cubic spin [z] is multiplicative up to the symbol (z/w). The analogous result on Z[ Proof. First note that since w and z are primary, it follows that all of u, r, wz, and ur−vs are primary, and 3|v and 3|s. If (u, v) = 1 or (r, s) = 1, then the claim is trivial since then both sides vanish. Assume then that (u, v) = (r, s) = 1. Let r 0 = (r, v) be primary, and denote r = r 0 r 1 , v = r 0 v 1 , so that (r 1 , v 1 ) = 1 (since r is primary we have (3, r 0 ) = 1 and we may pick a primary representative for r 0 ). By using s ≡ ur We now compute the above three factors separately.
We have s r 0 by Lemma 4 since ur 1 − v 1 s and r 1 are both primary. By a similar argument the second factor is For the third factor, since u, u 2 + v 2 , and ur − vs are primary, we have by two applications of Lemma 4 where λ is primary. Then, since 9|v 1 s, we see from the supplementary laws in Lemma 4 (note also that ε(λ, ur 1 − v 1 s) = 1 = ε(λ, ur 1 ) since ur 1 − v 1 s and ur 1 are primary) since r 0 and u are primary. Hence, we get Remark 5. It is perhaps surprising that our Lemma 5 is much simpler than the corresponding results in the quadratic case [1, Lemma 20.1] and [14,Proposition 8], where the equalities hold only up to a simple factor. The main reason for this seems to be that −1 is always a cube so that [−1/a] = 1 for all a ∈ Z[ζ 3 ], a fact that we used in the proof.
We will abbreviate the norm maps as follows. For any a = a (1) + a (2) The symbol (z/w) in Lemma 5 is completely multiplicative in the upper variable. Similar to [1,Sections 19 and 21], to handle Type II sums we need a multiplier rule in the lower variable also, which is given by the following. 12 ] be primary primitive and let ζ = r + is ∈ Z[ζ 12 ] be primary. Set For z = r+is primary primitive we get from Lemma 5 a reciprocity law (z/w) = (w/z) for any primary primitive w. Note that by definition w 1 w 2 2 /d is primary primitive. Therefore, we get by reciprocity (note that r 2 + s 2 and d are primary) Remark 6. In [1, Section 19] we have for primary z = r + is, w = u + iv ∈ Z[i] with w primitive In our case the middle equality does not hold but we have for z = r+is, w = u+iv ∈ Z[ζ 3 ] primary For every z ∈ Z[ζ 12 ] coprime to 3 there exists a unit µ such that µz ≡ 1 (3) (cf. [10,Exercise 7.4]). The next lemma shows that the subgroup of primary units is {±(iε 6 0 ) k : k ∈ Z}.
Since every number coprime to 3 is congruent to some unit modulo 3, this implies the claimed structure for (Z   (3), so that none of the values in the table are ≡ ±1, ±i (3).
Proof. There is some unit µ such that z = µz ′ . Since z and z ′ are primary, also µ must be primary. By Lemma 7 we see that µ = ±(iε 6 0 ) k for some k ∈ Z. Hence, by Lemma 5 , and similarly we see that where ν(z) depends only on the residue class z (3). We will not need it in the following but it might be interesting to give a simple closed formula for ν(z).
Remark 8. It is natural that we need the reciprocity laws to prove that the spin is welldefined on ideals. After all, the cubic reciprocity can be restated as a transformation rule for [z] under multiplication of z by a root of unity. This is because for z = r + is which by considering z 1 = r + i(1 − ζ 3 ) and z 2 = r + i3 covers also the supplementary laws.

Sieve argument
In this section we prove Theorem 1. We apply a sieve argument in Z[ζ 12 ] to decompose our sum into Type I and Type II sums. The argument is essentially same as in Harman's sieve method [5]. We could use [4, Proposition 5.2] directly but we give our on proof based on Buchstab's identity since it does not take much effort to inclue it here.
For two functions f and g with g ≥ 0, it is convenient for us to denote f (N) ≺≺ g(N) if f (N) ≪ ε N ε g(N). A typical bound we use is τ k (n) ≺≺ 1, where τ k is the k-fold divisor function. For multivariable functions such as sums over two variables we write We say that an arithmetic function f is divisor bounded if |f (n)|≪ τ k (n) for some k.
For the sieve we require the following arithmetic information, which is proved in Sections 7 and 8.   Remark 9. Note that [a] = 0 if a is not primitive. Thus, we may assume that the coefficients α and β in the above are supported on primitive ideals.
Remark 10. Our Type I information is very weak but this is compensated by the fact that the Type II bound is non-trivial as soon as M ≫ x η or N ≫ x η . Proof of Theorem 1. Let Z = x γ for some γ ∈ (0, 1/2) which we will optimize later on. By Buchstab's identity where the error term E consists of that part in S 2 (A) where the implicit variable n is divisible by another prime ideal of same norm N 12 (p) (we could also handle this part by fixing a complete ordering for prime ideals p but then we would later have to remove cross-conditions coming from this). We have trivially For the second sum we have (writing n = p 1 · · · p k ) by Proposition 10 once we remove the cross-conditions N 12 (p) < N 12 (p 1 ) and N 12 (pp 1 · · · p k ) ∼ x by Perron's formula (cf. for instance [5,Chapter 3.2], this works essentially the same in our situation since we apply it to the real quantities N 12 (p j )). Here the error termẼ consists of the part where pp 1 · · · p k is divisible by a square, so that we haveẼ ≪ xZ −1 by a similar argument as with E above.
For the first sum S 1 (A) we use the Möbius function to expand the condition (n, P 12 (Z)) = 1 to get For the first sum we apply Proposition 9 to get For the second sum we write d = p 1 · · · p k for N 12 (p 1 ) ≤ · · · ≤ N 12 (p k ) < Z. Since there are at most four prime ideals of the same norm and d is square free, by the greedy algorithm there is a unique ℓ ≤ k such that d = d 1 d 2 with where d ′ 1 := p 1 · · · p ℓ−j if j is the largest number such that N 12 (p ℓ−j+1 ) = N 12 (p ℓ ) (that is, we apply the greedy algorithm for groups of at most four prime ideals of the same norm). Hence, the second sum S 12 (A) can be partitioned as The cross-conditions N 12 (p ℓ ) < N 12 (p ℓ+1 ) and N 12 (d 1 )N 12 (d 2 n) ∼ x can now be removed by Perron's formula, so that by Proposition 10 we get Combining the bounds (4.1), (4.2), and (4.3), and choosing Z := x 5/71 (note that then

Fixing primary generators
For the proofs of our arithmetic information (Propositions 9 and 10) we need to fix primary generators of ideals of Z[ζ 12 ] in a consistent manner, and in such a way that the resulting conditions do not cause problems later on. Luckily fixing an embedding of Z[ζ 12 ] in C along with Lemma 7 allows us to do just this. We choose the embedding which maps ζ 12 to e 2πi/12 ∈ C. For any z = r +is ∈ Z[ζ 12 ], r, s ∈ Z[ζ 3 ] we let |z|:= |r +is| denote the norm of the complex number r + is ∈ C. Note that then Lemma 11. For every ideal a coprime to 3 there exists a unique generator z = r + is ∈ Z[ζ 12 ] of a such that z ≡ 1 (3) and Furthermore, for such a z = r + is we have |r|, |s| ≪ |z|.
In the summations we will denote this condition by ∧ , so that we may write, for example,

Truncated Poisson summation formula on Z[ζ 3 ]
In the proofs of Propositions 9 and 10 we will need a version of the Poisson summation formula on Z[ζ 3 ]. For the lemma we fix an embedding identifying ζ 3 with e 2πi/3 ∈ C, so that any element of Z[ζ 3 ] is viewed as a complex number. For z ∈ C denote is an additive character of Z[ζ 3 ]/qZ[ζ 3 ] (hereq denotes the complex conjugate). (1) , α (2) ). Then for any C, ε > 0 with
We make the change of variables so that xq + β is mapped to We get by making the translation (denoting z 0 := (x 0 + ζ 3 y 0 )) x → x + 1 KN 3 (q) 1/2 z 0q , that is, For all h 1 , h 2 we have the trivial estimate |c h 1 ,h 2 |≤ c 0,0 ≪ G 1 (note that x → xq/N 1/2 3 (q) is a rotation in C so that c 0,0 is independent of q). As usual, h 1 = h 2 = 0 gives us the main term, since by another double application of Poisson summation For |h 1 |> H or |h 2 |> H we can iterate integration by parts to show that the contribution from this part is ≪ C,ε K −C .

Type I sums
Using the notation of Section 5, for any primary w ∈ Z[ζ 12 ] define In this section we show (analogously to [1, Proposition 22.1]) the following proposition, which implies Proposition 9.
Proposition 13. We have For the proof we need a generalization of the Pólya-Vinogradov estimate for short character sums. Unfortunately on Z[ζ 3 ] we do not quite have the usual square-root bound since the estimate relies on the error term in counting lattice points; on a number field of degree d and a primitive non-principal character χ modulo q Landau's generalization gives However, for a smoothed version we have the Pólya-Vinogradov estimate in the usual form. Also, since we require the bound not only for short sums near 0 but in more general small sets, the smoothed version is more convenient for us. Unfortunately in our application we need to transition from a smoothed version to a sharp cut-off, which causes us to lose a power of x compared to the results in [1].
Proof of Proposition 13. By Lemma 5, if q := N 12/3 (w), then for some ω 2 + 1 ≡ 0 (q) denoting z = r + is, r, s ∈ Z[ζ 3 ] Shifting s by ωr to get [(r + ω(s + rω))/q] = [ω/q][s/q], we see by Lemma 11 that where I(r) is the domain in C defined by the conditions (so that I(r) is contained in the annulus |s − r(i − ω)| ≍ N 1/4 ). If rq is a perfect cube we use the trivial bound. In the remaining part we use a smooth finer-than-dyadic decomposition. Let K > 1 be a parameter to be optimized later. There exists a smooth partition of unity for a certain fixed compactly supported C ∞ -smooth function F . By scaling and squaring we get a smooth partition . Using this we can to partition I(r) into smoothed boxes of side-length ≍ K, obtaining ≪ N 1/2 /K 2 such boxes weighted with functions G K as in Lemma 14. On the boundary of I(r) we use the trivial bound K 2 and the fact that there are ≪ N 1/4 /K boxes that intersect with the boundary, so that assuming that rq is not a perfect cube. Hence, Choosing K = N 1/6 N 3 (q) 1/6 ≪ N 1/4 to optimize the bound we get Note that if K = N 1/6 N 3 (q) 1/6 ≫ N 1/4 then this bound is trivial.

Type II sums
The goal of this section is to prove Proposition 10. It turns out that the arguments in [1, Section 21] generalize to our case essentially verbatim. Our slightly better exponent is thanks to a small technical refinement (using a smooth weight) which is actually necessary in our situation.
By using Lemma 5 and the notation of Section 5, the claim is reduced to bounding (absorbing Proof. We may assume that N > M 4+η for some small η > 0, since otherwise by a trivial bound By Cauchy-Schwarz and by Lemma 11 we have Recall the notation of Lemma 6, that is, , e := (w 1 , σ(w 2 2 )), and d := N 12/3 (e), where σ is the conjugation σ(a+ib) = a−ib. By Lemma 6 we have (writing x := r 0 −ωs 0 , y := r 0 + ωs 0 ) unless both of d and q 1 q 2 2 /d are perfect cubes in which case we get To see this, recall that (3, q) = 1 and note that [y/(q 1 q 2 2 /d)] is a cubic character modulo q/d. Hence, we get
Remark 11. Note that the modulus q 1 q 2 2 being a cube is morally the same as q 1 q 2 being a square which explains why we get the same term M 1/2 N as in [1,Lemma 21.2].
Lemma 15 is non-trivial as soon as N ≫ M 4 ≫ 1. Similarly as in [1], we now use Hölder's inequality to extend this range so that we can handle the range M ≍ N. We get whereQ is of similar form as Q except that α w is replaced by some coefficientsα w with |α w |≤ 1 and β z is replaced by the divisor bounded coefficient Remark 12. The reason our bound is superior to that in [1, Proposition 21.3] is that we used the smooth function F √ N (r) instead of a sharp cut-off N 3 (r) ≤ √ N in the proof of Lemma 15. This allows us to improve the term M 2 N 3/4 appearing in Lemma [1, Lemma 21.2] to M 5 . Obviously the same refinement can be implemented to improve their bound to the same form. 9. Connection to primes of the form It is tempting to ask if the method of Friedlander and Iwaniec can be extended to produce primes of the form a 2 + b 6 on Z (cf. [12,Remarque 4.20]). Unfortunately there seems to be two large obstacles to this. Firstly, the sequence is too sparse for replicating the steps in [1,. The second problem is structural -the proofs in [1] rely on the law of quadratic reciprocity in multiple places, while for cubic residues we do not have a suitable reciprocity law on Z. To mend this we need to transfer the whole set-up to Z[ζ 3 ]. Unfortunately the first problem persist (cf. the paragraph around (9.4) below).
In this section we explain how the sum in Theorem 1 arises if we consider primes of the form α 2 + β 6 on Z[ζ 3 ], which was the original motivation for this manuscript. All of the discussion presented here is non-rigorous and for the sake of illustration we omit all of the technical issues that arise in [1]. The argument follows exactly the same lines as in [1].
The number of ideals a with N 12 a ≤ x that have such a generator β 3 + iα is which is very sparse (the inability of the Friedlander-Iwaniec method to handle sets of density < x −1/3 is also noted by Helfgott [9]). Due to this we are not able to handle the Type II sums, as we shall see soon below (see (9.4)). However, it should be possible to obtain an approximation to this problem by considering primes of the form α 2 + λ 2 β 6 where λ runs over elements of small norm N 3 (λ) ≤ x δ , to show that for some fairly small δ > 0 we get a lower bound of the correct order of magnitude for the number of such primes. For this the sieve of Friedlander and Iwaniec needs to be replaced by Harman's sieve (see [11] for a version of this on Number fields).
Using a sieve argument the main problem is to handle Type II sums of the form where MN = x, and α and β are bounded coefficients with β behaving like a Möbius function in terms of a Siegel-Walfisz type condition. The goal then is to show that S 1 ≪ C x 2/3 log −C x. We now pick a primary generator z of n according to Section 5, and then pick a generator σ(w) of m such that y = σ(w)z. The cross-condition N 12 (w) 1/4 N 12 (z) 1/4 ≤ |z||w|< N 12 (w) 1/4 N 12 (z) 1/4 |ε 0 | 6 (9.2) 1 (∆).
Unfortunately here we run into a problem with the sparseness of our sequence (note that also in [1] for the Type II sums this is the only part of the argument affected by the sparsity). We would like to apply Poisson summation (Lemma 12) to evaluate the smoothed sum over β 1 , β 2 . However, due to the diagonal contribution (9.3) we have essentially while the length of the sum is N 3 (β j ) ≪ x 1/6 (9.4) which is just narrowly too short (Poisson summation becomes ineffective if the length of the sum is less than square root of the size of the modulus). Due to this we are not able to evaluate the sum in any range of N. If we consider the aforementioned approximation version of the problem with α 2 + λ 2 β 6 for N 3 (λ) ≤ x δ , then the diagonal part gives a restriction N ≫ x 1/3−2δ/3 log C x which gives some room to work with. We have been able to evaluate the sum over β 3 1 λ 1 z 2 ≡ β 3 2 λ 2 z 1 (∆) in some ranges using a large sieve argument similar to that in Heath-Brown and Li [7] (although the argument required here is much more intricate).
Remark 14. Since the argument falls short barely, there is some hope that with a delicate estimate we could handle Type II sums in some very narrow non-trivial range for N = x 1/3+o (1) . This would suffice to break the parity barrier, that is, to show that there are infinitely many of α 2 + β 6 are a product of exactly two primes (one of size M and the other of size N).
Assuming that the sum over β 1 , β 2 could be computed, then the main term is essentially (up to a multiplicative factor and a smooth coefficient, and ignoring the fact that γ j may have common factors with ∆) (for the approximate version of the problem the congruence is ω 3 ≡ z 2 λ 1 z −1 1 λ −1 2 (∆), which is morally the same). Here we need to show only a little bit of cancellation, that is, S 5 ≪ N 2 log −C x. To evaluate the sum over cubic roots we make use of the Chinese Reminder Theorem and the cubic residue character to get (assuming ∆ is square-free, primitive, and ignoring the fact that 3|∆) Similarly as in [1, Section 10], we now split the sum into three parts U + V + W according to the size of δ 1 δ 2 , where in U we have N 3 (δ 1 δ 2 ) ≪ log C N, in W we have N 3 (δ 1 δ 2 ) ≫ N 3 (∆) log −C , and V is the remaining middle part. For U we get the required cancellation from the β z , which look like a Möbius function, by using a suitable Siegel-Walfisz type bound.
For V we have not checked in detail but we expect that the large sieve -type arguments in [1, generalize to our case.
For W the generic case is δ 1 δ 2 = ∆. To handle this we need the following analogue of [1, Lemma 17.1], which we will prove at the end of this section.