The initial correlation estimate

cp(P1, ..., PJ) = 1−J

p+Od,D,J

1 p²

when p is good. In practice we shall need a more sophisticated version of this fact, when certain complex weightsp^−Pj∈Szj are inserted into the right-hand side of (77); see Lemma 10.4.

10. The initial correlation estimate

To prove (73) and (74) we shall need the following initial estimate which handles general polynomial averages ofν over large scales, but with an error term that can get large if there are many “bad” primes present. More precisely, this section is devoted to proving the following result.

Proposition 10.1. (Correlation estimate) Let P1, ..., PJ∈Z[x1, ...,xD]have degree Remark 10.2. We only expect this estimate to be useful when the number of bad primes is finite. This is equivalent to requiring that the polynomials P1, ..., PJ are co-prime, and each one is linear in at least one variable. Because the sumP

p1/pis (very slowly) divergent (see (110)), the last error term can be unpleasantly large on occasion, but in practice we will be able to introduce averaging over additional parameters which will make the effect of the error small on average, the point being that the setsPtandPb

are generically rather small. The radius R^4J+1 is not best possible, but to lower it too much would require some deep analytical number theory estimates such as the Bombieri–

Vinogradov inequality, which we shall avoid using here. The upper boundR^logR(which was not present in earlier work) can also be lowered, but for our purposes any bound which is subexponential inR will suffice.

Remark 10.3. All the primes p<w will be bad (but not terrible); however, their contribution will be almost exactly canceled by the φ(W)/W term present in ν, and we do not need to include them into Pb. Even a single terrible prime will cause the main term 1P_t=∅ to vanish (basically because one of the Pj(x) will now be inherently composite and so will be unlikely to have a large value ofν), which will make asymptotics difficult; however, terrible primes are no worse than merely bad primes for the purposes ofupper bounds.

Proof of Proposition 10.1. Throughout this proof we fixD,J andd, and allow the implicit constants in theO(·) ando(·) notation to depend on these parameters. We will also always assumeR to be sufficiently large depending onD,J andd.

We expand the left-hand side using (72) as φ(W)

Here of course lcm(·) denotes least common multiple. Note that the presence of the µ andχfactors allows us to restrictm₁, ..., m⁰_J to be square-free and at mostR.

The first task is to eliminate the role of the convex body Ω, taking advantage of the large inradius assumption. LetM:=lcm(m1, m⁰₁, ..., mj, m⁰_j). ThusM is square-free and at mostR^2J. The functionx7!1_lcm(mj,m⁰

j)|W Pj(x)+bis periodic with respect to the lattice M·Z^D, and thus can be meaningfully defined on the groupZ^D_M. Applying Corollary C.3 (recalling that Ω is assumed to have inradius at leastR^4J+1), we thus have

E_x∈Ω∩ZD Let us first dispose of the error termO(1/R^2J+1). The contribution of this term to (79) can be crudely bounded byO(R^−2J−1), and so the contribution of this term to (79) can be crudely bounded by

Thus we may discard this error, and reduce to showing that φ(W)

Observe, from the Chinese remainder theorem, thatαis multiplicative, so that if lcm(m_j, m⁰_j) =Y squarefree, then therj’s are either 0 or 1, and we simplify further to

α_lcm(m1,m⁰

1),...,lcm(mJ,m⁰_J)=Y

p

cp((W Pj+b)r_j=1),

where the local factors cp are defined in Definition 9.1, and the dummy variable j is ranging over all the indices for whichrj=1. Also note that themj andm⁰_j are bounded by R, and thus we may certainly restrict the primes p to be less than R^logR without difficulty.

The next step is to replace theχ factors by terms which are multiplicative in the mj andm⁰_j. Sinceχis smooth and compactly supported, we have the Fourier expansion

e^xχ(x) = Z ∞

−∞

ϕ(ξ)e^−ixξdξ (81)

for some smooth, rapidly decreasing functionϕ(ξ) (so in particularϕ(ξ)=OA((1+|ξ|)^−A) for anyA>0). For future reference, we observe that (81) and the hypotheses on χ will imply the identity (see [19, Lemma D.2], or the proof of [32, Proposition 2.2]).

We follow the arguments in [36], [20], [32] and [19], except that for technical reasons (having to do with the terrible primes) we will be unable to truncate the ξ variables.

From (81) we have where we adopt the notational conventions

z_j:=1+ξ_j

logR and z⁰_j:=1+ξ_j⁰ logR. Our task is thus to show that

φ(W)

The left-hand side can be factorized as Z ∞

whereEp=Ep(z1, ..., z⁰_J) is the Euler factor

Note that if thez_j andz_j⁰ were zero, then this would just be the complementary factor

¯

c_p(W P₁+b, ..., W P_J+b) defined in Definition 9.1; see (77). Of course, z_j and z_j⁰ are non-zero. To approximateE_p in this case, we introduce the Euler factor

E_p⁰ := Y

j∈[J]

(1−1/p^1+zj)(1−1/p^1+z0j) 1−1/p^1+zj+zj0 . Note thatE⁰_p never vanishes.

Lemma 10.4. (Euler product estimate) We have Y

Proof. For p<w, we directly compute (since w is slowly growing compared to R, andW P_j+bis equal modulopto b, which is coprime top) that and hence (again becausewis slowly growing)

Y

Thus it will suffice to show that Y

Forpterrible, Lemma 9.5 gives the estimate E_p1

p1 pE_p⁰,

and so it will suffice to show that Y

Forpbad but not terrible, Lemma 9.5 gives the crude estimate pbad but not terrible

E_p

Thus it suffices to show that

Y

p(1+O(1/p²)) is convergent, andwgoes to infinity, it in fact suffices to show that

for all good primes larger thanw. But this easily follows from Lemma 9.5 and Taylor expansion (recall that the real parts ofzj andz⁰_j are 1/logR>0).

Now we use the theory of the Riemann zeta function. From (113) we have that Y On the other hand, from (108) we have

1 for any realξ, and hence

Y Applying Lemma 10.4, we conclude that

φ(W)

Thus, by the triangle inequality, to show (83) it will suffice to show that

But the first estimate follows from (82), while the second estimate follows from the rapid decrease ofϕ. This proves Proposition 10.1.

To illustrate the above proposition, let us specialize to the case of monic linear polynomials of one variable (this case was essentially treated in [18] and [14]).

Corollary 10.5. (Correlation condition) Let h₁, ..., h_J be integers, and let I⊂R be an interval of length at least R^4J+1. Then

E_x∈I∩Z Y

Proof. Apply Proposition 10.1 with P_j(x):=x+h_j and Ω:=I. Then there are no terrible primes, and the only bad primes larger than w are those which divide h_j−hj⁰

for some 16j <j⁰6J.

This can already be used to derive the “correlation condition” in [18]; a similar aplication of Proposition 10.1 also gives the “linear forms condition” from that paper.

We will also need the following variant of the above estimate.

Corollary10.6. (Correlation condition on progressions) Let h₁, ..., h_J be integers, q>1,a∈Zq and I⊂Rbe an interval of length at least qR^4J+1. Then

Proof. Apply Proposition 10.1 withPj(x):=(qx+a)+hj and with Ω :={x∈R:qx+a∈I}.

Then the bad primes are those which divideh_j−hj⁰ or which divideq. (There are terrible primes ifaandqare not coprime, but this will not affect the upper bound. One can get more precise estimates as in Corollary 10.5, but we will not need them here.)

This in turn implies the following result.

Corollary 10.7. (Correlation condition with periodic weight) Let h1, ..., hJ be integers, q>1,I⊂R be an interval of length at least qR^4J+1 and f:Z!R⁺ be periodic modulo q (and thus definable on Zq). Then

Ex∈I∩Zf(x) Y

j∈[J]

ν(x+hj) =OD,J,d

(Ey∈Z_qf(y)) exp

OD,J,d

X

p∈Pb