BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS

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1 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS ABEL CASTILLO, CHRIS HALL, ROBERT J LEMKE OLIVER, PAUL POLLACK, AND LOLA THOMPSON Abstract The Hardy Littlewood prime -tuples conjecture has long been thought to be completely unapproachable with current methods While this sadly remains true, startling breathroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem In this wor, we extend the Maynard-Tao method to both number fields and the function field F q t 1 Introduction and statement of results The classical twin prime conjecture asserts that there are infinitely many primes p such that p + is also prime While this conjecture remains completely out of reach of current methods, there has nevertheless been remarable recent progress made towards it, beginning with wor of Goldston, Pintz, and Yıldırım [4], who showed, if p n denotes the nth prime, that lim inf n p n+1 p n log p n = 0, so that gaps between consecutive primes can be arbitrarily small when compared with the average gap Expanding upon these techniques, Zhang [18] proved the amazing result that lim inf n p n+1 p n , ie, that there are bounded gaps between primes! The techniques of Zhang and Goldston, Pintz, and Yıldırım have subsequently been significantly expanded upon by Maynard [11], Tao, and the Polymath project [14], so that the best nown bound on gaps between primes, at least at the time of writing, is 5 Remarably, the techniques of Maynard and Tao also enable one to achieve bounded gaps between m consecutive primes, ie, that lim infp n+m 1 p n is finite The main idea in all of these results is to attac approximate versions of the Hardy Littlewood prime -tuples conjecture: Given a -tuple H = h 1,, h of distinct integers, we say that H is admissible if the set {h 1,, h } mod p is not all of Z/pZ for each prime p The Hardy Littlewood prime -tuples conjecture can then be stated as follows Conjecture Given an admissible -tuple H = h 1,, h, there are infinitely many integers n such that each of n + h 1,, n + h is prime This conjecture remains intractable at present note that the = case immediately implies the twin prime conjecture However, Maynard, Tao, and Zhang have recently succeeded in obtaining partial results that would have seemed incredible just a few years ago In particular, we have the following theorem of Maynard [11] and Tao CH was partially supported by a grant from the Simons Foundation RJLO was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship 1

2 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON Theorem Maynard Tao Let m There exists a constant 0 := 0 m such that, for any admissible -tuple H = h 1,, h with 0, there are infinitely many n such that at least m of n + h 1,, n + h are prime A result on bounded gaps comes from taing m = and providing an explicit admissible 0 -tuple of small diameter Indeed, Zhang s [18] main theorem is the m = case of the above, and he obtained 0 = Maynard [11] was able to tae 0 = 105, and the Polymath project [14] has reduced the permissible value to 0 = 51 In our wor at hand, we prove an analogue of the Maynard Tao theorem for number fields and the function field F q t, and we derive corollaries which we believe to be of additional arithmetic interest We begin by extending the Maynard Tao theorem to number fields, for which we must first fix some notation Given a number field K with ring of integers O K, we say that α O K is prime if it generates a principal prime ideal, and we say that a -tuple h 1,, h of distinct elements of O K is admissible if the set {h 1,, h } mod p is not all of O K /p for each prime ideal p Our first theorem is a direct translation of the Maynard Tao theorem Theorem 11 Let m There is an integer 0 := 0 m, K such that for any admissible -tuple h 1,, h in O K with 0, there are infinitely many α O K such that at least m of α + h 1,, α + h are prime Two remars: 1 As the proof of Theorem 11 will show, the numerology which produces 0 from m is similar to that in Maynard s paper, and is exactly the same if K is totally real In general, 0 will depend only upon m and the number of complex embeddings of K Another way of extending the Maynard Tao theorem to number fields was considered by Thorner [17], who proved the analogous result for rational primes satisfying Chebotarevtype conditions ie, primes p such that Frob p lies in a specified conjugacy-invariant subset of GalK/Q for some K/Q As an immediate corollary to Theorem 11, we can deduce bounded gaps between prime elements of O K, where the bound depends only on the number of complex embeddings of K As an example, we have the following corollary for totally real fields Corollary 1 If K/Q is totally real, then there are infinitely many primes α 1, α O K such that σα 1 α 600 for every embedding σ of K We now turn our attention to the function field F q t Here, the role of primes is played by monic irreducible polynomials in F q [t] We define a -tuple h 1,, h of polynomials in F q [t] to be admissible if, for each irreducible P, the set {h 1,, h } does not cover all residue classes of F q [t]/p Theorem 13 Let m There is an integer 0 := 0 m, independent of q, such that for any admissible -tuple h 1,, h of polynomials in F q [t] with 0, there are infinitely many f F q [t] such that at least m of f + h 1,, f + h are irreducible Remar Striingly, the independence of 0 from q passes even so far that Maynard s values of 0 m are permissible in this setting as well In particular, we may tae 0 = 105 As a corollary, we can deduce bounded degree gaps between irreducible polynomials In fact, one could already prove something stronger: if q 3, then any a F q occurs infinitely often as a gap see [6] for q > 3 and [13] for q = 3 These proofs are constructive, but the degrees of the irreducible polynomials produced lie in very sparse sets Our next result shows

3 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 3 that any a F q, and, indeed, any monomial, in fact occurs in many degrees Moreover, given any large degree, a positive proportion of elements of F q [t] of bounded degree occur as a gap Theorem 14 Let 0 := 0 from Theorem 13 and let q i For d 0, if n is sufficiently large and satisfies n d, q 1 = 1, each monomial in F q [t] of degree d occurs as a gap between monic irreducibles of degree n ii For d 0 and n sufficiently large, the proportion of elements of degree d that appear as gaps between irreducibles in degree n is at least q 1 The same conclusion holds if we restrict to monomials of degree d Remar The observation that our methods permit us to deduce the first part of Theorem 14 is due to Alexei Entin This paper is organized as follows In Section, we describe the Maynard Tao method in a general context, and we prove Theorems 11 and 13 simultaneously In Section 3, we consider the application of these theorems, and we prove Corollary 1 and Theorem 14 The general Maynard Tao method The Maynard Tao method for producing primes in tuples is very general, and relies upon a multidimensional variant of the Selberg sieve; indeed, the multidimensional nature of the sieve is the ey improvement over the wor of Goldston, Pintz, and Yıldırım, and that of Zhang Many of the steps in the method are essentially combinatorial, relying principally upon multiplicative functionology and elementary statements, rather than hard information about the structure of the integers or the primes It is only in a few ey places that deep information is used, and, indeed, these results can be assumed to be blac boxes As such, when proving our theorems, we proceed in a very general fashion We first define general notation and establish a dictionary which permits us to tal simultaneously about the integers the Maynard Tao theorem, number fields Theorem 11, and the function field F q t Theorem 13 This of course introduces some notational obfuscation, but we nevertheless consider this approach useful: first, it enables us to prove each theorem simultaneously, and, second, it elucidates what is needed to prove a Maynard Tao type result in a general setting In Section, we use this dictionary, together with the combinatorial arguments of Maynard, to lay down the proof of the Maynard Tao theorem, assuming the existence of the relevant blac boxes It is only in Section 3 that we remove ourselves from the general setting and specialize to the number field and function field settings where we have the necessary arithmetic information Accordingly, it is here that precise versions of Theorems 11 and 13 are proved 1 The dictionary We begin by letting A denote the set of integers that we are considering Thus, in the case of the Maynard Tao theorem, we will tae A = Z In the number field setting, we will tae A to be the ring of integers O K of some number field K/Q, and in the function field setting, we will tae A to be the polynomial ring F q [t] For any positive integer N, we let AN denote the box of size N inside A Over the integers, this is the interval N, N] In the polynomial setting, we let AN be the collection of monic elements of norm N; that is, if N = q n, then AN is the set of monic, degree n elements of F q [t] The definition of AN is slightly more complicated in the number field

4 4 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON situation We first define A 0 N as the set of α O K which satisfy 0 < σα N for all real embeddings σ : K C and satisfy σα N for all complex embeddings We then tae AN := A 0 N \ A 0 N Given a nonzero ideal q A, we define analogues of three classical multiplicative functions, namely the norm q := A/q, the phi-function ϕq := A/q, and the Möbius function µq := 1 r if q = p 1 p r for distinct prime ideals p 1,, p r and µq = 0 otherwise We define the zeta function of A by ζ A s := q A q s When A = O K, the function ζ A s is the usual Dedeind zeta function of K When A = F q [t], one has the closed form expression ζ A s = 1 This differs from the usual zeta function 1 q 1 s of F q t in that the Euler factor corresponding to the prime over 1/t has been removed We record here that the number of elements α AN satisfying a congruence condition α α 0 mod q is given by AN + O AN, q, q where { 1 if A = Z or A = Fq [t], AN, q 1 + AN q 1 1 d if A = O K and [K : Q] = d In fact, if A = F q [t] and AN q, then we can tae AN, q = 0 When A = Z or A = F q [t], these estimates for AN, q are trivial When A = O K, matters are more complicated but still relatively familiar One starts by embedding K into Minowsi space R r 1 C r Under this embedding, q goes to a lattice, while the constraints on AN correspond to a certain region of R r 1 C r The estimate for AN, q comes from estimating the number of translates of the fundamental parallelogram that intersect the boundary of that region Compare with the proof of [10, Lemma 1] For our purposes, what is important to tae away is that we always have a power savings in the error term: AN, q AN / q 1 ν for some positive ν = νa, as long as AN q Let P denote the prime elements of A and tae P N = P AN If A = Z, P is simply the set of primes, and, if A = O K, P is the set of generators of principal prime ideals If A = F q [t], P is the set of monic irreducible polynomials In all of these cases, we have a prime number theorem of the form P N c AN log N for some constant c, and we moreover have a prime number theorem for the set P N; q, α 0 of primes in the coprime residue class α 0 mod q of the form P N; q, α 0 = 1 ϕq P N + EN; q, α 0 For any individual q, we have the upper bound EN; q, α 0 = o q P N, and we say that P has level of distribution θ > 0 if, for any B > 0, the bound q Q max α 0 mod q α 0,q=1 EN; q, α 0 B AN log B N

5 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 5 holds for all Q AN θ and all sufficiently large N If A = Z, the Bombieri-Vinogradov theorem asserts that the primes have level of distribution θ for any θ < 1/ and the Elliott- Halberstam conjecture is that any θ < 1 is permissible see [] for more information A generalized form of the Bombieri-Vinogradov theorem due to Hinz [9] shows that the primes in O K have some level of distribution θ, the specific value depending only on the number of complex conjugate embeddings of K; in particular, any totally real field has level of distribution θ for any θ < 1/ Finally, in the function field setting, Weil s proof of the Riemann hypothesis for curves implies that we may tae any θ < 1/ Sieve manipulations: Multiplicative functionology We are now ready to describe the Maynard Tao method in general terms; our exposition follows that of Maynard [11], to which we mae frequent reference We say that a tuple h 1,, h A is admissible if it does not cover all residue classes modulo p for any prime ideal p of A The main objects of consideration are the sums S 1 := α AN α v 0 mod w d 1,,d : d i α+h i i λ d1,,d and S := α AN α v 0 mod w χ P α + h i d 1,,d : d i α+h i i λ d1,,d, where χ P denotes the characteristic function of P, λ d1,,d are suitably chosen weights, w := p <D 0 p for some D 0 tending slowly to infinity with N, say D 0 = log log log N, and v 0 is a residue class modulo w chosen so that each α + h i lies in A/w Because each summand is non-negative, if we can show that S > ρs 1 for some positive ρ, then there must be at least one α AN for which more than ρ of the values α+h 1,, α+h are prime This is our goal, and it is where the art of choosing the weights λ d1,,d comes into play We begin by maing some assumptions regarding their support In particular, given d 1,, d, define d := d i, and set λ d1,,d = 0 unless d, w = 1, d is squarefree, and d R, where R will be chosen later to be a small power of AN The main result of this section is the following Proposition 1 Suppose that the primes P have level of distribution θ > 0, and set R = AN θ/ δ for some small δ > 0 Given a piecewise differentiable function F : [0, 1] R supported on the simplex R := {x 1,, x [0, 1] : x x 1}, let F max := sup t1,,t [0,1] F t 1,, t + F x i t 1,, t If we set λ d1,,d := µd i d i r 1,,r d i r i i r i,w=1 i µr 1 r log ϕr i F r1 log R,, log r log R

6 6 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON whenever d 1 d < R and d 1 d, w = 1, and λ d1,,d = 0 otherwise, then and S 1 = 1 + o1ϕw AN c A log R w +1 I F S = 1 + o1ϕw P N c A log R +1 w +1 m=1 J m F where c A is the residue at s = 1 of ζ A s, I F := F x 1,, x dx 1 dx R and 1 J m F := F x 1,, x dx m dx 1 dx m 1 dx m+1 dx [0,1] 1 0 Before we can prove Proposition 1, we first show that, by diagonalizing the quadratic form, we can rewrite S 1 and S We begin with S 1 Lemma For ideals r 1,, r, let y r1,,r = µr i ϕr i d 1,,d r i d i i λ d1,,d d i, and set y max = sup r1,,r y r1,,r If R = AN 1/ δ for some δ > 0, then S 1 = AN w r 1,,r y r 1,,r y ϕr i + O max AN ϕw log R w +1 D 0 Remar The change of variables to y r1,,r is invertible, the proof of which relies only on elementary manipulations see [11, p 9] Proof of Lemma We begin by expanding the square and interchanging the order of summation to obtain S 1 = λ d1,,d λ e1,,e 1 e 1,,e d 1,,d = AN w d 1,,d e 1,,e λ d1,,d λ e1,,e [d i, e i ] α AN α v 0 mod w α h i mod [d i,e i ] i + O λ max AN, w d 1,,d e 1,,e [d i, e i ] Here the on the summation indicates it is to be taen over those d 1,, d, e 1,, e for which the congruence conditions modulo w, [d 1, e 1 ],, [d, e ] admit a simultaneous solution; note that in that case, w, [d 1, e 1 ],, [d, e ] are pairwise coprime Now recall from 1 that AN, q AN / q 1 ν for some ν > 0, provided that AN q This implies that the above error is λ max AN 1 ν 1 [d i, e i ] 1 ν d 1,,d e 1,,e

7 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 7 For each q, the number of ways of choosing d 1,, d and e 1,, e so that [d i, e i ] = q is at most τ 3 q Hence our error is λ max AN 1 ν µ qτ 3q λ q max AN 1 ν R ν 1 ν q R p R p λ max AN 1 ν R ν log R 3 = λ max AN log R 3 AN /R ν To go from the first line to the second, we used a version of Mertens theorem for global fields See, for example, [15] This sort of estimation of sums by Euler products will be used frequently in what follows without further comment Because R = AN 1/ δ, this error is negligible compared to the error claimed in the statement of the lemma We now focus our attention on the main term Following Maynard s manipulations to uncouple the interdependence of d i and e j and maing the change of variables indicated in the statement of the lemma, the main term becomes AN w u 1,,u µu i ϕu i s 1,,,s, 1 1 i,j i j µs i,j ϕs i,j y a 1,,a y b1,,b, and we note, for consideration of the error term, that λ max y max log R In the above, a i := u i j i s i,j, b j := u j i j s i,j, and the on the summation indicates it is to be taen over s i,j such that s i,j, u i = s i,j, u j = 1 = s i,j, s a,j = s i,j, s i,b for all a i, b j Moreover, considering the support of the y s, if some s i,j 1, then s i,j > D 0 owing to the fact that s i,j, w = 1 The contribution in that case is at most 1 y max AN w u R u,w=1 µu ϕu µs i,j ϕs i,j s i,j >D 0 s A µs 1 y max AN ϕw log R ϕs w +1 D 0 We may thus restrict our attention only to those terms arising from s i,j = 1 for all i j The lemma follows We now turn to S, the handling of which will require more delicate information than was needed for S 1 We first define, for 1 m, the component sums S m := α AN α v 0 mod w χ P α + h m d 1,,d : d i α+h i i λ d1,,d so that S = m=1 Sm To rewrite S m in a manner similar to what was done with S 1, we need information about how the primes are distributed in arithmetic progressions Specifically, we will need the assumption that P has level of distribution θ > 0,

8 8 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON Lemma 3 Assume that P has level of distribution θ > 0 and that R = AN θ/ ε Let y r m λ d1,,d 1,,r = µr i gr i ϕd i, d 1,,d r 1 d i i d m=1 where g is the multiplicative function defined by gp = p for all prime ideals p of A Let y max m = sup r1,,r y r m 1,,r Then, for any fixed B > 0 we have S m P N m yr = 1,,r ϕw r 1,,r gr i y m max ϕw AN log N y + O + O max AN w 1 D 0 log N B Proof We begin by expanding out the square and swapping the order of summation, obtaining S m = χ P α + h m α AN α v 0 mod w [d i,e i ] α+h i i λ λ d1,,d e1,,e d 1,,d e 1,,e α AN α v 0 mod w [d i,e i ] α+h i i As in Lemma, we rewrite the inner sum over a single residue class modulo q = w [d i, e i ], which we may do if the ideals w, [d 1, e 1 ],, [d, e ] are pairwise coprime The element α + h m will lie in a residue class coprime to the modulus if and only if d m = e m = 1, the trivial ideal Based on our choice of v 0 mod w, this is the only case that yields a contribution We find that 1 ν P N χ P α + h m = + O AN ϕq w [d + O EN; q, α 0, i, e i ] where we recall that P N = P AN The first O-term is needed in the number field case, since it is α that is restricted to AN instead of α + h m Letting EN; q := max α0,q=1 EN; q, α 0, we thus find that S m P N λ d1,,d = λ e1,,e ϕw ϕ[d i, e i ] + O λ λ d1,,d e1,,e EN; q d 1,,d e 1,,e d 1,,d e 1,,e e m=d m=1 + O λ max AN 1 ν d 1,,d e 1,,e 1 [d i, e i ] 1 ν The second error term is Oλ max AN log R 3 AN /R ν, by an argument already appearing in the proof of Lemma This is negligible for us Now consider the first O- term For any q, there are at most τ 3 q ways to choose -tuples d 1,, d, e 1,, e such that w [d i, e i ] = q Recall that λ max y max log R Moreover, by our assumptions on

9 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 9 the support of λ d1,,d, the modulus q satisfies q w R Thus, the error term in contributes no more than 3 ymaxlog R µq τ 3 qen; q q <R w We now recall that we have assumed that the primes P have level of distribution θ, and we have taen R = AN θ/ ɛ Using the trivial bound EN; q AN /ϕq along with the Cauchy-Schwarz inequality, we therefore find that µq τ 3 qen; q 1/ µq τ3q AN 1/ µq EN; q ϕq q <R w q <R w AN log N B q <R w for any large B Now that we have handled the error term, we are free to concentrate on the main term As in the proof of Lemma, we decouple d i and e j by introducing an auxilliary summation over ideals s i,j, and we define the function multiplicative function ga by gp = p, so that 1 ϕ[d i, e i ] = 1 gu i ϕd i ϕe i u i d i e i Our main term can thus be written as P N 4 gu i ϕw u 1,,u u m=1 s 1,,,s, 1 1 i,j µs i,j d 1,,d e 1,,e u i d i e i i s i,j d i e j i j d m=e m=1 λ d1,,d λ e1,,e ϕd iϕe i We now mae the change of variables indicated in the statement of the lemma This yields P N µu i µs i,j ϕw gu i gs i,j ym a 1,,a y m b 1,,b, u 1,,u u m=1 u <R u,w=1 s 1,,,s, 1 1 i,j i j where the a i s and b j s are defined as in the proof of Lemma When some s i,j 1, the contribution is ym max AN µu µs µs i,j ϕw log N gu gs gs s i,j s i,j >D 0 ym max ϕw AN log R 1 w 1 D 0 log N

10 10 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON Putting all of this together, we find that S m P N m u = 1,,u ϕw gu i + O as claimed u 1,,u y y m max ϕw AN log R D 0 w 1 We note that the quantities y m r 1,,r can be related to the variables y r1,,r Lemma 4 If r m = 1, then y m r 1,,r = a m y r1,,r m 1,a m,r m+1,,r ϕa m ymax ϕw log R + O w D 0 + y max AN, log N B Proof The proof of this result relies upon combinatorial manipulations and standard estimates, and, using the ideas in Lemmas and 3 can be deduced almost mutatis mutandis from Maynard s proof of Lemma 53 [11] We are now ready to mae a specific choice of our sieve weights In particular, by choosing y r1,,r to be determined by the values of a smooth function, we will be able to express S 1 and S m in particularly { nice terms Thus, let F : [0, 1] R be a piecewise differentiable function supported on x 1,, x [0, 1] : } x i 1 If r = r i satisfies µr = 1 and r, w = 1, set log r1 y r1,,r := F log R,, log r log R and set y r1,,r = 0 otherwise In order to evaluate the summations of y r1,,r, we will need the following lemma, which is an analogue of a result of Goldston, Graham, Pintz, and Yıldırım [3, Lemma 4] This result also appears as Lemma 61 in [11] Lemma 5 Suppose γ is a multiplicative function on the nonzero ideals of A such that there are constants κ > 0, A 1 > 0, A 1, and L 1 satisfying and L w p <z 0 γp p γp log p p 1 A 1, κ log z/w A, for any w z Let g be the totally multiplicative function defined on prime ideals by gp = γp/ p γp Let G: [0, 1] R be a piecewise differentiable function, and let G max = sup t [0,1] Gt + G t Then µd gdg d <z log d log z = S cκ A where c A := Res s=1 ζ A s and S = p log zκ Γκ 1 0 Gxx κ 1 dx + O A,A1,A,κ LGmax log z κ 1, 1 γp κ p p

11 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 11 Remar In both [3] and [11], the analogous error term is asserted to be OSLG max log z κ 1 In other words, there is a factor of S not present in our statement However, the proofs appear to support this stronger estimate only if one maes a further assumption on the size of z compared to L Fortunately, this discrepancy is of no importance in the applications, as this error term is always subsumed by larger errors Proof Let Gz := d <z µd gd A straightforward argument using partial summation along the lines of that given explicitly by Goldston et al in their proof of [3, Lemma 4] reduces the claim to showing that Gz = S cκ A log zκ Γκ O A,A 1,A,κLlog z κ 1 for all z 1 This last assertion is an exact analogue of what is shown by Halberstam and Richert in their proof of Lemma 54 in [5] In fact, following their argument [5, pp ] essentially verbatim, we find that Gz = clog z κ + OLlog z κ 1 for some constant c and all z 1 Compare with equations 310 and 311 on pages 150 and 151 of [5] It remains only to show that c = c κ A S/Γ + 1 The argument at the bottom of p 151 of [5] shows that 1 c = Γκ + 1 lim sκ 1 + gp s 0 + p s p To compute the limit, note that ζ A s + 1 = p 1 p s 1 and that s c A /ζ A s + 1 as s 0 + This implies that lim sκ 1 + gp = c κ s 0 + p s A lim 1 + gp 1 1 κ s 0 p + p s p s+1 p Our opening assumptions on γp imply uniform convergence of the final product for real s 0 The proof of this follows the proof of the first part of Lemma 53 in [5] Thus, lim 1 + gp 1 1 κ = 1 + gp 1 1 κ = S s 0 + p s p s+1 p p Hence, c = c κ A S/Γ + 1, which completes the proof of the lemma Proof of Proposition 1 With all of our earlier results in place, the proof of this result follows from exactly the same reasoning as Maynard s proofs of Lemmas 61 and 6 [11] Here our Lemma replaces his Lemma 51, our Lemma 3 replaces his Lemma 5, our Lemma 4 replaces his Lemma 53, and our Lemma 5 replaces his Lemma 61 In fact, we find that the asymptotic estimates for S 1 and S asserted in Proposition 1 hold with errors that are OF max AN ϕw log R w +1 D 0 3 Final assembly of theorems Proposition 1 allows us to obtain the following analogue of [11, Proposition 4] Corollary 6 Suppose that the set of primes P in A has level of distribution θ > 0 Let H = h 1,, h be an admissible -tuple Let I F and J m F be defined as in the statement of Proposition 1 Let S denote the set of piecewise differentiable functions p

12 1 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON F : [0, 1] R supported on R := {x 1,, x [0, 1] : x i 1} with I F 0 and F 0 for each m Let J m M := sup F S m=1 J m F I F θm and let r := There are infinitely many α A such that at least r of the α + h i 1 i are prime Proof We mimic the proof of [11, Proposition 4] Recall from that if S := S ρs 1 > 0 for a certain N, then there are more than ρ primes among the α + h i 1 i, for some α AN Consequently, if S > 0 for all large N, then there are infinitely many translates of h 1,, h containing more than ρ primes Put R = AN θ/ ɛ for a small ɛ > 0 Choose F 0 S K so that m=1 J m F 0 > M ɛi F 0 Using Proposition 1, we see we can choose the weights λ d1,,d so that where S = ϕw w AN c +1 A log R c A log R P N AN ϕw w +1 AN c A log R I F 0 := c A lim N θ ɛ m=1 P N log AN AN J m F ρi F 0 + o1 M ɛ ρ + o1, The existence of this limit will be shown momentarily If ρ = Θ M / δ, then choosing ɛ sufficiently small, we get that S > 0 for large N Since δ > 0 was arbitrary, there must be infinitely many α A such that at least Θ M / of the α + h i 1 i are prime We now show that = 1, which will complete the proof of the proposition We consider separately the cases when A = F q [t] and when A = O K If A = F q [t], then ζ A s = 1 and so c 1 q 1 s A = 1 On the other hand, for N = log q qn, we have AN = q n and P N = q n /n + Oq n/ /n For all of these facts, see, eg, [16, pp 11 14] Thus, P N AN log q/ log AN as N = q n, and so = 1 Suppose now that A = O K, where K is a number field with r 1 real embeddings and r pairs of complex conjugate embeddings Consider the region in Minowsi space corresponding to the conditions defining A 0 N: {x 1,, x r1, z r1 +1,, z r1 +r R r 1 C r : 0 x i N, z j N for all 1 i r 1 and r j r 1 + r } This has volume N d π r On the other hand, the image of O K under the Minowsi embedding is a lattice with covolume r DK, where D K denotes the discriminant of K It follows that A 0 N π r N d / D, as N Since AN = A 0 N \ A 0 N, AN πr N d 1 1/ d D

13 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 13 We turn now to the estimation of P N For this, we employ Mitsui s generalized prime number theorem [1], a special case which is that the number of primes in A 0 N is w K du 1 du r1 +r r 1 hk Reg K logu 1 u r1 +r, [,N] r 1 [,N ] r as N Here w K is the number of roots of unity contained in K, h K is the class number of K, and Reg K is the regulator of K The integral appearing here is asymptotic to N d / log N d, by [8, Lemma 6] Hence, P N w K 1 1/ d N d r 1 hk Reg K logn d Finally, Dedeind s class number formula asserts that c A = r 1 π r h K Reg K w K DK Referring bac to the definition of, we find after some algebra that indeed = 1 As shown by Maynard [11, Proposition 413], we have M > log log log for all large enough values of In particular, M as So Theorem 11 follows at once from Corollary 6 provided that the primes in O K always possess a positive level of distribution Similarly, Theorem 13 follows provided that the primes in F q [t] possess a positive level of distribution not depending on q Both provisos were already asserted to hold in 1 In fact, we have the following: Theorem 7 Hinz Let K/Q be a number field with r pairs of complex conjugate embeddings If r = 0 ie, K is totally real, the set P of primes of O K has level of distribution θ for any θ < 1 1 In general, P has level of distribution θ for any θ < r + 5 Theorem 8 If A = F q [t], then the set P has level of distribution 1 Indeed, for all q and N, we have the stronger pointwise error estimate max α 0 mod q α 0,q=1 EN; q, α 0 log q AN 1/ Theorem 7 is contained in the somewhat more general main theorem of [9] Theorem 8 is a consequence of Weil s Riemann Hypothesis and was first deduced by Hayes [7] see also [1] 3 Applications 31 Bounded gaps in totally real number fields The proof of Corollary 1 maes use of the following simple observation Lemma 31 Suppose that H is an admissible tuple in Z Then H is also an admissible tuple in O K for every number field K Proof The reduction of H modulo p always lands in the prime subfield of O K /p and so cannot cover O K /p unless p has degree 1 But if p has degree 1, then O K /p = Z/pZ for some rational prime p, and H fails to cover Z/pZ since H is admissible in Z

14 14 A CASTILLO, C HALL, R LEMKE OLIVER, P POLLACK, AND L THOMPSON Suppose now that K is totally real By Theorem 7, the primes in K have level of distribution θ for any θ < 1 Maynard [11, Proposition 43] has shown that the number M 105 in Corollary 6 satisfies M 105 > 4 Corollary 1 follows now from Corollary 6, Lemma 31, and the result of Engelsma that there exists an admissible 105-tuple h 1 < h < < h 105 of rational integers with h 105 h 1 = Gap densities in F q t We now turn our attention to Theorem 14, which we recall concerns gaps between monic irreducibles of fixed large degree n Proof of Theorem 14 Let 0 := 0 from Theorem 13, and assume that q i We wish to show that any monomial a t d F q [t] occurs as a gap between monic irreducibles of degree n for every sufficiently large n satisfying n d, q 1 = 1 For any q, the tuple {αt d : α F q } is admissible, and so, because q 0 + 1, we may apply Theorem 13 We thus see that, for each sufficiently large n, some monomial c t d occurs as a gap between monic irreducibles of degree n; call these irreducibles f 1 t and f t If n d, q 1 = 1, there is an ω F q such that ω n d = c/a, and we note that the polynomials f 1 ωt/ω n and f ωt/ω n are monic and irreducible We then compute that f 1 ωt ω n f ωt ω n = c ωtd ω n = c ω n d td = a t d ii We now turn our attention to the second part of Theorem 13 concerning the proportion of degree d polynomials that appear as gaps in degree n Let Z, d, n denote the assertion that, for any admissible -tuple h 1,, h such that each of h 1,, h and h 1 h, h 1 h 3,, h 1 h is of degree d, there is an f F q [t] of degree n such that at least two of f + h 1,, f + h are monic and irreducible; we note that Theorem 13 implies that Z 0, d, n holds for any d provided that n is sufficiently large We will prove by induction on 0 that if Z, d, n holds, then the proportion of polynomials 1 of degree d appearing as gaps in degree n is at least 1 1 q 1 If =, the assertion is clear: Z, d, n implies that every non-zero polynomial of degree d appears as a gap For 3, we note that either Z 1, d, n holds or it doesn t If we are in the former case, then, as 1/ 1 is decreasing, the conclusion follows On the other hand, if Z 1, d, n does not hold, then there must be h 1,, h 1 as above for which there is no f F q [t] of degree n such that two of f + h 1,, f + h 1 are monic irreducibles Now q 0 +1 > ; thus, for any h of degree d with each difference h h 1,, h h 1 also of degree d, the tuple h 1,, h 1, h is admissible Since we are assuming Z, d, n holds, there must be an f of degree n for which f + h and some f + h i are both monic irreducibles; hence, h h i occurs as a gap Varying over the q 1 1 q d such h, each gap can appear at most 1 times, whence the number of distinct gaps is at least q 1 1 q d 1 Noting that there are q d q 1 elements of degree d, the claim follows Lastly, the assertion about monomials comes from only looing at tuples h 1,, h with each h i a distinct monomial of degree d

15 BOUNDED GAPS BETWEEN PRIMES IN NUMBER FIELDS AND FUNCTION FIELDS 15 Acnowledgements This wor began at the American Institute of Mathematics worshop on arithmetic statistics over finite fields and function fields We would lie to than AIM for providing the opportunity for us to wor together References [1] M Car Distribution des polynômes irréductibles dans F q [T ] Acta Arith, 88: , 1999 [] J Friedlander and H Iwaniec Opera de cribro, volume 57 of American Mathematical Society Colloquium Publications American Mathematical Society, Providence, RI, 010 [3] D A Goldston, S W Graham, J Pintz, and C Y Yıldırım Small gaps between products of two primes Proc Lond Math Soc 3, 983: , 009 [4] D A Goldston, J Pintz, and C Y Yıldırım Primes in tuples I Ann of Math, 170:819 86, 009 [5] H Halberstam and H-E Richert Sieve Methods Dover boos on mathematics Dover Publications, 011 [6] C Hall L-functions of twisted Legendre curves J Number Theory, 1191:18 147, 006 [7] D R Hayes The distribution of irreducibles in GF[q, x] Trans Amer Math Soc, 117:101 17, 1965 [8] J G Hinz On the theorem of Barban and Davenport-Halberstam in algebraic number fields J Number Theory, 134: , 1981 [9] J G Hinz A generalization of Bombieri s prime number theorem to algebraic number fields Acta Arith, 51: , 1988 [10] D A Kaptan A generalization of the Goldston-Pintz-Yildirim prime gaps result to number fields Acta Math Hungar, 1411-:84 11, 013 [11] J Maynard Small gaps between primes Preprint available at [1] T Mitsui Generalized prime number theorem Jap J Math, 6:1 4, 1956 [13] P Pollac An explicit approach to Hypothesis H for polynomials over a finite field In Anatomy of integers, volume 46 of CRM Proc Lecture Notes, pages Amer Math Soc, Providence, RI, 008 [14] Polymath8 Bounded gaps between primes title=bounded_gaps_between_primes [15] M Rosen A generalization of Mertens theorem J Ramanujan Math Soc, 141:1 19, 1999 [16] M Rosen Number theory in function fields, volume 10 of Graduate Texts in Mathematics Springer- Verlag, New Yor, 00 [17] J Thorner Bounded gaps between primes in Chebotarev sets Preprint available at [18] Y Zhang Bounded gaps between primes Ann of Math, 179: , 014 Department of Mathematics, University of Illinois at Chicago, Chicago, IL address: acasti8@uicedu Department of Mathematics, University of Wyoming, Laramie, WY address: chall14@uwyoedu Department of Mathematics, Stanford University, Palo Alto, CA address: rjlo@stanfordedu Department of Mathematics, University of Georgia, Athens, GA address: pollac@ugaedu Department of Mathematics, Oberlin College, Oberlin, OH address: lolathompson@oberlinedu