AVERAGES OF EULER PRODUCTS, DISTRIBUTION OF SINGULAR SERIES AND THE UBIQUITY OF POISSON DISTRIBUTION

Transcription

1 AVERAGES OF EULER PRODUCTS, DISTRIBUTION OF SINGULAR SERIES AND THE UBIQUITY OF POISSON DISTRIBUTION EMMANUEL KOWALSKI Abstract. We discuss in some detail the general roblem of comuting averages of convergent Euler roducts, and aly this to examles arising from singular series for the k-tule conjecture and more general roblems of olynomial reresentation of rimes. We show that the singular series for the k-tule conjecture have a limiting distribution when taken over k-tules with distinct) entries of growing size. We also give conditional arguments that would imly that the number of twin rimes or more general olynomial rime atterns) in suitable short intervals are asymtotically Poisson distributed.. Introduction Euler roducts over rimes are ubiquitous in analytic number theory, going back to Euler s roof that there are infinitely many rime numbers based on the behavior of the zeta function ζs) as s. As defining L-functions of various tyes, Euler roducts are articularly imortant, and their roerties remain very mysterious. In this aer, we consider the issue of the average or statistical behavior of another imortant class of Euler roducts, the so-called singular series, arising in counting roblems for certain atterns of rimes singular series also occur in many roblems of additive number theory or diohantine geometry, but we do not consider these here). The first tye of rime atterns are the rime k-tules, which are the subject of a famous conjecture of Hardy and Littlewood. Let k be an integer and let h = h,..., h k ) be a k-tule of integers with h i for all i. Let then πn; h) = {n N n + h i is rime for i k} be the counting function for rimes reresented by this k-tule; note that, for instance, h =, 3) leads to the function counting twin rimes u to N. For any rime number, let ν h) denote the cardinality of the set {h,..., h k } mod ) of the reductions of the h i modulo. Note that ν h) mink, ) for all, and that if we assume as we now do) that the h i s are distinct, then ν h) = k for all sufficiently large. 200 Mathematics Subject Classification. P32, N37, K65. Key words and hrases. Singular series, rime k-tules conjecture, Bateman-Horn conjecture, limiting distribution, Euler roduct, moments, Poisson distribution.

2 2 EMMANUEL KOWALSKI The singular series associated with h is defined as the Euler roduct.) Sh) = ν ) h) ) k = ν h) ) ) k which is absolutely convergent as will be checked again later; here and throughout the aer, as usual, is restricted to rime numbers). The significance of this value is found in the Hardy-Littlewood rime k-tule conjecture originally stated in [HL]), which states that we should have N.2) πn; h) = Sh) + o)), log N) k as N +, and in articular, if Sh) 0, there should be infinitely many integers n such that n + h,..., n + h k are simultaneously rime. Of course, if k 2, this is still comletely oen, but let us mention that from sieve methods, it follows that πn; h) 2 k N k! + o))sh) log N) k as N + see, e.g., [IK, Th. 6.7] or [HR, Ch. 4, Th. 5.3]), showing that the singular series does arise naturally. Also some other reviously inaccessible additive roblems with rimes, related to counting arithmetic rogressions of fixed length) of rimes are currently being attacked with striking success by B. Green and T. Tao see [GT]). More generally, one considers olynomial rime atterns. First, a finite family f = f,..., f m ) of olynomials in Z[X] of degrees degf j ) is said to be rimitive if the f j are distinct, and each f j is irreducible, has ositive leading coefficient, and the gcd of its coefficients is. If f is rimitive, we say that an integer n is an f-rime seed if f n),..., f m n) are all ositive) rimes. Then we denote by πn; f) = {n N n is an f-rime seed} for N the counting function for those rime seeds. Moreover, let m egf) = degf j ). j= A generalization of the k-tule conjecture, due to Bateman and Horn [BH], states that.3) πn; f) egf) Sf) N log N) m, as N +, if Sf) 0, where 2.4) Sf) = ν f) ) ) m, with ν f) being now the number of x Z/Z such that f j x) = 0 for some j, j m. The qualitative version of which is due to Schinzel [S]. 2 Here, excet in the secial case where all fj are linear, the singular series Sf) is not absolutely convergent see below for more details on this; the roblem is that ν f) is only equal to m on average over, and not for all large enough, excet if each f j is linear); the roduct is thus defined as the limit of artial roducts over rimes y.

3 AVERAGES OF EULER PRODUCTS 3 The Hardy-Littlewood conjecture for a k-tule h is equivalent with this conjecture for the rimitive family f = X + h,..., X + h k ) for which ν h) as defined reviously does coincide with ν f). Our goal is to study various averages of singular series, for which there is undoubted arithmetic interest. A result of Gallagher [Ga] states that.5) lim h + h k Sh) =, for any fixed k, as h +, where h = max h i and restricts to k-tules with distinct comonents. This roerty was used by Gallagher himself to understand the behavior of rimes in short intervals see also the recent work by Montgomery and Soundararajan [MS]), and it is also imortant the remarkable results of Goldston, Pintz and Yıldırım concerning small gas between rimes see [GPY] or the survey [K]). Our first question is to ask about finer asects of the distribution of Sh). To aly the method of moments, we first rove the following: Theorem.. Let k be fixed. For any comlex number m C with Rem) 0, there exists a comlex number µ k m) such that lim h + h k Sh) m = µ k m). Moreover, for m, k both integers, we have the symmetry roerty.6) µ k m) = µ m k) ; in addition, we have µ m) = for all integers m, and hence µ k ) = for all k. The last statement µ k ) = ) is of course Gallagher s theorem.5); our roof is not intrinsically different, but maybe more enlightening. These results are in fact quite straightforward, and only the final symmetry in k and m is maybe surrising. However, its origin is not articularly mysterious: it is a local henomenon, and it can be guessed from.2) by a formal comutation. We will also find estimates for the size of the moments which are good enough to imly the existence of a limiting distribution of Sh) for k-tules k fixed): Theorem.2. Let k be fixed. There exists a robability law ν k on R + = [0, + [ such that Sh), for h with h h and h +, becomes equidistributed with resect to ν k, or equivalently lim h + h k for any bounded continuous function on R. fsh)) = ft)dν k t) R + The second question we exlore is the generalization to other rime atterns of the result of Gallagher based on.5)) that shows that a uniform version of the rime k-tule conjecture imlies that for a fixed λ > 0, the distribution of

4 4 EMMANUEL KOWALSKI πx + λ log x) πx) is close to a Poisson distribution of arameter λ as x +, i.e., it imlies that λm.7) {n N πn + h) πn) = m} e λ, as N +, N m! for any integer m 0. It turns out that, indeed, under a general uniform version of the Bateman-Horn conjecture, for any fixed rimitive family f, the number of f- rime seeds in short intervals of fair length i.e., intervals around n in which.3) redicts that, on average, there should be a fixed number of f-rime seeds) always follows a Poisson distribution. As for the symmetry roerty of the higher moments for the singular series related to k-tule conjecture, this turns out to deend rimarily on local identities, but we found this rigidity of atterns to be quite surrising at first sight. Precisely: Theorem.3. Assume that the Bateman-Horn conjecture holds uniformly for all rimitive families with non-zero singular series, in the sense that.8) πn; f) = egf) Sf) N cf) ε )) log N) m + O log N holds for all rimitive families f, all ε > 0, and all N 2, where cf) = Hf j ), Ha 0 + a X + + a d X d ) = max a i, i j m and the imlied constant deends at most on the degrees of the elements of f and on ε. Let f be a fixed rimitive family with Sf) 0. For N, let δn, f) = egf) Sf) log N)m. Then for any λ > 0 and any integer r 0, we have λr lim {n N πn + λδn, f); f) πn; f) = r} = e λ N + N r!. In other words, for N large, the number of f-rime seeds in an interval around N of length λlog N) m is asymtotically distributed like a Poisson random variable with mean given by Sf) egf) λ. The final urose of this aer is to emhasize the fact that Theorems. and.3 are secial cases of the roblem of comuting the average of some families of values of Euler roducts, and because here the Euler roducts are absolutely convergent or almost so) the outcome is consistent with the heuristic that the -factors are indeendent random variables, so the average of the Euler roduct is the roduct of local averages. All this is a fairly common theme in analytic number theory, but our resentation is maybe more systematic than usual. The works of Granville- Soundararajan [GS] and Cogdell-Michel [CM] also resent this oint of view very successfully for values of certain families of L-functions at the edge of the critical stri, and Y. Lamzouri [La] has develoed this tye of ideas in a quite general context. Although this is not really relevant from the oint of view of singular series, we just mention that Euler roducts built of local averages still make sense inside the critical stri for many families of L-functions, and are closely related to their distribution as one can see, e.g., from the work of Bohr and Jessen [] for the Riemann zeta function). On the critical line, renormalized Euler roducts still

5 AVERAGES OF EULER PRODUCTS 5 occur in the moment conjectures for L-functions see, e.g., [KS]), although other factors conjecturally linked to Random Matrices) also aear. In the next section, we state in robabilistic terms a general result on averages of random Euler roducts. Then we use it to rove Theorem. and Theorem.2 in Sections 3 and 4. In Section 5, we rove Theorem.3. Notation. As usual, X denotes the cardinality of a set. By f g for x X, or f = Og) for x X, where X is an arbitrary set on which f is defined, we mean synonymously that there exists a constant C 0 such that fx) Cgx) for all x X. The imlied constant is any admissible value of C. It may deend on the set X which is always secified or clear in context. On the other hand, f g as x x 0 means f/g as x x 0. We use standard robabilistic terminology: a robability sace Ω, Σ, P ) is a trile made of a set Ω with a σ-algebra and a measure P on Σ with P Ω) =. A random variable is a measurable function Ω R or Ω C), and the exectation EX) on Ω is the integral of X with resect to P when defined. The law of X is the measure ν on R or C) defined by νa) = P X A). If A Ω, then A is the characteristic function of A. For k-tules h = h,..., h k ), we recall that h = max h i ). When different values of k can occur, we sometimes write h k to indicate the number of comonents of h, in articular a sum such as ah) h k h is a sum over k-tules of ositive integers) with comonents h. 2. A robabilistic statement We assume given a robability sace Ω, Σ, P ), and two sequences of random variables X, Y : Ω C which are indexed by rime numbers. We assume that Y ) is an indeendent sequence; recall that this means that P Y A,..., Y k A k ) = P Y i A i ) i k for all choices of finitely many distinct rimes,..., k, and all measurable sets A i C, and that a consequence is that when the exectation makes sense), we have EY Y k ) = EY ) EY k ). We now extend the family to all integers by denoting X q = q X, Y q = q for any squarefree integer q, and X q = Y q = 0 if q is not squarefree. We will consider the behavior of the random Euler roducts Z X = + X ), Z Y = Y, and in articular their exectations EZ X ) and EZ Y ). + Y )

6 6 EMMANUEL KOWALSKI For this urose, we assume that the roducts converge absolutely almost surely). More recisely, exand formally + X ) = X q, where restricts the sum to squarefree numbers. Then we assume that 2.) q>x q X q R X x) where R X x) is an integrable non-negative random variable such that R X x) 0 almost surely as x +. It then follows that Z X is almost surely an absolutely convergent infinite roduct. We moreover assume that the roduct 2.2) + EY ) ) converges absolutely). By indeendence of the Y ), we know that ) EY q ) = E Y = EY ) and so exanding again in series, we obtain that 2.3) EY q ) = EY ) = + EY ) ) < +. q q q q Our goal is to show that if X ) is distributed more or less like Y ), but without being indeendent, the exectation of Z X is close to + EY )). In articular, we will tyically have X ) deend on another arameter say h), in such a way that X,h converges in law to Y which will remain fixed) when h +, and this will lead to the relation ) lim E + X,h ) = + EY )) h + in a number of situations. We interret this as saying that when alicable) the average of the Euler roduct Z X is obtained as if the factors were indeendent, and taking the roduct of the local averages + EY ) of the model random variables defining Z Y. Here is the recise and almost tautological) finitary statement from which alications will be derived. Proosition 2.. Let X ), Y ) be as above. Then for any choice of the auxiliary arameter x > 0, we have EZ X ) = + EY )) + O ER X x)) + EX q Y q ) + ) EY q ), where the imlied constant is absolute, and in fact has modulus at most. q q x q>x

7 AVERAGES OF EULER PRODUCTS 7 Proof. This more or less roves itself: for any x, write first + X ) = X q = X q + X q, q then use 2.) to estimate the second term, and take the exectation, which leads to q x q>x EZ X ) = q x EX q ) + OER X x))). Next, we insert Y q by writing X q = Y q + X q Y q ), getting and then use q x EY q ) = EZ X ) = EY q ) + EX q Y q ) + OER X x))) q q x EY q ) + O q>x q x ) EY q ) = + EY )) + O q>x ) EY q ), to conclude the roof. Remark 2.2. Observe that by 2.3), the last term in the remainder tends to zero as x +. Moreover, if R X x) is dominated by an integrable function as x +, the assumtion that R X x) 0 almost surely imlies that the first term also tends to zero. Thus to conclude in ractical alications, one needs to control the middle term. In terms of the extra arameter h mentioned before the statement of the roosition, we may tyically hoe for uniform estimates for ER X x)), in terms of h, say if we also have a bound of the tye ER X x)) h α x β, α, β > 0; 2.4) EX q ) = EY q ) + Oq γ h δ ), γ, δ > 0, or if this holds on average over q < x, which may often be easier to rove, as is the case for the error term in the rime number theorem, as shows the Bombieri- Vinogradov theorem), this leads to a remainder term which is h α x β + x +γ h δ + εx) with εx) 0 as x +, uniformly in h. Then we can conclude that 2.5) lim h + EZ X) = + EY )) by choosing x suitably as a function of h, rovided we have α β < δ γ +. We will see this in action concretely in the next sections. Notice that if α can be chosen arbitrarily small i.e., R X x) is bounded almost uniformly in terms of h), then this condition can be met.

8 8 EMMANUEL KOWALSKI Remark 2.3. If we assume, instead of 2.2), that the roduct of + E Y ) converges, which is stronger, it follows that Y < + almost surely its exectation being finite), and hence the infinite roduct defining Z Y converges absolutely almost surely. Also, since we have ) E + Y ) = + EY )) P P for all P, we would obtain EZ Y ) = + EY )). rovided Z Y converges dominatedly, for instance. This formula is also valid if Y 0, by the monotone convergence theorem. It rovides an interretation of the right-hand side of 2.5). 3. Moments of singular series for the k-tule conjecture In this section, we rove Theorem., which includes in articular Gallagher s theorem, in a way which may seem somewhat comlicated but which clarifies the result. We first assume an integer k to be fixed. We rewrite.) as Sh) = + k ν h) k ) k ) ) k. It is therefore natural to define a, ν) = k ν k ) k ) k for all rimes and real numbers ν, 0 < ν omitting the deendency on k). We then define a m, ν), for m C with Rem) 0, by requiring that + a m, ν) = + a, ν)) m, with the convention 0 m = 0 if Rem) = 0; the condition ν imlies that + a, ν) 0, so this is well-defined indeed. If we assume ν <, we may extend this to all m C). We first need a technical lemma. Lemma 3.. For m C with Rem) 0, write m + = 0 if Rem) <, and m + = m otherwise. For all rime and ν with ν min, k), we have 3.) a m, k) m )) m O 2, 3.2) a m, ν) m m + + O, )) if ν < k, where the imlied constants deend only on k. Proof. Notice first that, in the stated range, we have a, k) 2, a, ν), if ν < k,

9 AVERAGES OF EULER PRODUCTS 9 where the imlied constants deend only on k, and then write a m, ν) = + a, ν)) m = ma, ν) and estimate directly. 0 + ta, ν)) m dt We are now going to rove Theorem.. Fix h though h will tend to infinity at the end). We first interret the m-th moment of the singular series in robabilistic terms, then introduce the source of its limiting value in the framework of the revious section. Consider the finite set again, deending on k) Ω = {h = h i ) h i h, h i distinct}, with the normalized counting measure. Denoting h k = Ω, notice that 3.3) h k = h k + Oh )) for h, the imlied constant deending only on k. We will denote by E and P the exectation and robability for this discrete sace. So we have, for instance, that P ν = ν) = h {h Ω ν h) = ν}. k Our goal is to find the limit as h + of the average Sh) m = E Sh) m ) h k h i distinct notice that, by 3.3), if the limit exists, it is also the limit of h k Sh) m, h i distinct as h + ). We write X h) = a, ν h)) and X m, h) = a m, ν h)), so that + X m, h)) = Sh) m by construction. Now consider a second sace Ω 2 = Z/Z) k with the roduct measure of the robability counting measures on each factor. We denote by ω = h ) the elements of Ω 2. To avoid confusion with ν defined for h Ω, we introduce the random variables { Ω 2 {,..., k} ρ : ω = h ) number of distinct h i in Z/Z, which satisfy ρ mink, ).

10 0 EMMANUEL KOWALSKI We can now define random singular series using Ω 2, writing Y = a, ρ ) and considering the Euler roduct + Y ), and similarly with Y m) = a m, ρ ) and m. + Y m)) = + Y )) We denote by P 2 and E 2 the robability and exectation for this sace. By construction of Ω 2, the random variables ρ ) are indeendent, and so are the Y ), and the Y m)) for a given m. Note also that the comonents h are equidistributed: for any rime and any a Z/Z) k, we have 3.4) P 2 h = a) = k. We now use Proosition 2. to comare the average E Sh) m ) with E2 + Y ) m ). Although this roosition is hrased with a single robability sace Ω on which both Euler vectors are defined, this is not a serious issue and the statement remains valid, rovided the exectations are suitably subscrited and one writes E X q m)) E 2 Y q m)) on the right-hand side instead of EX q m) Y q m)). 3 We start by estimating the tail Rx) = R Xm) x) of the Euler roduct defining Sh) m. In keeing with robabilistic conventions, we omit the argument h Ω in many laces. Denoting h) = h i h j ), and noting that ν = k unless, we have from Lemma 3. the bound, ) )) m + X m) m + O, ) 2 i<j for some C > 0 deending only on k) and all h, m with Rem) 0) and, the imlied constant deending only on k this justifies, in articular, the convergence of the Euler roduct Z X for every h). Hence, taking the roduct over q for a squarefree integer q, we get 2 X q m) m B) ωq) q, )q 2 q + C, ) ) m We could also simly consider Ω = Ω Ω 2 with the roduct measure, or equivalently and maybe more elegantly) assume that we start with some sace Ω and two vectors X ), Y ), distributed according to the rescrition of Ω and Ω 2 resectively, i.e., with P X = a) = h {h Ω a, ν h)) = a}, k P Y = a) = k {h Z/Z)k a, ρ h)) = a}.

11 AVERAGES OF EULER PRODUCTS for some constants B > 0 and C 0 deending only on k. Since is bounded by 3.5) 2h) k2, a standard comutation with sums of multilicative functions leads to q>x X q m) x log 2hx) D for x 2 and some constant D 0, deending on k and m. The next ste is to justify the analogue of the convergence of 2.2); more recisely, we have 3.6) + E 2 Y m) )) < +. Indeed, Lemma 3. leads to E 2 Y m) ) 2 + P 2 ρ < k) 2 for 2, where the imlied constant deends on k and m, since it is clear that we have 3.7) P 2 ρ < k) kk ) 2 for all rimes and k write that the event {ρ < k} is the union not necessarily disjoint of the kk )/2 events h i = h j with i j, each of which has robability / by uniform distribution 3.4)). By indeendence, we then also get 3.8) E 2 Y q m) ) A ωq) q 2. for all squarefree integers q and some constant A, which deends only on k and m. Finally, it remains to estimate E X q m)) E 2 Y q m)). We claim that, for any a C, we have q k ωq) ) 3.9) P X q m) = a) = + O P 2 Y q m) = a) + O h)) h where the imlied constants deend only on k. Assuming this, and noting that X q m) and Y q m) take the same finitely many values at most k ωq) distinct values, which are F ωg) q where the imlied constant and F deend on m and k), it follows that E X q m)) = q + O E 2 Y q m)) + O h)) G ωq) where G deends on m and k, leading in turn to E X q m)) E 2 Y q m)) q h E 2 Y q m) ) + Gωq) h h ), Eωq) h see 3.8)), where the imlied constant deends only on k and m, as does E.

12 2 EMMANUEL KOWALSKI Summing over q < x, it then follows from Proosition 2. that ) h Sh) m = E + X m)) = + E 2 Y m)))+ k h O xh log 2hx) B + x log 2hx) D) for some B deending on k and m. Choosing for instance x = h /2 leads to the existence of the m-th moment of singular series, with limiting value given by 3.0) µ k m) = + E 2 Y m))) = ) km { k h Z/Z) k ρ h) ) m }. It only remains to rove 3.9). Note that this is clearly an exression of quantitative equidistribution or convergence in law) of X q to Y q as h +. 4 The roof is quite simle. First of all, given arbitrary integers s with q, we have P ν h) = s for q) = h k = h k ν h)=s for q ρ h )=s h Z/Z) k h h mod q) where there are as many outer sums in the last line as there are rimes dividing q, and the last sum involves summation conditions for all q). This inner sum is 3.) = + Oh k ) h h mod q) h h mod q) h h mod q) where the imlied constant deends on k i.e., we now forget the condition on h to have distinct comonents). Lattice-oint counting leads to = hk q q k + O h)) where the imlied constant deends again only on k. In view of the equidistribution of h for h ) Ω 2, we therefore derive from the above the following quantitative equidistribution result: 3.2) P ν h) = s for q ) = P 2 ρ h ) = s for q ) q +O +O. h)) h) Now to derive 3.9), we need only observe that Y q m) and X q m) are identical functions of ρ and ν resectively for q). Hence 3.2) imlies 3.9) by summing over all ossible values of s ) q leading to a given a, using the fact that there are at most k ωq) such values the latter being a very rough estimate!). It remains to rove the symmetry roerty.6) to finish the roof of Theorem.. We note in advance that since Sh) = for all -tule h, we have 4 It can also be interreted as a form of sieve axiom.

13 AVERAGES OF EULER PRODUCTS 3 µ m) = for all m, and hence µ k ) = for all k, which is Gallagher s result.5). The symmetry turns out to be true locally, i.e., the -factor of the Euler roducts 3.0) defining µ k m) and µ m k) coincide for all and integers k, m. There are different ways to see this, and the following seems to encasulate the origin of the henomenon. Given a finite set F which will be Z/Z), consider the following obviously symmetric exression of m and k: F m+k x F m, h F k {x i} {h j}= which is the robability, for the normalized counting measure on F k+m, that a air of a k-tule and an m-tule, both of elements of F, do not contain a common element). Then it can be interreted either as F m m τ= x F m ρx)=τ F k h F k {h j} {x i}= = F m m τ= x F m ρx)=τ = F m x F m or by the same comutation with m and k reversed) as F k h F k ρh) F ) m, τ ) k F ρx) F using ρ ) to denote the number of distinct elements in F of an m-tule, then of a k-tule). Alied with F = Z/Z, u to the symmetric factor /) km in 3.0), the first is the -factor for µ m k), and the second is the -factor for µ k m), showing that they are indeed equal. Remark 3.2. Quantitatively, we have roved that Sh) m = µ k m)h k + Oh k /2+ε ), for any ε > 0, where the imlied constant deends on k and m. For m =, Montgomery and Soundararajan [MS, 7),. 593] have obtained a more refined exansion with contributions of size h k log h and h k, and error term of size h k 3/2+ε. Remark 3.3. The fact that µ k ) = can be used to recover the combinatorial identities used by Gallagher [Ga,. 7 8] instead of the robabilistic hrasing above. We review this for comleteness: in order to rove µ k ) =, it suffices to show that the average of a, ρ ) is zero. We have h Z/Z) k a, ρ h)) = ) k a, ν) {h Z/Z) k ρ h) = ν} ν=

14 4 EMMANUEL KOWALSKI and on the other hand, we have {h Z/Z) k ρ h) = ν} = ν ){ k ν where { k ν} is the number of surjective mas from a set with k elements to one with ν elements 5 ; indeed, a k-tule h with ν distinct values is the same as a ma {,..., k} Z/Z with image of cardinality ν, i.e., the set of such tules is the disjoint union of those sets of surjective mas {,..., k} I over I Z/Z with order ν. Therefore, Gallagher s result follows from the identity ){ } k a, ν) = 0 ν ν ν= ν= which is roved in [Ga,. 7], and which we have therefore reroved. Similarly, the identities ){ } k ){ } k = k, ν = k+ ) k, ν ν ν ν ν= of [Ga,. 8] can be derived from the roof that the -factor for µ k ) is. Remark 3.4. From.2), one can guess that µ k m) = µ m k) for m integer, by comuting ) m Λn + h i ) = Λn j + h i ) n N i k }, h k h n m N i k j m where n is an m-tule), which is a symmetric exression in n and h, excet for the ranges of summation, and which should be asymtotic to either µ k m)h k N m or µ m k)h k N m by a uniform k-tule conjecture. In fact, the comutation we did amounts to doing the same argument locally i.e., looking on average over h at the distribution of integers such that, for a fixed rime, n + h,..., n + h k are not divisible by ). This symmetry µ k m) = µ m k), desite the simlicity of its roof, is a very strong roerty, as ointed out to us by A. Nikeghbali. Indeed, write X k = Z Y,k, the random variable given by the random singular series. Since we have µ k m) = t m dν k t) = EXk m ), R + the symmetry imlies that the sequence EXk m)) k, for a fixed value of m, is the sequence of moments of a robability distribution of [0, + [, which is a highly non-trivial roerty. We refer to the survey [Si] of the classical theory surrounding the moment roblems, noting that from Theorem of loc. cit. it follows that, for any fixed m, we have α i ᾱ j µ i+j m) > 0, α i ᾱ j µ i+j+ m) > 0, 0 i N 0 j N 0 i N 0 j N 5 This is denoted σk, ν) in [Ga], and it is not the standard notation, which would write r! { k r instead. }

15 AVERAGES OF EULER PRODUCTS 5 for any N and any comlex numbers α i ) C N {0}. It would be quite interesting to know what other tyes of natural sequences of random variables or robability distributions) satisfy the relation EXk m) = EXm). k One fairly general construction is as follows this was ointed out by A. Nikeghbali and P. Bourgade): just take X n = Z n for Z a random variable such that all moments of Z exist, or a bit more generally, take a sequence X n ) of ositive random variables such that the Xn /n are identically distributed. But note that the variables we encountered are not of this tye. Examle 3.5. Let m = 2. We find using the symmetry roerty) that the meansquare of Sh) is given by lim h + h k Sh) 2 = µ k 2), where µ k 2) = ) 2 ) k + ) k ) 2k. ) In articular, we find using Pari/GP for instance): µ 2 2) = µ 3 2) = µ 4 2) = µ 5 2) = µ 6 2) = Note that the second and higher) moments increase quickly with k as roved in Proosition 4. in the next section). This is exlained intuitively by the fact that Sh) is often zero: for instance, the 2-factor of Sh) is zero unless all h i are of the same arity, which haens with robability 2 k only see Examle 4.3 for a more recise estimate). For those, of course, the 2-factor is very large equal to 2 k ). 4. Growth and distribution of moments of singular series In this section, we will rove Theorem.2, using the methods of moments. For this, we consider the roblem which has indeendent interest) of determining the growth of µ k m). We look at the deendency on m for fixed k, or equivalently the deendency on k for fixed m, by symmetry as in Examle 3.5). The result is that the moments grow just a bit faster than exonentially. Proosition 4.. For any fixed k, we have log µ k m) = km log log 3m + Om), for m, where the imlied constant deends on k. Proof. We use the formula 3.0), written in the form µ k m) = ) kme2 We will rove first that ρ log µ k m) km log log 3m + Om), ) m ). for m, with an imlied constant deending on k, before roving the corresonding uer bound.

16 6 EMMANUEL KOWALSKI We start by checking that all terms in the Euler roduct are, i.e., for all rimes, all integers k and all real numbers m, we have 4.) E 2 ρ ) m ) mk. ) Indeed, by the symmetry between the -factor for µ k ) and for µ k), we have ) k = E2 ρ ), while raising to the m-th ower and alying Hölder s inequality gives E 2 ρ )) m E2 ρ ) m ). From this we can bound µ k m) from below by any subroduct, and we look at µ km) = ) kme2 ρ ) m ). m The robability that ρ is is clearly equal to k ) there are only k-tules with this roerty). Hence we have crude lower bounds E 2 ρ ) m ) k ) k and µ k m) µ km) m + ) km ) k. The logarithm of this exression is easily bounded from below as follows: log µ k m) km ) log + ) k ) log m m = km log log 3m + Om), for m 2, the imlied constant deending only on k, by standard estimates, and we can incororate trivially m = also. To rove the corresonding uer bound, we slit the Euler roduct 3.0) into two ranges: we write µ k m) = µ ) k m)µ2) k m), where µ ) k m) is the roduct over rimes < km which includes the range used for the lower bound), while µ 2) m) is the roduct over the other rimes km. We will show that k log µ ) k m) km log log 3m + Om), log µ2) k m) m log 2m, with imlied constants deending on k, and this will conclude the roof. We start with small rimes, and simly bound the exectation of ρ/) m by the trivial bound ; this leads to log µ ) k m) km <km log ) = km log log 3m + Om), where the imlied constant deends on k, again by standard estimates.

17 AVERAGES OF EULER PRODUCTS 7 Next, we estimate µ 2) k m) more carefully. The logarithm say Lx)) of the roduct restricted to km x is given by Lx) = km log ) + log E 2 ρ ) m ). km x km x Using 3.7), we write first, for km, the uer bound E 2 ρ ) m ) with A k = k 3 2k 2 ) since = k ) m P 2 ρ < k)) + P 2 ρ < k) k ) m + P 2 ρ < k) k ) m ) k ) m mk 2 k ) mk mx x) m mx + mm ) k mk2 k ) 2 2, = mk + m2 k ma k 2 2 mm ) x 2 for 0 x, m. 2 Moreover, we have log x) x x 2 /2 for 0 x <, and hence after some rearranging, we obtain log E 2 ρ ) m ) mk + m2 k ma k 2 2 mk 2 m2 k ma ) 2 k 2 2 = mk + m3 k 2 3 m4 k ma k 2 2 m2 ka k 2 3 m2 A 2 k 2m3 k 2 A k 8 4, the terms involving m 2 k 2 )/2 2 ) having cancelled out. Summing over km x, we can let x go to infinity in all but the first resulting term since they define convergent series; bounding the tail by σ km) σ log 2km), leads to km x >km log E 2 ρ ) m ) km km x m ) + O log 2m for all m and x km, where the imlied constant deends on k. Finally, log Lx) km + log )) m ) + O, log 2m km< x and since + log ) defines an absolutely convergent series with tail for > y) decreasing like y log y), we obtain the desired bound for log µ 2) k m) = lim Lx). x +

18 8 EMMANUEL KOWALSKI The existence of a limiting distribution Theorem.2) is an easy consequence of this. Corollary 4.2. Let k be a fixed integer. As h goes to infinity, the singular series Sh) for h Ω, i.e., such that h h, converges in law to the random singular series Z Y = Z Y,k = ) k ρ on Ω 2. In other words, there exists a robability law ν k on [0, + [, which is the law of Z Y, such that Sh), for h h, becomes equidistributed with resect to ν k, or equivalently lim h + h k ) fsh)) = ft)dν k t) R + for any bounded continuous function on R. Moreover we have 4.2) µ k m) = E 2 ZY m ) = t m dν k t). R + Proof. First of all, using 3.0), the monotone and dominated convergence theorems and 3.6) imly that we have 4.3) µ k m) = E 2 Z m Y ) for all integers m. Now a standard result of robability theory the method of moments ) states that given a ositive random variable X and a sequence of ositive random variables X n ), such that EX m ) < +, EX m n ) < + for all n and m, the condition lim n + EXm n ) = EX m ) for all m imlies the convergence in law of X n to X, if the moments EX m ) do not grow too fast a sufficient, but not necessary condition). In fact, it is enough that the ower series i m EXm ) t m m! m 0 have a non-zero radius of convergence, which in our case holds with X = Z Y ) by the almost exonential uer bound for µ k m) in Proosition 4.. Finally, the formula 4.2) follows from 4.3). Examle 4.3. As a corollary of Proosition 4. and symmetry, we have log µ k 2) = 2k log log 3k + Ok) for k. Combined with the classical lower bound for non-vanishing arising from Cauchy s inequality, it follows that for every fixed k, we have lim inf h + h k {h h h and Sh) 0} µ k) 2 ex 2k log log 3k + Ok))). µ k 2)

19 AVERAGES OF EULER PRODUCTS 9 This is close to the truth, as one can check by noting that we have in fact 6 lim h + h k {h h h and Sh) 0} = P 2Z Y,k 0) = P 2 ρ < ) k using the almost sure absolute convergence of the random Euler roduct Z Y,k. We have the bounds ) k )k k P 2 ρ < ) k since, for k, a k-tule will have ρ < only if it omits at least one value in Z/Z; the lower bound follows by looking at those omitting 0, for instance, and the uer one is a union bound over the ossible omitted values), from which we get i.e., we have k log log 3k + Ok) log P 2 Z Y,k 0) k k log log 3k + Ok), P 2 Z Y,k 0) = ex k log log 3k + Ok)). It follows from this that if we relace the sace Ω of all k-tules with distinct entries by the much smaller one Ω = {h Ω Sh) 0}, which still deends on h, with cardinality h k ), the singular series still has a limiting distribution when interreted as a random variable on Ω with h + : indeed, this is the distribution ν k given by since, for any integer m, we have Sh) h m = k h Ω ν k A) = ν ka ]0, + [), ν k ]0, + [) h k h k E Sh) m ) µ km) P 2 Z Y,k 0) = t m d ν k t), [0,+ [ as h +. Of course, those moments do not satisfy the symmetry roerty enjoyed by µ k m). Remark 4.4. Before going on to the second art of this aer, the following question seems natural: are there arithmetic consequences ossibly conditional, similarly to Gallagher s roof of.7)) of the existence of m-th moments of the singular series for k-tules? 5. Poisson distribution for general rime atterns In this section, we rove Theorem.3, essentially by following Gallagher s reduction to averages of Euler roducts, which turn out to be easily comutable after alication of Proosition 2.. We fix a rimitive family of olynomials f with Sf) 0 the reader may want to review the notation in the introduction for what follows). To aly Gallagher s method, we also require some auxiliary families of olynomials, indexed by k-tules. 6 This does not follow directly from convergence in law for Sh), but from the absolute convergence and local structure of the singular series.

20 20 EMMANUEL KOWALSKI Thus let k be an integer and h a k-tule of integers. For our fixed rimitive f, we denote f h = f j X + h i )) j m, i k which is a family of km integer olynomials. Technical difficulties will arise because this family may not be rimitive, even if the comonents of h are distinct which is a necessary condition), i.e., we may have an equality f j X + h i ) = f j2 X + h i2 ), for some i i 2, j j 2. For instance, we have X, X + 2) 3, ) = X + 3, X +, X + 5, X + 3) in the case of twin rimes). However, we will show that these degeneracies have no effect for the roblem at hand. Moreover, f h is rimitive whenever h has distinct arguments, in the following quite general situations: if m = ; if the degrees of the f j are distinct; if no two among the olynomials f j are related by a translation X X + α, for some α Z. This means that the reader may well disregard the technical roblems in a first reading for the twin rimes, see also Examle 5.9 which exlains a secial reason why the degeneracies have no consequence then). The following lemma is already a first ste, and we will need it before roving the full statement. Lemma 5.. Let f be a rimitive family and k. Then for any h, we have {h h k h, f h is not rimitive} h k where the imlied constant deends only on k and m. Proof. Let I be the set of k-tules h with distinct comonents such that f h is not rimitive. If h I, then there exists at least one relation of the tye 5.) f j X + h i ) = f j2 X + h i2 ), i i 2, j j 2, hence f j X) = f j2 X + h i2 h i ), so the two olynomials differ by a shift. Let R be the set of airs j, j 2 ) for which f j X) = f j2 X + δj, j 2 )) for some integer δj, j 2 ) 0. Because the olynomials involved are non-constant, this integer is indeed unique. The cardinality of R is bounded in terms of m only, and from the above, any k-tule h I must satisfy at least one relation h i h i2 = δj, j 2 ), for some i i 2 and j, j 2 ) R. Each such relation is valid for at most h k among the k-tules with h h. We will deduce Theorem.3 from the following unconditional) result, which is another instance of average of Euler roducts:

21 AVERAGES OF EULER PRODUCTS 2 Proosition 5.2. Let f = f,..., f m ) be a rimitive family and k an integer. Then we have lim h + h k Sf h) = Sf) k, where here restricts the summation to those k-tules for which f h is rimitive. Remark 5.3. Taking f = X), with Sf) = and f h = X + h,..., X + h k ), we recover once more Gallagher s result.5). We have the following comlementary statement, which is also unconditional recall that, in many cases, it holds for trivial reasons; it does not follow trivially from Lemma 5. because although fewer k-tules are concerned, the number of rime seeds increases when f h is not rimitive). Lemma 5.4. Let f = f,..., f m ) be a rimitive family with Sf) 0, and k an integer. Then for any N 2, if h λlog N) m for some λ > 0, and for any ε > 0, we have h k h f h not rimitive πn; f h) N log N) ε where restricts the sum to those k-tules with distinct entries, and where the imlied constant deends only on k, f, λ and ε. Here is the roof of the conditional) Poisson distribution, assuming those two results. Proof of Theorem.3. The argument is essentially identical with that of Gallagher, but we reroduce it for comleteness, and so that the necessary uniformity in the Bateman-Horn conjecture becomes clear. Because the Poisson distribution is characterized by its moments, it is enough to rove that for any fixed integer k, we have N n N πn + λδn, f); f) πn; f)) k EP k λ ), as N +, where P λ is any Poisson random variable with mean λ. Write h = λδn, f). Exanding the left-hand side, we obtain ) N n N n<m i n+h m i f-rime seed where there are k sums over m,..., m k. Write m i = n + h i, so that h i h, and the condition becomes that f j n + h i ) is rime for all i and j, i.e., that n be an f h-rime seed. Exchanging the order of summation, we get πn; f h). N h k h Before alying.8), we need to account for the k-ules which do not necessarily have distinct comonents, and for those where f h is not rimitive. For this, observe first that πn; f h) only deends on the set containing the comonents of the k-tule h. This justifies the fact that the reorderings that

22 22 EMMANUEL KOWALSKI follow are ermissible. For each r, r k, and each r-tule h with distinct comonents, the set of those k-tules for which the set of values is given by the set of comonents of h has cardinality deending only on r and k, but indeendent of h, and in fact it is given by { k r} one can assume that h =,..., r), and obtain a bijection { {suitable k-tules} {surjective mas {,..., k} {,..., r}} between the two sets). Then we can write N h f : i h i ) h k h πn; f h) = N k r= { } k πn; f h ) r! r h r h where we divide by r! because we sum over all r-tules instead of only ordered ones, and restricts to r-tules with distinct entries. Now, for each r, we searate the sum over r-tules for which f h is rimitive from the other subsum. Alying.8) and using the easy bound cf h ) cf) h max degfj) r, where the imlied constant deends on r and f) the first sum still denoted ) is equal to k r= r! { } k r egf) r log N) rm h r h h Sf h ε )) ) + O, log N for any ε > 0, where the imlied constant deends on f, k and ε. Using Proosition 5.2 and the choice of h = λ egf)sf) log N) m, this converges as N + to the limit k r= λ r r! { } k, r which is well-known to be the k-th moment of a Poisson distribution with mean λ this is checked by Gallagher for instance, see [Ga, 3]). Hence, to conclude the roof, we need only notice that Lemma 5.4 alied with k = r for r k) imlies taking ε = /2 for concreteness) that the comlementary sum is bounded by N k r= r! { } k r h r h f h not rimitive πn; f h ) log N) /2 for N 2, where the imlied constant deends on k, f and λ. Hence this second contribution goes to 0 as N +, as desired. We now rove Proosition 5.2. This is the conjunction of the two following lemmas, where we use the same notation as in Section 3, but change a bit the definition of robability saces. Precisely, Ω 2 = Z/Z) k

23 AVERAGES OF EULER PRODUCTS 23 is unchanged, but we let Ω = {h = h,..., h k ) h i h, f h is rimitive} with the counting robability measure note that the condition forces h to have distinct coordinates). By Lemma 5., note that we have 5.2) Ω h k as h +. The next lemma shows that the average of Euler roduct involved can be comuted as if the comonents where indeendent: Lemma 5.5. Let Sf) = f,..., f m ) be a rimitive family with Sf) 0. Then for any k, we have lim h + h k Sf h) = lim E ) km ν )),f h + where = E 2 ) km ρ,f )), ν,f h) = ν f h) for h = h,..., h k ) with h i, ρ,f h) = {x Z/Z f j x + h i ) = 0 for some i, j} for h Z/Z) r. The second lemma comutes the limit locally: Lemma 5.6. Let f = f,..., f m ) be a rimitive family. Then for any k and any rime, we have E 2 ) km ρ )),f = ) km ν ) f) k. Looking at the definition.4) of Sf), both lemmas together rove Proosition 5.2. We start by roving Lemma 5.6 because Lemma 5.5 is certainly lausible enough in view of Section 3, and the reader may be more interested by the final formal flourish. Proof of Lemma 5.6. It suffices to comute E 2 ρ ),f since the other factor is the same on both sides. We argue robabilistically, although one can also just exand the various sums and do the same stes in a different language, as we did when roving the symmetry.6)). We can write ρ,f = Z/Z M where M Z/Z is the random) subset of those x Z/Z such that f j x+h i ) = 0 for some i and j. We write Z/Z M = χ M x)) x Z/Z where χ M x) is the random variable equal to one if x M and zero otherwise. We have χ M x) = {fjx+h i)=0}) = ξ f,i x), i k j m i k

24 24 EMMANUEL KOWALSKI say. Since ξ f,i x) only involves the i-th comonent of the random h Ω 2, the family ξ f,i x)) is an indeendent k-tule of random variables. Consequently we derive E 2 ρ ),f = ) E 2 ξ f,i x) = x Z/Z x Z/Z i k i k E 2 ξ f,i x)). To conclude we notice that for every x and i, h x+h i is identically uniformly) distributed, so that all ξ f,i x) are identically distributed like ξ f = ξ f, 0) = {fjh )=0}). j m Hence all x give the same contribution, and we derive that E 2 ρ ),f = E 2 ξ f ) k = P 2 f h ) f m h ) 0) k = ν ) f) k, since h is uniformly distributed in Z/Z. To rove Lemma 5.5, we wish to aly Proosition 2.. A comlication is that, if egf), the singular series Sf h) are not defined by absolutely convergent roducts, and therefore the result is not directly alicable. However, we can byass this difficulty here without significant work because of the following fact: all the relevant Euler roducts can be uniformly renormalized to absolutely convergent ones. This is the content of the next lemma. Lemma 5.7. Let f be a rimitive family with Sf) 0, and let k be an integer. There exist real numbers γ f) > 0, for all rimes, such that the roduct γ f) converges, and such that the following hold: ) For all rime, and all k-tule h Z/Z) k, we have ) km ρ,f h) ) = γ f) + X,f h)) for some coefficients X,f h), and for all k-tule of integers h such that f h is rimitive, the roduct 5.3) + X,f h)) is absolutely convergent. 2) We have lim h + h k h k h + X,f h)) = + E 2 X,f )), where the sum is over k-tules with f h rimitive.

25 AVERAGES OF EULER PRODUCTS 25 Proof. ) To define γ f), let θ j, j m, be a comlex root of the irreducible olynomial f j, and let K j = Qθ j ) be the extension of Q of degree degf j ) generated by θ j. Then ut γ f) = ) krj) ) i m where r j n), for n, is the number of rime ideals of norm n in the ring of integers of K j. In view of this definition, to check first that the roduct of γ f) converges, we can do so for each f j searately. Then the statement follows, after taking the logarithm of a artial roduct over X, from the well-known asymtotic formula X r j ) = X + ck j) + Olog X) ) for X 2, where ck j ) is a constant deending only on K j, and the imlied constant also deends only on K j. It therefore remains to rove that the roduct 5.3) is absolutely convergent for any k-tule of integers h with f h rimitive. To do so, we claim that there exists an integer Dh) which may also deend on f) such that, for Dh), we have m m 5.4) ρ,f h) = k ν f j ) = k r j ). j= j= The desired convergence then follows from that of γ f) ) km ρ,f h) ) = Dh) Dh) ) ρ,f h) ρ,f h and the latter is clear since the -factor can be written + O 2 ), where the imlied constant deends only on k and f. The existence of Dh) is easy; first, let D h) = Resf j X + h i ), f j X + h i )), i,j) i,j ) where Res, ) is the resultant of two olynomials. By comatibility of the resultant with reduction modulo, we have D h) if and only if, for some i, j) i, j ), there exists a common zero x Z/Z of f j X + h i ) and f j X + h i ). By contraosition, we first obtain m ρ,f h) = kν f) = k ν f j ), for D h) the sets of zeros modulo of the comonents of f h are then distinct, and obviously there are as many, namely the sum ν f) of the ν f j ), for each of the k shifts h i ). Next, it is a standard fact of algebraic number theory that for each j, there exists an integer j such that ν f j ) = r j ) for j. Thus we can take Dh) = D h) to obtain the second equality in 5.4). j= j m j ),

26 26 EMMANUEL KOWALSKI Note that Dh) is non-zero hence ) because otherwise, there would exist a common zero θ C of f j X +h i ) and f j X +h i ), and because those are irreducible integral rimitive 7 olynomials with ositive leading coefficient, this is only ossible if f j X + h i ) = f j X + h i ), which is excluded by the assumtion that f h be rimitive. Note in assing the estimate Dh) 2 h k ) 2k2 m degf j) for all h, where the imlied constant deends only on f; this follows straightforwardly from the determinant exression of the resultant in D h) see, e.g., [L, V.0]). 2) With the bounds we have roved on X,f h) leading to an analogue of Lemma 3.), and the estimate on Dh) analogue of 3.5)), together with Lemma 5. to ensure that the equidistribution of k-tules modulo squarefree integers q remains valid comare with 3.)), we can retty much follow the stes of the roof of Theorem.. We also use 5.2) to go from the limit of the exectation on Ω to summing over k-tules normalized by /h k and taking h +. The details are left to the reader. Proof of Lemma 5.5. We have first h k Sf h) = ) h k γ f) + X,f h)) ) γ f) + E 2 X,f )) as h +, by the above, and then we can simly write this limit as ) γ f) + E 2 X,f )) = E 2 γ f) + X,f )) E 2 ) km = ρ,f h) We conclude with the last remaining art of the roof, namely Lemma 5.4. The following roof can almost certainly be imroved, but although the statement becomes fairly clear after checking one or two examles, the author has not found a cleaner way to deal with the aarent ossibilities of combinatorial comlications. The oint is that as f h becomes less rimitive i.e., there are less distinct elements among the km olynomials involved), the number of rime seeds N should increase by a ower of log N)), but also the number of k-tules with this roerty diminishes by a ower of h λlog N) m ), and this gain has to comensate for the loss. Proof of Lemma 5.4. We first quote a standard sieve uer-bound for an individual rimitive family f with m elements), which is uniform, and which allows us to )). 7 In the sense that the gcd of their coefficients is.

27 AVERAGES OF EULER PRODUCTS 27 rove the lemma unconditionally: for N 2, for any k-tule h with distinct elements for which f h contains l distinct comonents, we have 5.5) πn; f h) log log 3 h ) km N log N) l, where the imlied constant deends only on k and f. Precisely, 5.5) for k-tules follows immediately from, e.g, Th. 2.3 in [HR], and it is easy to adat this to the case at hand since uniformity is only asked with resect to h. Note also that, since the alication we give is conditional on much stronger statements like.8), we could also aly the latter for this urose. Now, as in the roof of Lemma 5., we denote by I the set of k-tules h with distinct comonents such that f h is not rimitive. Recall R is the set of airs j, j 2 ) for which f j X) = f j2 X + δj, j 2 )) for some unique) integer δj, j 2 ) 0. We continue as follows: for an h I, let Γ h be the grah with vertex set {,..., k} and with unoriented) edges i, i 2 ) corresonding to those indices for which the relation 5.6) h i h i2 = δj, j 2 ) holds for some j, j 2 ) R; the roof of Lemma 5. shows that there is at least one edge. Because the number of ossibilities for Γ h is clearly bounded in terms of k only, and we allow a constant deending on k in our estimate, we may continue by fixing one ossible grah Γ and assuming that all h I satisfy Γ h = Γ. This being done, we first estimate from above the number of k-tules which lie in I under the above assumtion that the grah is fixed!). We claim that 5.7) {h I h h} h c where c = π 0 Γ) is the number of connected comonents of Γ. To see this, notice that each connected comonent C corresonds to a set of variables which are indeendent of all others, so that I is the roduct over the connected comonents of sets I C of C -tules satisfying the relations 5.6) dictated by C. Now we have {h I C h h} h, because C is connected: if we fix some vertex i 0 of C, then for any choice of h i0, the value of h i is determined by means of the relations 5.6) for all vertices i of C, using induction on the length of a ath from i 0 to i which exists by connectedness). Taking the roduct over C of these individual uer bounds, we obtain the desired estimate 5.7). We next need to estimate from below the number of distinct elements in the family f h for a fixed h I still under the assumtion that the grah Γ h = Γ is fixed). Let again C be a connected comonent of the grah Γ. We consider the set say {f h} C ) of olynomials of the form f j X +h i ), where j m and i is a vertex of C. We claim this set contains at least m+ distinct olynomials if C has at least 2 vertices, and m if C is a singleton. Indeed, fixing a vertex i 0 of C, the set contains the olynomials f j X + h i0 ), which are distinct since f is a rimitive family. This already takes care of the case where C is a singleton, so assume now that C contains