Small gaps between primes

Transcription

1 Small gas between rimes Andreea Mocanu March 5, 5 Abstract In this aer we study the wor of James Maynard [], in which he roves that lim inf n n+m n <, thus establishing that there are infinitely many intervals of finite length that contain a fixed number of rimes. We use Sieve theory, in articular Selberg s sieve, and we loo at reliminary wor in order to better understand this result. To conclude, we discuss the imlications of this result. Contents Introduction Sieve theory 3. Sieve of Eratosthenes Selberg s sieve Admissible sets Level of distribution of rimes The Goldston Pintz Yıldırım theorem 8 4 The Maynard theorems 7 4. Main roosition Selberg sieve maniulations Smooth choice of y Otimisation and consequences Main results Conclusions and further directions 59

2 Introduction The study of rime numbers has intrigued eole since antiquity. It reresents the basis of Analytic Number Theory. In 896 the rime number theorem was roved indeendently by de la Vallée Poussin and Hadamard, describing the asymtotic behaviour of rimes. Their wor was based on that of Riemann and, with the clever use of Comlex Analysis contour integration, they roved that lim x πx x/ log x =. We write this as πx x/ log x, where we define πx := x, the number of rimes less than or equal to x. Equivalently, we can weight each rime with a weight log to obtain the estimate ϑx := log x. x This imlies that the average ga between consecutive rimes less than or equal to a given n is aroximately log n, i.e. rimes become less common as they become larger. Peole were concerned with investigating gas between rime numbers more thoroughly, and since then many conjectures have arisen. We focus on small gas between rimes and when we say small we mean in comarison to the average ga. There are sectacular results in this area in the recent years. In a aer rinted in 5 [7], Goldston et al. roved, among other things, that: lim inf n n+ n log n =, where n is the n th rime, thus showing that there are infinitely many consecutive rimes that have an arbitrarily small ga comared to the average ga. In their roof they use Sieve Theory, which is a comilation of techniques used to count a large set of integers which satisfy certain roerties usually involving rime numbers. Furthermore, under the Elliott Halberstam conjecture regarding the level of distribution of rimes, they rove that: lim inf n n+ n 6, in other words there are infinitely many airs of rime numbers that differ by 6 or less.

3 In 3, Yitang Zhang ublished a aer [6] in which he established the first finite bound on gas between rime numbers, showing that lim inf n n+ n 7 7. His method was a refinement of that of Goldston et al. and the result was a breathrough as the roof is unconditional. The olymath roject [] reduced the ga to 468, by further develoing Zhang s techniques. The aim of this roject is to study the wor of James Maynard, who, a few months after Zhang, used a generalisation of the methods used in [7] to rove that lim inf n n+m n <. In other words, there are infinitely many intervals of finite length that contain m + or more rimes. He also imroved Zhang s estimate considerably by obtaining lim inf n n+ n 6, and, under the Elliott Halberstam conjecture, that lim inf n n+ n and lim inf n n+ n 6. We will first discuss some sieving techniques which are useful in studying gas between rimes. Then we tal about admissible sets and the Elliott Halberstam conjecture, as well as the level of distribution of rimes, at which oint we state the Bombieri Vinogradov theorem. In Section 3 we discuss the methods used in the Goldston et al. aer, which will then hel us discuss Maynard s techniques and results in Section 4. Sieve theory Sieve Theory is an imortant tool used in Number Theory, whose modern version emerged more or less years ago. To this day, there are a handful of sieves that are used more often than others. Our focus will be on the sieve of Eratosthenes, as an introduction, and the Selberg sieve, which we use in our analysis of Maynard s wor. Let us begin with a formal definition of a sieve, as stated in []: Definition.. Let A be a finite set of objects and let P be an indexed set of rimes such that to each P we have associated a subset A of A. We define the set SA, P := A \ P A The urose of sieve theory is to estimate the size of this set from above and below. 3

4 In general, A is a set of ositive integers and A is a subset of A consisting of elements lying in secific congruence classes modulo. For examle, if A =, N, P = {, N} and A = {n A, n mod and n+ mod }, then SA, P is the set of all twin rimes N that are not in P. We now introduce the sieve of Eratosthenes, which was written down in the form we resent by A.M. Legendre in 88, in the second edition of his boo Théorie des Nombres.. Sieve of Eratosthenes The sieve of Eratosthenes is a simle way of sifting out rimes u to a certain uer bound, x. Informally, we mae a list of all the integers, 3,..., x, where by x we denote the greatest integer less than or equal to x. We start by calling a rime and crossing all of its multiles off the list. As 3 is uncrossed, we call it a rime, too, and roceed by crossing off all of its multiles. We then ic the next uncrossed number and reeat the algorithm until the next uncrossed number is greater than x. All the numbers that are uncrossed by the end are rime. Formally, we wish to study the number Φx, z := #{n x : n is not divisible by any rime < z}, 3 where x, z are ositive real numbers. Let P z := <z. We can use the following result about the Möbius function, µ, to write Φx, z = n x µd = d n d n,p z = x d P z µd d { if n =, otherwise. µd = = x µd d d P z µd d P z n x d n + Oz = x + O z, <z 4 5 4

5 where in the first line we used the fact that #{n x : n modulo d} = x/d = x/d + O. We can relate this result to πx. We have πx = πx πz + πz Φx, z + πz Φx, z + z. 6 Since x e x holds for ositive x, which can be shown by derivation, it imlies that ex. <z <z In addition to this, we use the following result: z log log z + O, which we rove using artial summation from the following theorem of Chebycheff.4.5 in []: Theorem.. z log log z + O. We do the summation by arts: z = z log log = log z z log z + O + log z z = + O + = + O log z z log + z t log / t log dt t log t + O t log dt t z log z t log t dt + O t log t dt + O + log log z, log z which gives the required result. Using this, we can obtain an uer bound for Φx, z and, by substituting in 6, for πx. We choose z := c log x for a small ositive constant c to get the result: 5

6 Proosition.. πx x log log x. We now want to study a more general setting. Let A be a set of natural numbers lest than or equal to x and let P be a set of rimes. To every P we assign a number ω of distinct residue classes modulo. Remar. In Proosition., ω = and we fix the residue class mod. Let A denote the set of elements of A belonging to at least one of these distinct residue classes modulo, let A := A and for any squarefree integer d comosed of rimes in P we define A d := d A and ωd := d ω. The Chinese remainder theorem ensures ωd is well defined: given two distinct rimes and and a residue class modulo each of these rimes, there exists a unique residue class modulo. We roceed by induction. Let z be a ositive real number and define P z := P <z We define SA, P, z to be the number of elements of the set A \ A. P z Assume that there exists an X such that for some R d.. #A d = ωd d X + R d Theorem.3 The Sieve of Eratosthenes. With the above setting, suose the following conditions hold:. R d = Oωd; 6

7 . for some κ, P z ω log κ log z + O; 3. for some ositive real number y, #A d = for every d > y. Then SA, P, z = XW z + O X + y log z κ+ ex log y, log z log z where W z := P <z ω. Remar. As we assign more residue classes to each rime ω increases, κ becomes bigger and, as a consequence, so does the error term in the third condition. Remar 3. In Proosition., κ = and the second remar becomes <z log log z + O, which was roven by Mertens see [] for alternative roof. The roof requires the following lemmata, which we quote from [] without roof: Lemma.4. With the setting and hyotheses of Theorem.3, let Then d P z d>y F t, z := d t d P z ωd. F t, z = O tlog z κ ex log t. log z Lemma.5. With the setting and hyotheses of Theorem.3, ωd = O log z κ+ ex log y. d log z 7

8 Proof of Theorem.3. We want to use the inclusion-exclusion rincile from combinatorics, namely n n B i = + B i... B i. = i <...<i n Using this and the first and third hyotheses of Theorem.3, we get SA, P, z = d P z d y = X µd#a d = d P z µd ωd d d P z d y µd Xωd d d P z d>y + OF y, z µd ωd + OF y, z. d We write the first sum inside the main term as its Euler roduct and we aly Lemma.5 to the second one and Lemma.4 to the error term to obtain SA, P, z = XW z + O X + y log z κ+ ex log y. log z log z. Selberg s sieve In 947, Selberg came with a clever contribution in estimating Φx, z, as defined in 3. His idea was to relace the Möbius function in 5 with a quadratic form, which can then be minimised for otimal results. This method was then used with small modifications by Goldston et al. in [7] and subsequently by Maynard, in a multi-dimensional setting, to obtain the results mentioned in the Introduction. Selberg made the observation that for any sequence λ d of real numbers such that λ =, the following holds: µd λ d. d This is because the left-hand side of the inequality is equal to if = and to otherwise, while the right-hand side is if = and greater than or equal to otherwise. d 8

9 Hence, from 5 we get Φx, z n x d n,p z = n x = x λ d d,d n,p z d,d P z λ d λ d d,d P z λ d λ d n x [d,d ] n λ dλ d [d, d ] + O d,d P z λ d λ d where by [a, b] we ve denoted the least common multile of a and b. Since our sequence λ d was chosen arbitrarily aart from the one condition, we may mae the assumtion that λ d = for d > z and obtain Φx, z x λ d λ d [d, d ] + O λ d λ d. 7 d,d z d,d z, Our urose is to estimate the main term so we loo at d,d z λ d λ d [d, d ] 8 as a quadratic form in λ d d z and we try to minimize it. We use the fact that [d, d ]d, d = d d, where we ve denote by a, b the greatest common divisor of a and b, and that δ d φδ = d, where φ is Euler s totient function, i.e. φδ is the number of ositive integers less than or equal to δ that are relatively rime to δ. We write 8 as d,d z λ d λ d [d, d ] = d,d z = δ z φδ λ d λ d d d d, d = d,d z δ d,d λ d λ d d d so we have diagonalised our quadratic form to φδu δ, δ z 9 d,d z = δ z λ d λ d φδ d d δ d,d λ d φδ, d d z δ d

10 under the linear transformation u δ := d z δ d λ d d. 9 This transformation is invertible and we can use the dual Möbius inversion formula..3 in [] to obtain λ δ δ = d µ u d. δ d z δ d Bearing in mind our conditions on λ d, we have u δ = for δ > z and µdu d = λ =. Using this, we can write d z φδu δ = δ z δ z φδ u δ µδ + φδv z V z, where V z := d z µ d/φd. We want to minimise this quadratic form under the constraint. It is easy to see that we have a minimal value of /V z, which occurs when u δ = µδ φδv z = λ δ = δ δ d µd/δµd φdv z. Using this, we can rove that λ d for any d see.7 in [] for roof. Substituting our results into 7, we obtain Φx, z x V z + Oz. Remar 4. We can use this to obtain a better estimate for πx than Proosition., namely: πx x log x, which is Chebyscheff s uer bound for πx. The roof is again based on writing πx Φx, z + z

11 and we use to estimate Φx, z. We have d z µd φd d z µd d = d z d d, d z where the d z denotes that we are summing over non squarefree d. We now quote without roof the following roosition from [].3.3: Proosition.6. d z d z d = log z + O. We use this and the fact that d 4 d z/4 since if d has a squarefree divisor, say, then d z imlies d/ z/4 and also d 4d/. We obtain πx d, x log z + z. We then need to choose an aroriate z again, which in this case will be z := x/ log x /. This gives the result. We now discuss the general Selberg sieve. Let A be a finite set of natural numbers and let P be a set of rimes. For every P, let A be a subset of A notice that, unlie the sieve of Eratosthenes, A does not necessarily deend on fixed residue classes modulo. Let A := A and for any squarefree integer d comosed of rimes in P we define A d := d A. Let z be a ositive real number and define P z and SA, P, z as before. Theorem.7 Selberg s sieve. With the above setting, suose there exists X > and a multilicative function f which satisfies f > for any rime P, such that for any squarefree integer d comosed of rimes in P we have #A d = X fd + R d

12 for some real number R d. Write fn = d n f d, 3 where f is a multilicative function which is uniquely determined by f using the Möbius inversion formula, in other words f n = d n µdfn/d. Set Then SA, P, z V z := d z d P z µ d f d. X V z + O R [d,d ]. d,d z d,d P z Remar 5. Note the inequality sign in Selberg s sieve, comared to the equality we had in Theorem.3. The main terms in the two sieves have the same order of magnitude both are Ox/ log z, but there is an imrovement in the error term. Remar 6. In, in the formula for V z, we have f to be the Euler function, hence fn = d n φd = n is the identity function. The roof of this theorem is analogous to that of : Proof of Theorem.7. As before, we start with a sequence λ d of real numbers satisfying λ = and λ d = for d > z. We want to obtain a similar estimate for SA, P, z as we did for Φx, z. By the inclusion-exclusion rincile, we have SA, P, z = a A a/ A P z We define, for any a A, = d P z µd a A d = a A Da :=, P a A µd d P z a A d and if a / A for any P, then Da :=. We have µd = µd λ d = d P z d P z,da d P z,da a A d λ d d P z a A d..

13 Substituting in our formula for SA, P, z, we obtain: SA, P, z a A = d,d z = X λ d d P z a A d d,d z d,d P z = λ d λ d #A [d,d ] a A λ d λ d f[d, d ] + O d,d P z a A [d,d ] d,d z d,d P z λ d λ d λ d λ d R [d,d ] using. We loo at the sum in our main term as a quadratic form in λ d d z, which we want to diagonalise and then minimise in order to finish the roof. For that, we need to use the fact that for a multilicative function f and two ositive squarefree integers d and d, f[d, d ]fd, d = fd fd, 4 which is similar to what we have used before. Using this and 3, we obtain d,d z d,d P z λ d λ d f[d, d ] = d,d z d,d P z = d,d z d,d P z = δ z δ P z f δ = f δ δ z δ P z λ d λ d fd fd fd, d λ d λ d fd fd d,d z d,d P z δ d,d d z d P z δ d δ d,d f δ λ d λ d fd fd λ d fd. Hence our quadratic form is reduced to the diagonal form f δu δ, δ z δ P z 3,

14 under the linear transformation u δ := λ d fd, d z d P z δ d which is invertible and by the dual Möbius inversion formula we have λ δ fδ = d µ u d. δ d P z δ d Bearing in mind our conditions on λ d, we have u δ µdu d = λ =. Using this, we can write d z d P z δ z δ P z f δu δ = δ z δ P z f δ u δ µδ + f δv z V z, = for δ > z and from which we deduce that our quadratic form has a minimal value of /V z, which occurs when µδ u δ = f δv z, since the coefficients aearing in the quadratic form, f d, are ositive by multilicativity of f..3 Admissible sets In Maynard s aer, the main focus is on -tules of rime numbers of the form {n + h,..., n + h }. He roves that there are infinitely many intervals of finite length that contain rimes. This is something number theorist have been woring on for a long time. In 849, de Polignac conjectured that for every ositive even natural number there are infinitely many consecutive rime airs { n, n+ } such that n+ n =. Zhang roved that this holds for some < 7 7 and the result is being constantly imroved with the hel of sieve theory and comuter rograms. The case = is the famous twin rime conjecture. We start by considering a set H = {h,..., h } of natural numbers. For each rime we loo at the set Ω = {n mod : n h i mod for some h i H} of residue classes in H modulo. 4

15 Definition. Admissible set. We call a set H defined as above admissible if, for every rime, Ω <. Remar 7. We want our set H to generate -tules of rimes of the form {n + h,..., n + h }. Therefore, if Ω = for some, at least one of the n + h i would be congruent to modulo and therefore could not be a rime. Remar 8. Given a set H, it is enough to chec if it is admissible u to the biggest rime less than or equal to, since for a rime greater than it is imossible to cover all residue classes modulo said rime. Examle.. The set H = {, 3, 5} is not admissible. We need to chec for rimes less than or equal to 3: 3 5 mod, so Ω = {}, however mod 3, 3 mod 3 and 5 mod 3, so Ω3 = {,, 3}. G. H. Hardy and J. E. Littlewood [3] stated the following conjecture regarding -tules of rimes: Conjecture.8 Hardy-Littlewood. Let H be an admissible set. Then there exist infinitely many -tules of rimes of the form {n+h,..., n+h } and in fact we can count them asymtotically: x #{n x : n + h,..., n + h all rime} GH log x, where we define GH := Ω 5 and we call it the singular series. Remar 9. We can see that for =, the conjecture is equivalent to saying that #{n x : n + h rime} G{h}x/ log x, and in this case Ω = for all, imlying G{h} =. Thus we get a restatement of the rime number theorem,. Given an admissible set H = {h,..., h }, we can use Selberg s sieving technique to get an uer bound for the set S = {n : n + h,..., n + h all rime and n [N, N]}, for some large N, which will later become of interest to us. Let P n := n + h... n + h. 6 5

16 We want to use the same tric as the one in the Selberg sieve. To estimate the size of our set S we need a function which is equal to if all of the n+h,..., n+h are rime and non-negative otherwise, to relace the Möbius function. We define a sequence λ d with λ = and λ d = for d > R for some R. Then S, N n N d P n as long as we fix R to be less than N. This is because if this holds, when all of the n + h i s are rime, the only non-zero term in the inner sum is for d =. We want to extend Ω multilicatively similar to how we extended ω multilicatively in Subsection. as n Ωd if and only if n Ω for all d. We can use the Chinese remainder theorem to show that this imlies Ωd = d Ω. We follow the same stes as in the roof of to get Ω[d, d ] S x λ d λ d + O Ω[d, d ] λ d λ d. [d, d ] d,d d,d R Note that, comared to 7, there is an extra factor of Ω[d, d ] aearing both in the error term and in the main term. This is because if n Ωd for some d, then so does n + d, so when we slit the interval [N, N] into x/d intervals of length d, each interval contains exactly Ωd integers n Ωd. To minimise the main term, we now from [4] that the best choice for diagonalising λ d is log R/d λ d µd. 7 log R With this choice, we get S O N!GH, log N which is a factor of! away from the result stated in the conjecture..4 Level of distribution of rimes We will now tal about rimes in arithmetic rogressions and we will state the Bombieri Vinogradov theorem. We define πx;, a to be the number of rimes x with a mod. The definition maes sense only when a is corime to, as otherwise would λ d 6

17 be divisible by a and hence would not be a rime, unless a =. Note that the case = is simly πx. The rime number theorem for rimes in rogression states: πx;, a lix φ, 8 where lix := x log t dt. When =, we get πx lix and it is nown that lix x/ log x, thus we get the rime number theorem. Note that a does not aear in the right-hand side of 8, imlying that rimes in arithmetic rogressions are evenly distributed, i.e. an arithmetic rogression of the form a mod contains asymtotically as many rimes as one of the form b mod whenever a and b are corime to. It is roven in [5] that, for any fixed N >, πx;, a = lix φ + Ox ex c log x holds for all x and all integers, a with a, = and log x N and c is an absolute constant. We now define the level of distribution of rimes: Definition.3 Level of Distribution. Given θ >, we say that rimes have level of distribution θ if, for any A >, the following holds: max πx a,= πx;, a φ x A log x. 9 A x θ Under the Generalised Riemann hyothesis GRH, it can be shown that πx;, a = πx φ + Ox/ log x, so if we allow to vary u to x / and tae the maximum over it, this imlies that rimes have level of distribution θ for θ < /. Bombieri and Vinogradov validated this claim indeendently in 965 without the use of the GRH, by roving the following theorem: Theorem.9 Bombieri Vinogradov. For any A > there exists B = BA > such that max liy y x a,= πy;, a φ x A log x. A max x/ log x B 7

18 Remar. We allow to vary u to x / /log x B since we can find ε > such that x / ε x / /log x B x /. Remar. We are also taing the maximum over all y x, which is an imrovement to our definition. Their roof was based estimating certain L-functions in various rectangles with the use of the large sieve method, which we quote from []: Theorem. The large sieve. Let A be a set of natural numbers lest than or equal to x and let P be a set of rimes. For every P suose we are given a set {w,,..., w ω, } of ω distinct residue classes modulo. Let z be a ositive real number and define P z as we have before. We define SA, P, z to be the number of elements of the set {n A : n w i, mod i ω, P z}. Then where SA, P, z z + 4πx, Lz Lz := d z µ d d ω ω. It was conjectured by Elliott and Halberstam [3] that rimes have level of distribution θ for θ <, while Friedlander and Granville [5] roved that 9 does not hold if we relace x θ with x/log x B for any fixed B, lie the result obtained in Theorem.9. We can also define an equivalent of the function ϑ for rimes in rogressions, by ϑ y; q, a := ϖn, y<n y n a mod q where we define ϖn to be equal to log n if n is a rime and otherwise. It can be shown that is equivalent to max a,q= ϑ y; q, a y φq x log x. A max y x q x θ 3 The Goldston Pintz Yıldırım theorem As we mentioned in the Introduction, Goldston et al. Selberg s sieve to rove the following theorem: use a variation of 8

19 Theorem 3. GPY. lim inf n n+ n log n =. This shows that there are infinitely many consecutive rimes that have an arbitrarily small ga between them, comared to the average ga imlied by. We will now highlight some of the ideas in their roof. Proof of Theorem 3.. We start by introducing some basic notation. Let N be a arameter that increases monotonically to infinity and we fix H and R such that H log N log R log N. By and l we denote arbitrary ositive integers that are bounded. Remar. We will see later on that the condition that and l are chosen arbitrarily is crucial, because the correct choice enables us to comlete the roof. Let H = {h,..., h } be an admissible set defined as before and we imose on it the condition that H [, H]. We next define a sequence λ R d; a in a similar way to 7, by { if d > R, λ R d; a = µd log R a if d R. a! d Our first aim is to evaluate the quantity N<n N n Ωd λ R d; + l 3 Note that n Ωd d P n with P n defined as in 6. We let Λ R n; H, a := n Ωd λ Rd; a and we claim that the following holds: Lemma 3.. With the above notation, for R N / /log N C, where C > deends only on and l, Λ R n; H; + l = GH l Nlog R +l + l! l N<n N with GH defined as in 5. + ONlog N +l log log N c, 9

20 Setch of roof of Lemma 3.. We can exand the square in 3 and write our quantity as NT + O Ωd λ R d; + l, with d T = Ω[d, d ] λ R d ; + lλ R d ; + l. [d, d ] d,d To estimate the error term, note that if d is a squarefree integer which can be written as the roduct of m distinct rimes, we have that τ d = m, where τ is the generalised divisor function, τ n = {a,..., a : a... a = n}. Then, since H = and we require Ω to be less than be definition, we have We have Ωd = Ω... Ω m m = τ d. ζ s = n= τ n n s. We can then use Perron s discontinuous integral and the Leibniz rule for differentiation to get τ n = n x xlog x! + Oxlog x, hence we can write our error term as R a O τ d log = ORlog R log R a d d = OR log R c. 4 To estimate the main term, we use contour integration along cleverly chosen vertical lines. We concentrate on the main ideas and ignore the error terms. First, note that we can write λ R d; a = µd πi s R ds, 5 d sa+ as long as a, where by α we denote the vertical line assing through α in the comlex lane. We use this to write T as T = R s +s F s πi, s ; Ω s s ds ds +l+, 6

21 with F s, s ; Ω = µd µd Ω[d, d ] [d, d ]d s, d,d d s which can also be written as F s, s ; Ω = Ω in the region of absolute convergence. We then let Gs, s ; Ω := F s, s ; Ω + s s s +s, ζs + ζs +, ζs + s + which is regular and bounded for Rs, Rs > /. We get the singular series in the main term from G, ; Ω = Ω = GH, by definition. We have Gs, s ; Ω = Oex c log N σ log log log N, with σ := minrs, Rs,, which we obtain by writing out log Gs, s ; Ω exlicitly. We can use G to estimate T by shifting both contours in 6. We let U = ex log N and we shift the s -contour to the vertical line c log U + it and the s -contour to c log U +it, where t R and c is a sufficiently small ositive constant. Next, we truncate the contours to t U and t U/ and we denote the results by L and L. With these choices 6 becomes T = πi L L Gs, s ; Ω + Oex c log N. ζs + s + R s +s ζs + ζs + s s ds ds +l+ We then shift the s -contour to L3 = c log U + it, for t U and we encounter a ole of order l + at s = and a ole of order at s = s. We use Cauchy s residue theorem to bound Res asymtotically and obtain s = s T = πi L { Res s = } ds + Olog N +l / log log N c, 7

22 for some constant c. In order to estimate this, we define Zs, s = Gs, s ; Ω s + s ζs + s +, s ζs + ζs + which is regular and bounded in a neighbourhood of the oint,, since G is regular in that area, while the oles of ζs i + and the zeros given by s i at s i = cancel each other in the denominator. Cauchy s residue formula gives us l { } Res = Rs Zs, s s = l!s l+ s s + s Rs. s = We substitute this into 7 and we shift the contour to L 4 = c log U + it, for t U/, which gives us an error term of size Oex c log N again. We have a new ole at s = and we use Cauchy s residue formula to obtain T = Res Res + Olog s = s = N+l = Zs, s R s+s πi s + s s s ds ds l+ + Olog N +l, 8 C C where we denote by C and C the circles s = ρ and s = ρ, resectively, where ρ > is chosen to be small. We write s = s and s = ξs and substitute into the main term of 8 to obtain T = Zs, ξsr πi C 3 C sξ+ ξ + ξ l+ s dsdξ + Olog +l+ N+l, where C 3 is the circle ξ =. We first evaluate I = Zs, sξr πi C sξ+ ds. s +l+ By Cauchy s integral formula, we have We note that I = m Zs, ξs s m +l Zs, ξsr sξ+. + l! s s= = m! πi s= C,δ Zs, ξs s m+ ds = Olog log N c,

23 where C, δ is the circle centered at of radius δ and we obtain the asymtotic by choosing δ = /log log N. On the other hand, m R sξ+ = log R m ξ + m, s m s= and we can use Leibniz rule for differentiation to obtain Z, ξ + T = log R+l πi + l! C3 l dξ + Olog N +l log log N c. ξ l+ We estimate the integral I = ξ + πi C3 l dξ = ξ l+ l! l ξ + l = ξ l ξ= We obtain T = l. l GH l log R +l + Olog N +l log log N c 9 + l! l Observe that the estimate we obtained in 4 gets absorbed into the error term given by 9 when we multily through by N, because of the initial conditions imosed on R and N. Hence, the roof is comlete. We next want to introduce weights for rimes in our calculations. We define the function ϖ to be ϖn := log n for n rime, and otherwise. We now want to evaluate the quantity ϖn + hλ R n; H, + l, 3 N<n N where h is an arbitrary ositive integer less than or equal to H. We claim that the following holds. Lemma 3.3. Suose that the following assumtion holds: There exists an absolute constant < θ < such that, for any fixed A >, holds. Then, with the notation from Subsection.4, for R N θ/, we have ϖn + hλ R n; H, + l N<n N = GH {h} l Nlog R +l + l! l + ONlog N +l log log N c if h H, GH l + Nlog R +l+ + l +! l + + ONlog N +l log log N c if h H. 3

24 Setch of roof of Lemma 3.3. First, note that we may assume that h H since the factor n + h does not affect the comutation of Λ R n; H; + l. Then, let δx = if x = and otherwise. As before, we exand the square and write 3 as λ R d ; +lλ R d ; +l δb+h, [d, d ]ϑ N; b+h, [d, d ], d,d b Ω[d,d ] with an error term of size OR log N c, which we can obtain in the same way as 4. Note that we introduced the function δ in the inner sum because ϑ N; b + h, [d, d ] = when b + h and [d, d ] are not corime. We now use this and to write our quantity as N T N + O, log N A/3 where T = d,d λ R d ; + lλ R d ; + l φ[d, d ] b Ω[d,d ] δb + h, [d, d ]. We obtain the error term by comaring Ω to the generalised divisor function τ again. To evaluate the main term, we write the inner sum in the exression of T as δb + h, = Ω +, [d,d ] b Ω [d,d ] where Ω + is defined to be the function Ω for the set H {h}. This is because δb + h, = h Ω, meaning that δb + h, = for each residue class from H excet for one. We use 5 again to get with T = πi R s +s F s, s ; Ω s s ds ds +l+, F s, s ; Ω = [d µd µd,d ] Ω+ φ[d, d ]d s d,d d s, We can write F equivalently as F s, s ; Ω = Ω+ 4 + s s s +s,

25 in the same region of absolute convergence as that of F. We define G s, s ; Ω = F ζs + ζs + s, s ; Ω, ζs + s + and we consider two cases, according to whether h H or h H. If h H, for > H we have Ω + = +. If H + is admissible, we have, as before, G, ; Ω = GH +, and we can estimate T = GH+ + l! l l log R +l + Olog N +l log log N c. On the other hand, if H + is not admissible, we have GH + = and the main term in the revious estimate vanishes, while the error term remains the same. If h H, the above calculations hold with the translation, l l +. This is because for a rime > H, Ω + is equal to instead of + and + l + = + l. Hence, the roof is comlete. We are now ready to bring everything together. We want to evaluate the exression ϖn + h log 3N λ R n; H, + l, 3 H [,H] H = N<n N h H We want to rove that it is ositive for sufficiently large N, using Lemmata 3. and 3.3, so we set R = N θ/. If this turns out to be true, then the inner sum must be ositive, meaning that there exists an integer n N, N] such that ϖn + h log 3N >. h H This in turns imlies that there exists a subinterval of length H in N, N+H] which contains at least two rimes. That is because, if there are no rimes, then the sum taes a negative value and if there is only one rime, say, occurring in the interval, then we have log log 3N log N + H log 3N. We now from our initial conditions that H log N < N, so N +H < 3H, maing our exression negative. If, however, there were at least two rimes in the subinterval, we would have min r+ r H. 3 N< r N+H 5

26 In order to rove that 3 is ositive, we need to quote the following result from [6]: GH = + oh, 33 H [,H] H = as H tends to infinity. Using this and Lemma 3., 3 becomes + ϖn + hλ R n; H, + l N<n N h H h H H [,H] H = h H h H l NH log Nlog R +l + onh log N +l+. + l! l Using Lemma 3.3 and 33 again, this is asymtotically equal to l NH + log R +l + l! l l + + NH log R +l+ + l +! l + l NH log Nlog R +l + l! l { } l + = H + log R log N + l + l + l NH log R +l. + l! l So 3 is ositive as long as H log N + ε l + θ + l + l +, for any ε >. If we choose l =, it is enough to require H log N + ε θ. By Bombieri Vinogradov theorem, we can tae θ = / ɛ for any ɛ >, so we want H log N + ε + ɛ. 6

27 Choose ɛ = ε to get that we need Going bac to 3, we have H log N 3ε. 34 r+ r r+ r min < min N< r N+H log r N< r N+H log N H log N, 35 We now from our initial conditions that H = Olog N, so we can ic any δ > and ic H = δ log N. Then, choose ε = δ/3, so that 34 holds. Also, 35 becomes r+ r min N< r N+H log N δ, which in turn roves the theorem. Furthermore, under the Elliott Halberstam conjecture, Goldston et al. use the admissible 6-tule {7,, 3, 7, 9, 3} to show lim inf n n+ n 3 7 = 6. 4 The Maynard theorems We now loo at Maynard s wor, which revolves around the following conjecture: Conjecture 4. Prime -tules conjecture. Let H = {h,..., h } be admissible. Then there are infinitely many integers n such that all of n + h,..., n + h are rime. He roves the following theorems: Theorem 4.. holds for any integer m. lim inf n+m n m 3 e 4m, n This shows that there are infinitely many intervals of finite length that contain at least m + rimes. 7

28 Theorem 4.3. Let m N. Let r N be sufficiently large deending on m, and let A = {a, a,..., a r } be a set of r distinct integers. Then #{{h,..., h m } A : for infinitely many n all of n + h i are rime} #{{h,..., h m } A } m. In other words, a ositive roortion of admissible m-tules satisfy the rime m-tule conjecture for every m. Theorem 4.4. We have lim inf n+ n 6. n We note that the roof of this last theorem is based on the Bombieri Vinogradov theorem, namely on the fact that rimes have level of distribution θ for every < θ < /. If, however, we assume that rimes have level of distribution θ for every θ <, we can rove the following: Theorem 4.5. Under the assumtion that rimes have level of distribution θ for every θ <, we have lim inf n+ n, n lim inf n+ n 6. n To rove these results, Maynard uses an imroved GPY sieve method, by looing at the following setting: fix > and fix an admissible set H = {h,..., h } of size. Consider the quantity SN, ρ = χ P n + h i ρ w n, 36 N n<n where χ P is the characteristic function of the rimes, i.e. χ P n = if n is rime and otherwise, ρ > and w n are non-negative weights. As in Theorem 3., we want to rove that this quantity is ositive. In that case, at least one term in the sum over n must be ositive. Since w n is ositive, we must have that at least ρ+ of the n+h i are rime. So, as we let N, we get that there are infinitely many bounded length intervals containing ρ + rimes bounded by max h. Unlie the weights Λ R n; H, a used in Theorem 3., we consider our weights to be of the form w n =. 37 d i n+h i i 8 λ d,...,d

29 We will choose our λ d,...,d to loo lie λ d,...,d µd i fd,..., d, for a suitable smooth function f, which is a multi-dimensional generalisation of our choice for λ R d; a in the revious section. We begin by introducing some notation. For convenience, we choose our weights w n to be unless n ν mod W, where ν is a fixed residue class mod W, and W = D, where it suffices to choose Then, D = log log log N. log W = D log = ϑd, which, by calculations in [].8, is OD. Hence, W = Olog log N. Since H is admissible, we can choose ν such that ν + h i is corime to W, by alying the Chinese remainder theorem to the rimes dividing W. When n ν mod W, we define our weights as in 37. In order to estimate SN, ρ, we loo at the following two sums S =, 38 S = N n<n n ν mod W N n<n n ν mod W d i n+h i i λ d,...,d χ P n + h i as we did with 3 and 3 in Theorem Main roosition We aim to rove the following: d i n+h i i λ d,...,d, 39 Proosition 4.6. Let the rimes have level of distribution θ > and let R = N θ/ δ for some small fixed δ >. Let λ d,...,d be defined in terms of a smooth function F by µ λ d,...,d = µd i d r i log i φr i F r log R,..., log r, log R r,...,r d i r i i r i,w = i 9

30 whenever d, W =, and let λ d,...,d = otherwise. Moreover, let F be suorted on R = {x,..., x [, ] : x i }. Then we have rovided I F and J m S = + oφw Nlog R W + I F, S = + oφw Nlog R + W + log N I F = m= F for each m, where... J m F, F t,..., t dt... dt, J m F =... F t,..., t dt m dt... dt m dt m+... dt. Remar 3. Note that in the exressions for S and S there is no deendency on the actual elements of the admissible set H, only on its size, whereas in Lemma 3. and Lemma 3.3, the singular series aears in the main term, which clearly deends on the h i s. 4.. Selberg sieve maniulations Our aim is to introduce a change of variables to rewrite S and S in a simler form. We restrict the suort of λ d,...,d to tules for which the roduct d := d i is less than R, similarly to how we had λ R d; a = for d > R in the revious section. We also demand d, W = and µd =, imlying that d is squarefree and so d i, d j = for all i j. We want to rove the following lemma: Lemma 4.7. Let y r,...,r = µr i φr i r,...,r d,...,d r i d i i λ d,...,d d. i Let y max = su r,...,r y r,...,r. Then S = N y r,...,r y W φr i + O max Nlog R. 4 W D 3

31 Remar 4. This change of variables is a multi-dimensional generalisation of the change of variables 9 we erformed on the quadratic form λ d in Subsection.. Proof. We exand the square and swa the order of summation in 38 to obtain S = λ λ d,...,d e,...,e. d,...,d e,...,e N n<n n ν mod W [d i,e i ] n+h i i We can use the Chinese remainder theorem again, this time to write the inner sum as a sum over a single residue class modulo q = W [d i, e i ], as long as W, [d, e ],..., [d, e ] are airwise corime. In that case, we have that the inner sum is N/q + O. If W, [d, e ],..., [d, e ] are not airwise corime, then the inner sum has no contribution: suose some [d i, e i ] has a common factor a, say, with W. Then a n + h i and n ν mod a, imlying a n ν. So a h i + ν, contradicting the corimality condition between W and h i + ν. On the other hand, if some [d i, e i ] and [d j, e j ] have a common factor b, say, then n + h i and n + h j are both divisible by b, contradicting the fact that they are distinct rimes. Hence, we have S = N λ d,...,d λ e,...,e W [d + O λ d,...,d λ e,...,e, i, e i ] d,...,d e,...,e d,...,d e,...,e where denotes the corimality restriction on W, [d, e ],..., [d, e ]. We now λ d,...,d is non-zero only when d i < R, so if we denote by λ max = su d,...,d λ d,...,d, we can estimate the error term as O τ d = Oλ maxr log R, 4 λ max d<r by the exact same argument as in Section 3.. To deal with the main term, we want to remove the deendencies between the d i and the e j variables. We start by using once again the fact that [d i, e i ]d i, e i = d i e i and that δ d φδ = d to write [d i, e i ] = d i e i u i d i,e i φu i, so that our main term becomes N λ d,...,d φu i λ e,...,e W d i e i. u,...,u d,...,d e,...,e u i d i,e i i 3

32 We now that λ d,...,d is non-zero only when W, d i =, so we may dro from the summation the requirement that W is corime to each [d i, e i ], and we also now that we can assume that the d i and the e i are all airwise corime for the same reason. Thus we are only left with the condition that d i, e j = for all i j, which we can remove by multilying our exression by s i,j d i,e j µs i,j, since 4 holds. Our main term then becomes N W u,...,u φu i s,,...,s, i,j i j µs i,j d,...,d e,...,e u i d i,e i i s i,j d i,e j i j λ d,...,d λ e,...,e d i e i. We can restrict the s i,j to be corime to u i and u j because otherwise one of the airs {d i, d j },{e i, e j } would have a common factor, maing the corresonding λ vanish. By a similar argument, we can restrict our sum so that s i,j is corime to s i,a and s b,j for all a j and b i. We denote the summation over the s i,j with these restrictions by. We now want to introduce a change of variables, which will mae it easier for us to diagonalise our quadratic form. y r,...,r = µr i φr i d,...,d r i d i i This change of variables is in fact invertible, since y r,...,r φr i = µr i r,...,r d i r i i r,...,r d i r i i r i e i i r,...,r d i r i i = λ e,...,e e,...,e e i r,...,r r i /d i e i /d i i r,...,r d i r i i r i e i i d i λ d,...,d d. 4 i e,...,e r i e i i λ e,...,e e i µr i We tae a closer loo at the inner sum. We have ri µr i = µ µd i = µd i m,...,m m i n i i µm i = µd i m i n i i µm i, 3

33 where we let m i to be r i /d i and n i to be e i /d i. The inner roduct is equal to if n i = for all i and otherwise, or in other words it is equal to when e i = d i and otherwise. Thus, we obtain r,...,r d i r i i y r,...,r φr i = λ d,...,d µd id i 43 since /µn = µn holds for all n. Since r i d i for all i, bearing in mind our conditions on the d i, we have that y r,...,r are non-zero only when the following conditions hold: the r i are airwise corime, r i, W = for all i and r i < R. We now want to bound λ max asymtotically as a function of y max, where we let y max = su r,...,r y r,...,r. We tae r = m i and we use the fact that µ = µ, µn = if n is squarefree and otherwise and that n = d n φd n = φn e n /φe to write λ max su y max d i d,...,d d i squarefree u<r y max y max su d,...,d d i squarefree su d,...,d y max u<r d d i µu τ u, φu µd φd d i φd i r,...,r d i r i i r i<r r i squarefree r<r/ d i r, d i= r<r/ d i r, d i= µr τ r φr µr i φr i µr τ r φr where in the last line we too u = dr and used the fact that τ dr τ r by definition. We now tae a closer loo at the inner sum without the Möbius function. We now from the revious section that u<r τ u c Rlog R, for some constant c. We have τ u log u log R R c Rlog R u R + log t c tlog t dt t R c log R + c c log R + c log R +. log t log t dt t 33

34 We have φu = u u and so φu = u + u u d u d u u = u d = d u log u, u u Where in the last ste we used the following theorem from []: σu := d u d u log u. 44 So /φu log u/u and u<r τ u φu τ u log u u u<r log R +. Therefore, the error term given by 4 is OymaxR log N 4. We use this and we substitute our change of variables 4 into the main term to obtain S = N φu i µs i,j W u,...,u s,,...,s, i,j i j µa i µb i y a,...,a φb i φa i y b,...,b + OymaxR log R 4, where a j = u j i j s j,i and b j = u j i j s i,j. We bear in mind that we have restricted the s i,j to be corime to all the other terms in the exressions for a i and b j. By definition, the y are equal to when the a i and the b j are not squarefree, so we can write µa j = µu j i j s j,i, and the same holds for φa i, µb j and φb j. Therefore, we can write S = N W u,...,u µu i φu i s,,...,s, i,j i j µs i,j y φs i,j a,...,a y b,...,b + Oy maxr log R Bearing in mind the corimality conditions between the a i and b j and W, we have that the s i,j must be corime to W, otherwise the y vanish. Since 34

35 W = <D, we only need to consider s i,j = or s i,j > D. When the latter holds, the contribution from the main term in 45 is y maxn W u<r u,w = µu µs i,j φu φs i,j s i,j >D s µs, 46 φs where we note that the last term aears because we have choices for the air i, j inside the innermost roduct of 45. We now loo at each sum searately. The last sum is a convergent series and so it is O, since we now from [] that the following holds σnφn = n φn 4σn n log n = n 4 n 4 log n, n where in the last ste we used 44. We now estimate the middle factor in 46, using Chen s result again: µn φn φn 4 σn. 47 n 4 n D n D n D We now from Ramanujan s Identity..9 in [] with a = b = that σn = ζsζs ζs. n s ζs n= The right-hand side has a ole of order at s = 3, so by the Tauberian theorem, Ax := n x σn cx 3, for some constant c. We can now use summation by arts in 47 to obtain µn φn AM AD M At + 4 M 4 D 4 D n M D t dt 5 c 4 M c M. D t D By letting M, we obtain that the middle factor in 46 is O/D. Next, we want to rove: u<r u,w = µu φu φw log R. 48 W 35

36 First, we note that the µ in the numerator ensures we are woring with squarefree u s. We then loo at: u AR = φu. u<r u,w = Given that φu = u u holds for any u, we get AR = = u<r u,w = u W u<r u. 49 The first roduct in the right-hand side of 49 is recisely φw /W and what is left is u<r u/φu. It is roved by Murty 4.4. in [] that: u φu R. u<r So we have shown: AR φw R. 5 W We now do the summation by arts: φu = u φu u u<r u,w = u<r u,w = = R R AR + At t dt. Using 5, the first term of the sum becomes O and is dominated by the second one, which is: R φw W t φw dt = t W R φw dt = log R. t W Bringing everything together, 46 can be estimated as y O max φw Nlog R. W + D Going bac to our main estimate, we are left to consider the case when s i,j = for all i j. We have a i = b i and the u i are squarefree, so µu i =. Therefore S = N W u,...,u y u,...,u y φu i +O max φw Nlog R W + D 36 + ymaxr log R 4 5

37 We have R = N θ/ δ, so R = N θ δ N δ, since < θ. We also have φw /W < and W = exϑd = W exc log log log N = log log N c, where to obtain the inequality we used that θn n log for roof see.8 in []. So the first error term dominates. Since φw /W <, the lemma is roved. We now want to estimate for S. We write S = m= Sm, where S m = N n<n n ν mod W χ P n + h m d,...,d d i n+h i i λ d,...,d. 5 We want to estimate S m in a similar way to how we estimated S. We rove the following lemma: Lemma 4.8. Let y m r,...,r = µr i gr i d,...,d r i d i i d m= λ d,...,d φd i, 53 where g is the totally multilicative function defined on rimes by g =. Let y max m = su r,...,r y r m,...,r. Then for any fixed A > we have S m = N φw log N y m + O m yr,...,r r,...,r gr i max φw Nlog N W D y + O max N. 54 log N A Proof. We exand the square and swa the order of summation in 5 to obtain S m = λ λ d,...,d e,...,e χ P n + h m. d,...,d e,...,e N n<n n ν mod W [d i,e i ] n+h i i As before, we can write the inner sum as a sum over a single residue class modulo q = W [d i, e i ], as long as W, [d, e ],..., [d, e ] are airwise corime. If either one air of W, [d, e ],..., [d, e ] have a common factor, 37

38 then the inner sum has no contribution for the same reasons as in the revious lemma. In that case, we will have a contribution in the inner sum only when n+h m is rime, i.e. when it will lie in a residue class corime to the modulus. This haens if and only if d m = e m =. The inner sum will then contribute X N /φq + OEN, q, where EN, q = + su a,q= N n<n n a mod q χ P n φq N n<n χ P n, is an alternative definition for the level of distribution of rimes and X N = χ P n. Therefore we have S m = X N φw d,...,d e,...,e e m=d m= N n<n λ d,...,d λ e,...,e φ[d i, e i ] + O λ λ d,...,d e,...,e d,...,d e,...,e EN, q with q = W [d i, e i ]. We now estimate the error term. We now that the λ d,...,d are nonzero only when q < W R. Then, given a squarefree integer r, we have at most τ 3 r choices of d,..., d, e,..., e for which W [d i, e i ] = r, since [d i, e i ] deends on d i, e i and d i, e i. We use this and the fact that λ max y max log R + to estimate the error term as y max log R + µr τ 3 ren, r. r<r W Note that the Möbius function ensures we are woring with squarefree r. We have the trivial bound EN, r N/φq and we also assumed that the rimes have level of distribution θ, so we can use the Cauchy-Schwarz inequality to obtain the estimate the error term as y maxlog R + r<r W µr τ3r N / / µr EN, r, φr r<r W which gives an error term of size Oy maxn/log N A for any fixed A >. To estimate the main term, we remove the condition d i, e j = by multilying our exression by s i,j d i,e j µs i,j, lie we did in the revious 38,

39 roof. We can again restrict s i,j to be corime to u i, u j, s i,a and s b,j for all a j and b i. We denote by the summation with these restrictions. Next, we claim that φ[d i, e i ] = φd i φe i u i d i,e i gu i, where g is the totally multilicative function defined on rimes by g =. Using 4, we now that φ[d, e] = φdφe/φd, e, for squarefree d and e. We are left to show that φc = u c gu, for squarefree c. Since Euler s totient function is multilicative, it is enough to chec that this is true when c is a rime. Indeed, φ =, while the right-hand side is equal to +. Our main term becomes X N λ d,...,d gu i µs i,j λ e,...,e φw φd i φe i. Let u,...,u y m r,...,r = s,,...,s, i,j µr i gr i d,...,d e,...,e u i d i,e i i s i,j d i,e j i j d m=e m= d,...,d r i d i i d m= λ d,...,d φd i, Since d m = and r m d m, r m must be equal to. We substitute this into our estimate to obtain X N µu i µs i,j y m φw gu i gs i,j a,...,a y m b,...,b, u,...,u s,,...,s, i,j i j where a j = u j i j s j,i and b j = u j i j s i,j. As before, the y are equal to when the a i and the b j are not squarefree, so we can write µa j = µu j i j s j,i, and the same holds for φa i, µb j and φb j. Again, we consider the two cases of s i,j = and s i,j > D. When the latter holds, we get a contribution which is ym max N φw log N u<r u,w = µu gu 39 µs i,j gs i,j s i,j >D s µs. 55 gs

40 We have, for squarefree n, gn = n n /. We consider, again for squarefree n, the Dirichlet series fs = gn n = +. s s n x We then loo at the quotient fs + ζs + = s The corresonding Dirichelt series is s+ s = s+ = + s+ s. s+ s+ s s+ s+, which converges for Rs >. Thus we may write fs = ζs + hs, or equivalently fs = ζshs, with hs regular for Rs >. The Tauberian theorem gives n gn c x, n x and we can aly similar calculations to the ones in the revious lemma to obtain estimates for the last term of 55. We obtain that 55 is ym max φw Nlog R, W D log N where we have bounded X N by N/ log N, since X N = πn πn = X N = N/ log N + ON/log N. When s i,j =, we obtain S m = X N φw y m + O u,...,u y m u,...,u gu i max φw Nlog R W D y + O max N, 56 log N A since log R/ log N = θ/ δ <. We aly our estimate ofx N to this and obtain an error term of size ym max N µu ym max φw Nlog R 3, φw log N gu W u<r u,w = where we used the fact that we can estimate the inner sum as OφW /W log R the same way we obtained 48 and that R = N θ/ δ again. This is absorbed by the first error term in 56, which comletes the roof. 4

41 We now want to relate our variables y m r,...,r to the variables y r,...,r from Lemma 4.7. We rove the following lemma: Lemma 4.9. If r m = then y m r,...,r = a m y r,...,r m,a m,r m+,...,r φa m d,...,d r i d i i d m= ymax φw log R + O. W D Proof. We start by substituting 43 into 53 and we assume that r m =. We have y r m µd i d i y a,...,a,...,r = µr i gr i φd i φa i. a,...,a r i a i i a,...,a d i a i i We swa the summation between the d and the a variables and obtain y r m y a,...,a,...,r = µr i gr i µd i d i φa. i φd i d,...,d d i a i,r i d i i d m= We evaluate the sum over the d variables exlicitly. We want to rove the following: µd i d i = µa i r i φd i φa i. 57 i m d...d d i a i,r i d i d m= We consider the -dim case of 57 and rove that instead. The result follows by induction: µdd φd = µar φa. 58 d a r d We see that the condition r d a is equivalent to d/r a/r and we relabel d/r = s. Using this and the multilicativity of the Möbius and of the Euler totient functions, 58 becomes: s a/r µsµrsr φsφr = µar φa s a/r µss µrr φs φr = µar φa, and by searating the terms inside the sum that do not deend on s we get 58 s a/r µss φs = µar φr φa µrr. 4

42 Using the multilicativity of µ and φ again and relabelling a/r = n, we see that we are in fact trying to rove: s n µss φs = µn φn. 59 Let F n = s n µss φs. This is a multilicative function by definition, as all the functions inside the sum are multilicative. Hence, bearing in mind we are woring with square-free variables in 57 and 58, it is enough to rove 59 for rimes. Indeed, F = φ φ = = = µ φ, and so 59, 58 and 57 hold. Note that 59 does not hold for non squarefree numbers, as µ t = for t, imlying F = µ/φ always. So we have y m r,...,r = µr i gr i a,...,a r i a i i y a,...,a µa i a i φa i φa i. i m From the suort of the y, we now that we can restrict the summation over the a j so that a j, W =, so we either have a j = r j or a j > D r j. When j m, the contribution from when a j > D r j is µa j y max gr i r i φa j a j >D r j gr i r i φr i y max φw log R W D a m<r a m,w = y maxφw log R W D, µa m µa i φa m φa i i r i a i i j,m where we got from the first ste to the second by using the same bounds as in 46 and, in order to obtain the last estimate, note that the roduct in the second line is over a finite number of i s, so it is O. Therefore, when a j = r j for all j m, we have y m r,...,r = gr i r i φr i a m y r,...,rm,am,rm+,...,r φa m 4 ymax φw log R + O. W D