aa r X i v : . [ m a t h . N T ] N ov ON THE GREATEST PRIME FACTOR OF ab + 1 ´ETIENNE FOUVRY Abstract.
We improve some results on the size of the greatest prime factorof the integers of the form ab + 1 where a and b belong to some general givenfinite sequences A and B with rather large density. Introduction
Let N be an integer > A and B be two sets of integers, both includedin [1 , N ]. With these two finite sets, we build the set C = C ( A , B ), defined by C = C ( A , B ) := (cid:8) ab + 1 ; a ∈ A , b ∈ B (cid:9) . Let P + ( n ) be the greatest prime factor of the integer n if n > P (1) = 1. Theobject of this paper is to give a lower bound for the integer Γ + ( A , B , N ) := max c ∈C P + ( c ) , in terms of N and of the cardinalities |A| and |B| . The interest of this question isthat we suppose no condition of regularity for the sets A and B , but we only imposesome lower bound for |A| and |B| . The purpose of the present paper is to improvethe following Theorem A. (See [27, Theorem 2] ) For any positive ǫ , there exist positive con-stants c , c and c , depending at most on ǫ , in an effective way, such that, for any N > c , for any subsets A and B of [1 , N ] , satisfying the inequalities |A| , |B| > c N ((log N ) / log log N ) , we have the inequality (1) Γ + ( A , B , N ) > min n N − ǫ )(min( |A| , |B| ) /N ) , c ( N/ log N ) o . It is important to note that, in the particular case where A and B are densesubsets of [1 , N ], (which means that they satisfy |A| , |B| > δN , for some fixedpositive δ and N → + ∞ ), we then have Γ + ( A , B , N ) ≫ δ N δ , where δ is apositive function of δ . However the relation (1) never produces a lower boundbetter than N .Actually, much more is conjectured since in [24, Conj.1], the authors propose thefollowing Date : July 23, 2018.1991
Mathematics Subject Classification.
Primary 11R29; Secondary 11R11 .
Key words and phrases. greatest prime factor, primes in arithmetic progressions.The author benefited from the financial support of Institut Universitaire de France.
Conjecture 1.
For every ǫ satisfying < ǫ < , there exists N ( ǫ ) and C ( ǫ ) > ,such that, for every integer N > N ( ǫ ) , for every A and B ⊂ [1 , . . . , N ] satisfying (2) |A| , |B| > ǫN, we have the inequality (3) Γ + ( A , B , N ) > C ( ǫ ) N . Such a conjecture becomes false if we only impose the lower bounds |A| and |B| > ǫ ( N ) · N , where ǫ ( N ) is a function of N tending to 0 as slowly as we want, when N tends to infinity. To see this, choose p a prime satisfying (2 ǫ ( N )) − p ǫ ( N ) − and consider A = { a N ; a ≡ p } and B = { b N ; b ≡ − p } . Forsuch A and B , we easily see that Γ + ( A , B , N ) ( N + 1) /p = o ( N ).Before stating our results, we first give some general considerations on the set C .1.1. The subset of Linnik–Vinogradov.
Let LV ( N ) := { n ; n N , n = ab with 1 a, b N } . Hence LV ( N ) is the set of (distinct) products of two integers N . The study ofthe cardinality of this set is not an easy task at all, this a question due to Linnikand Vinogradov. K.Ford [8, Corollary 3] has now solved this problem by proving(4) | LV ( N ) | ≍ N (log N ) c (log log N ) , where c has the value c = 1 − = 0 .
086 07 . . . . (for a slightly weakerresult see [16, Theorem 23]). The relation (4) shows that LV ( N ) is a sparse subsetof [1 , N ], but only by a tiny power of log N .Hence, for any A and B , we have the trivial relation C ( A , B ) ⊂ LV ( N ) + { } , which shows in which sparse subset of [1 , N + 1], the set C lives obligatorily.To complete the description of the scenery of our problem, we recall the basicproperties of the classical function Ψ( x, y ), which counts the integers less than x with all their prime factors less than y . In other words, we define S ( x, y ) := { n x ; P + ( n ) y } and Ψ( x, y ) := | S ( x, y ) | . We only appeal to the rather easy resultΨ( x, y ) = xρ (cid:16) log x log y (cid:17) + O (cid:16) x log y (cid:17) , uniformly for x > y > ρ isthe Dickman function (see [28, p.370]), This function quickly goes to zero, since itsatisfies ρ ( u ) / Γ( u + 1) , ( u > . Using the above formula, the Stirling formula, and the inclusion–exclusion prin-ciple, we see that (cid:16) LV ( N ) + { } (cid:17) ∩ (cid:16) [1 , · · · , N + 1] \ S ( N + 1 , y ) (cid:17) = ∅ , N THE GREATEST PRIME FACTOR OF ab + 1 3 as soon as N is sufficiently large and y satisfies y > exp (cid:16) c log N log log log N log log N (cid:17) , where c is some absolute positive constant. This means that, with a naive ap-proach, we proved that the shifted Linnik–Vinogradov set LV ( N ) + { } , containsan element divisible by a prime(5) p > N c N log log N . We now state our results. They correspond to three different situations, whichappear to be more and more difficult. We can already feel the depth of Conjecture1 in the very particular case A = B = [1 , . . . , N ] (this corresponds to the condition(2) with ǫ = 1, and > replaced by > ). This very particular situation will be theobject of Theorem 1.1.2. The case A = B = [1 , . . . , N ] . Our first step will be to prove
Theorem 1.
For every
A > , there exists N = N ( A ) , such that, for every N > N , the interval [(1 − (log N ) − A ) N , N ] contains a prime p of the form p = ab + 1 , where a and b are integers satisfying a, b N .In particular, for N > N ( A ) we have the inequality Γ + ( A , B , N ) > (cid:16) − N ) A (cid:17) N , under the constraints A = B = [1 , . . . , N ] . Such a result has to be compared with the weak result given in (5) and it isfar from being trivial by the tools which will be involved. Theorem 1 implies thatthe inequality (3) of Conjecture 1 is true for any C ( ǫ ) <
1, in the particular case A = B = [1 , . . . , N ] . Its proof will be given in §
5, one of its qualities is to givea first idea of the difficulty of the proof of Conjecture 1, if such a proof exists.In our proof, we shall appeal to the Siegel–Walfisz Theorem concerning primes inarithmetic progressions. This fact prevents to produce an effective value for N ( A ),above. The same remark applies to Theorems 2 and 3 below.A very delicate question is to find the asymptotic expansion of the cardinalityof the set of primes belonging to LV ( N ) + { } . This question was treated in [17,Corollary 3], [8] and finally in [19, Corollary 1.1] which gives the asymptotic orderof magnitude of this cardinality.1.3. The case A = [1 , . . . , N ] and B general. This is the second step in ourgraduation of difficulty. In § Theorem 2.
Let δ satisfying < δ < . There exist an absolute constant c ,independent of δ , and a constant c = c ( δ ) , such that, for any N > c , for anysubset B of [1 , . . . , N ] satisfying the inequality (6) X b ∈B (1 − δ ) N N log N · (log log N ) c , there is a prime p in the interval ](1 − δ ) N , (1 − δ ) N ] of the form p = ab + 1 with a and b N and b ∈ B . ´ETIENNE FOUVRY In particular, if A = [1 , . . . , N ] and if B satisfies (6), we have Γ + ( A , B , N ) > (1 − δ ) · N , for any sufficiently large N . The condition (6) is not artificial at all for the following reason: the order ofmagnitude of the prime number p is almost N , hence, in the equality p = ab + 1,both a and b must be close to N . This implies that the set B must contain manyelements in the neighborhood of N .1.4. The case A and B general. In the more general situation, we shall prove
Theorem 3.
For every real number < δ < , there exists a function ̟ δ : N → R ∗ tending to zero at infinity, such that, for every N > , for every subsets A and B of [1 , . . . , N ] , satisfying (7) |A| > |B| > N (log N ) δ , the following inequality holds (8) Γ + ( A , B , N ) > N |A| /N )(1 − ̟ δ ( N )) . It is possible to describe the function ̟ δ more precisely, but this description willdepend on non explicit constants (see the comment after Theorem 1). Comparedwith (1), we see two advantages in Theorem 3. When A is more and more dense,the exponent of N in (8) tends to 2. This gives some consistency to Conjecture 1.In the other direction, if A and B satisfy |A| ∼ |B| ∼ N/ (log N ) δ with 0 < δ < / Γ + ( A , B , N ) > N − ǫ )(log N ) − δ , for N > N ( ǫ ). By (1), we would have have the same lower bound but with δ replaced s by 2 δ , thus Theorem 3 represents a valuable improvement for sparsesequences.Actually, after a talk given by the author at the Congress Activit´es Additives etAnalytiques (Lille, june 2009) where he exposed the results of the present paper,C. Elsholtz kindly turned our attention on a preprint of K. Matom¨aki on the samesubject. This work is now published ([21]) from which we extract the followingcentral result
Theorem B. ( [21, Theorem 2] Let C and c be positive. Then for every c satisfying c < − c − C , there exists N ( c ) such that, for every N > N ( c ) , for every A and B ⊂ [1 , . . . , N ] ,satisfying |A| > C N log N and |B| > |A| N c |A| /N , we have Γ + ( A , B , N ) > N √ c |A| /N . N THE GREATEST PRIME FACTOR OF ab + 1 5 When writing the proof of Theorem B, the author was unaware of [27], this isthe reason why she only refers to the older and weaker result [26]. Hence it is worthcomparing the strength of Theorems A & B. Theorem B really takes into accountthe situation where B is much thinner than A (a typical situation being |A| ≍ N and |B| ≍ |A| · N − δ with δ > Γ + ( A , B , N ) better than Γ + ( A , B , N ) > N √ − ǫ = N . ··· , insteadof Γ + ( A , B , N ) > N . ··· by Theorem A.It is also worth noticing that in [21, §
4] the author expresses the presentimentof the importance of the work of Bombieri, Friedlander and Iwaniec [4] to improveher results. The present paper confirms this intuition.
Acknowledgements.
The author is grateful to C. Elsholtz for letting himknow the existence of [21]. He also warmly thanks C. Stewart for his stimulatingconversations on the subject.2.
Tools from analytic number theory
Primes in arithmetic progressions.
In the rest of this paper, we reservethe letter p to prime numbers. We shall also systematically write L = log 2 x, where x >
1, is a real number we consider as tending to infinity.The proofs of Theorems 1, 2 & 3 are based on deep properties of the classicalfunction in prime number theory π ( x ; q, a ) = X p xp ≡ a mod q , when a and q are coprime integers. In particular, its behavior has to be comparedwith the function π ( x ) /ϕ ( q ), where π ( x ) is the cardinality of the set of primes x and ϕ ( q ) is the Euler function of the integer q . We recall some properties of thiscounting function on average in arithmetic progressions. The most classical one isthe Bombieri–Vinogradov Theorem (see [1], [29], [2, Th´eor`eme 17], [18, Theorem17.1], ... for instance) Proposition 1.
For every A , there exists B = B ( A ) such that (9) X q Q max y x max ( a,q )=1 (cid:12)(cid:12)(cid:12) π ( y ; q, a ) − π ( y ) ϕ ( q ) (cid:12)(cid:12)(cid:12) = O A (cid:0) x L − A (cid:1) , uniformly for Q x L − B and x > . In many applications, this proposition replaces the Riemann Hypothesis ex-tended to Dirichlet’s L –functions. The best constant for the moment is B = A + 1.But much more is conjectured: it is largely believed that (9) is true for Q = x − ǫ ,for any ǫ > O –constant depending now on A and ǫ ). This is the content ofthe Elliott–Halberstam Conjecture (see [7]). The proof of the Bombieri–VinogradovTheorem (see [18], for instance) is now presented as an elegant and deep consequenceof the large sieve inequality for multiplicative characters and of the combinatorialstructure of the characteristic function of the set of primes or of the van Mangoldtfunction Λ( n ) (see Lemma 7 below). It is a challenge to improve the value of Q in(9), even by modifying the way of summing the error terms π ( y ; q, a ) − π ( y ) ϕ ( q ) or evenby approaching the characteristic function of the set of primes by the characteristic ´ETIENNE FOUVRY function of another set of the same, but easier, combinatorial structure. The firstbreakthrough in that direction is due to Fouvry and Iwaniec [14] (see also [9]) and itwas followed by several papers of Bombieri, Fouvry, Friedlander and Iwaniec ([10],[11], [15], [13], [3], [4], [5]...) Also see [2, §
12 p.89–103] for an introduction to thesetechniques, based on Linnik’s dispersion method and on several types of bounds forKloosterman sums.For the problem we are studying in the present paper, we shall restrict to twopoints of view. The first one is
Proposition 2. (See [12, Corollaire 1] & [3, Theorem 9] ) For every non zero integer a , for every A , we have the equality (10) X q Q ( q,a )=1 (cid:16) π ( x ; q, a ) − π ( x ) ϕ ( q ) (cid:17) = O a,A (cid:0) x L − A (cid:1) , uniformly for Q x L − A − and x > . Note that in (10), we are summing the error terms, without absolute value, onconsecutive moduli q . Hence we benefit from oscillations of the signs of this errorterm. In the proof, this oscillation is exploited by kloostermania i.e by the studyof sums of Kloosterman sums with consecutive denominators. This is the heart ofthe work of Deshouillers and Iwaniec [6].To continue the presentation of our tools we recall the classical functions in primenumber theory θ ( x ), ψ ( x ) and ψ ( x ; q, a ) given by θ ( x ) = X p x log p, ψ ( x ) = X n x Λ( n ) and ψ ( x ; q, a ) = X n xn ≡ a mod q Λ( n ) . We also introduce the notation(11) q ∼ Q to mean that q satisfies the inequalities Q q < Q .In the second variation in the thema of Proposition 1 we sum the error termswith absolute values giving Proposition 3. (see [4, Main Theorem] ) There exists an absolute constant B withthe following property :For every integer a = 0 , for every x , y , and Q satisfying x > y > , Q xy ,we have the inequality X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) ψ ( x ; q, a ) − ψ ( x ) ϕ ( q ) (cid:12)(cid:12)(cid:12) ≪ x (cid:16) log y log x (cid:17) · (log log x ) B , where the constant implied in ≪ depends on a at most. The trivial upper bound for the quantity studied in Proposition 3 is O ( x ). Thesame is also true for the quantities which are majorized in Propositions 1 and 2.Hence, when Q = x , the upper bound given in Proposition 3 is non trivial onlyby a factor (log x ) − (log log x ) B . It will be sufficient for our proof, however.Proposition 3 is also interesting for Q = x + ǫ ( x ) , with ǫ ( x ) → x → ∞ ,giving an asymptotic expansion of ψ ( x ; q, a ) ∼ ψ ( x ) ϕ ( q ) , for almost all q ∼ Q , satisfying( q, a ) = 1. The technique of proof of Proposition 3 was followed up in [5], leading N THE GREATEST PRIME FACTOR OF ab + 1 7 to the relation ψ ( x ; q, a ) ≍ δ ψ ( x ) ϕ ( q ) , for almost all q ∼ Q , satisfying ( q, a ) = 1, with Q x + δ and where δ is a tiny positive constant.Actually, we shall use Proposition 3 under the form(12) X q ∼ Q ( a,q )=1 (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) ϕ ( q ) (cid:12)(cid:12)(cid:12) ≪ x log x · (cid:16) log y log x (cid:17) · (log log x ) B . This is a standard consequence of the inequality0 ψ ( x ) − θ ( x ) ≪ x L , and of the Abel summation formula written under the form π ( x ) = Z x t [d θ ( t )] , and a similar formula for π ( x ; q, a ). The equality (12) is well suited to the proof ofTheorem 2 but is not sufficient for the proof of Theorem 3. In §
4, we shall adaptthe original proof of Proposition 3 to prove
Theorem 4.
There exists an absolute constant B with the following property :For every integer a = 0 , for every x , y , P , P and Q satisfying y x , Q xy and P P , we have the inequality X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) X X P
We first recall some results concerning the average behavior of the divisor func-tions. The following subsection contains the results of Lemmas 11–15 of [4]. ´ETIENNE FOUVRY
Lemmas on divisor functions.
Let ℓ > n > τ ℓ ( n ) := X n = n ··· n ℓ , (this is the generalized divisor function of order ℓ ), then τ ( n ) = τ ( n ) , is the classical divisor function. Of course τ ( n ) = 1 if and only if n = 1, otherwise,its value is 0. Again some notations: • E is the characteristic function of the given subset of integers E , • z ( n ) is the characteristic function of the set of integers n divisible by no primefactor < z , where z > τ kℓ ( n ) and of z ( n ) τ ℓ ( n ). Lemma 1.
Let k > and ℓ > be integers and let ǫ > . We then have theinequality X x − y
It is well known that the main part of the divisor function τ ℓ ( n ) comes from thesmall divisors of n . Hence the summatory functions of z ( n ) τ ℓ ( n ) and τ ℓ ( n ) havedifferent behaviors when z becomes larger and larger. This is the object of the nextlemma. Lemma 2.
Let j > be an integer. We have the six relations ( i ) X n x z ( n ) τ j ( n ) ≪ x log 2 x · (cid:16) log 2 xz log 2 z (cid:17) j , ( i ′ ) X n x z ( n ) τ j ( n ) n − ≪ (cid:16) log 2 xz log 2 z (cid:17) j , ( i ′′ ) X w
For j = 0, all the results are trivial. For j >
1, the items ( i ) and ( ii ) areexactly [4, Lemma 13]. Also note that ( i ) can also be seen as a direct consequenceof Shiu’s result ([25, Theorem 1]) concerning sums of multiplicative functions, withthe adequate remarks concerning the uniformity of this result (see [22, p.258] or[23, p.119]).The item ( i ′ ) is a direct consequence of Mertens formula. The inequality ( i ′′ ) isa trivial consequence of ( i ′ ).In the items ( iii ) and ( iv ), we impose no sifting condition on the variable t . Thisexplains the change in the asymptotic order. We pass from ( i ) to ( iii ) by writing X nt x z ( n ) τ j ( n ) = X t x X n x/t z ( n ) τ j ( n ) ≪ (cid:16) log 2 xz log 2 z (cid:17) j X t x x/t log(2 x/t ) , and summing over t .Finally for ( iv ), we decompose X x
Lemma 3.
Let j , j , j and j be integers > . We then have X X X X n n n n xw n n n n n yn , n yn z ( n n n n ) τ j ( n ) τ j ( n ) τ j ( n ) τ j ( n ) ≪ x log 2 w (cid:16) log 2 y log 2 x (cid:17) (cid:16) log 2 xyz log 2 z (cid:17) j + j + j + j , (15) uniformly for x , y , z , w > . The constant implied in ≪ only depends on j , j , j and j . Similarly we have X X X X X tn n n n xw n n n tn n yn , tn yn z ( n n n n ) τ j ( n ) τ j ( n ) τ j ( n ) τ j ( n ) ≪ (log log 3 xyz ) · x log 2 w · (log 2 y ) log 2 x · (cid:16) log 2 xyz log 2 z (cid:17) j + j + j + j . (16) Finally, the relation (16) remains true if the summation is replaced by each of thethree following ones (17)
X X X X X tn n n n xw n n tn n n yn , n tyn , X X X X X tn n n n xw n tn n n tn yn , n yn or X X X X X tn n n n xw tn n n n n tyn , n yn . Proof.
The upper bound (15) is exactly [4, Lemma 14]. Remark that in (16) & (17),we are dealing with sums in dimension five since we have replaced the variable n i in (15) by tn i . In that case, we say that the variable t is glued to n i . This extravariable t , without sifting conditions, explains why the upperbound in (16) is largerthan the corresponding one in (15) by a log 2 x –factor. In our application the valueof the exponent of the log log–factor has no importance. It remains to adapt theproof of [4, Lemma 14] to obtain (16) by appealing to Lemma 2 and the upperbound (15) of Lemma 3.We now give all the details for the proof of (16), which corresponds to thecase where t is glued to n . By dyadic subdivision, we restrict the summation to x/ < tn n n n x. Playing with the conditions of summation in the left part of(16), we deduce that the variables n , n and n satisfy(18) n n n x , n n x , w n x and x/ y < n n n x. We first assume that(19) y x . We first sum on t and n , by using Lemma 2 ( iii )(20) X X t, n tn x/ ( n n n z ( n ) τ j ( n ) ≪ xn n n · (log log 3 xyz ) · (cid:16) log 2 xyz log 2 z (cid:17) j . Then, by Lemma 2 ( ii ), by (18) and the restriction (19), we have(21) X √ x/ yn n Lemma 4. Let x , y , z , w be real numbers > , let s = 5 or and let j , . . . , j s be integers > . We then have (29) X · · · X n ··· n s xw n s ··· n n s − yn s z ( n · · · n s ) τ j ( n ) · · · τ j s ( n s ) ≪ x log 2 x (cid:16) log 2 y log 2 w (cid:17) (cid:16) log 2 xyz log 2 z (cid:17) j + ··· + j s , where the constant implied in ≪ depends at most on j , . . . , j s . Similarly, we havefor s = 5 or and ν s , the inequality X · · · X ( t,n ,...,n s ) ∈E ( s,ν ) z ( n · · · n s ) τ j ( n ) · · · τ j s ( n s )(30) ≪ x · (log log 3 xyz ) s · (cid:16) log 2 y log 2 w (cid:17) · (cid:16) log 2 xyz log 2 z (cid:17) j + ··· + j s , where E ( s, ν ) denotes the set of s + 1 –uples ( t, n , . . . , n s ) satisfying the inequalities (31) n · · · ( tn ν ) · · · n s xw n s · · · ( tn ν ) · · · n ,n s − yn s , if ν = s and s − ,tn s − yn s , if ν = s − ,n s − ytn s , if ν = s. Proof. Actually this lemma is also true for s = 4, but we shall only use it in thecases s = 5 or s = 6 (see the end of § E ( s, ν ), by gluing (as we defined afterLemma 3) the variable t to the variable n ν .We now give the proof of (30) in the particular case s = ν = 5 (in other words,this is the case where t is glued to n ) since the other ten cases are similar. Wewrite the inequality X · · · X ( t,n ,...,n ) ∈E (5 , X w tn x z ( n ) τ j ( n ) X tn n ytn z ( n ) τ j ( n )(32) × X tn n ytn z ( n ) τ j ( n ) X n n x / ( n n tn ) z ( n ) τ j ( n ) × X n x/ ( n n n tn ) z ( n ) τ j ( n ) . By Lemma 2 ( i ) we have(33) X n x/ ( n n n tn ) z ( n ) τ j ( n ) ≪ x/ ( n n n tn )log 2 x · (cid:16) log 2 xyz log 2 z (cid:17) j . By Lemma 2 ( i ′ ), we get(34) X n n x / ( n n tn ) z ( n ) τ j ( n ) n − ≪ (cid:16) log 2 xyz log 2 z (cid:17) j . By Lemma 2 ( ii ), we have, for i = 3 or 4, the inequality(35) X tn n i ytn z ( n i ) τ j i ( n i ) n − i ≪ log 2 y log 2 tn · (cid:16) log 2 xyz log 2 z (cid:17) j i , and finally(36) X w tn x z ( n ) τ j ( n )( tn ) − (log 2 tn ) − ≪ log x (log 2 w ) · (log log 3 xyz ) · (cid:16) log 2 xyz log 2 z (cid:17) j , by Lemma 2 ( iv ) and the lower bound log(2 tn ) ≫ log 2 w . Gathering (32),...,(36),we deduce (30) in the particular case ( s, ν ) = (5 , (cid:3) Convolution of two sequences in arithmetic progressions. We continueto follow the notations of [4], in order to quote the necessary results from this paper.Let f an arithmetic function with finite support. We define k f k := (cid:16)X n | f ( n ) | (cid:17) . For a and q coprime integers, we introduce∆( f ; q, a ) := X n ≡ a mod q f ( n ) − ϕ ( q ) X ( n,q )=1 f ( n ) . N THE GREATEST PRIME FACTOR OF ab + 1 13 Hence ∆( f ; q, a ) measures the distribution of the sequence f ( n ) in the arithmeticprogression n ≡ a mod q . We shall be mainly concerned by the situation where f is the arithmetic convolution product f = α ∗ β , of two complex sequences α = ( α m ) m ∼ M and β = ( β n ) n ∼ N , with M N = x say and M, N > x ǫ . (See (11)for the meaning of ∼ ). We shall also study the convolution of three sequences.The following assumption is crucial in the context of dispersion technique. Let B > κ : R → R a real function. Now consider the condition( A ( B, κ )) concerning β = ( β n ) n ∼ N ( A ( B, κ )) For any A > , for any integers d, k > , ℓ = 0 , ( k, ℓ ) = 1 we have (cid:12)(cid:12)(cid:12) X n ≡ ℓ mod k ( n,d )=1 β n − ϕ ( k ) X ( n,dk )=1 β n (cid:12)(cid:12)(cid:12) κ ( A ) k β k τ B ( d ) N / (log 2 N ) − A . Of course any ( β n ) n ∼ N satisfies ( A ( B, κ )) by chosing for κ a huge function of N and A (for intance κ ( A ) = (log 2 N ) A )). This is an uninteresting case. The situationis quite different when we deal with sequences ( β n ) n > , which satisfy Siegel–Walfisztype theorem (for instance the characteristic function of the set of primes). If, inthat case, we consider the truncated sequence β = ( β n ) n ∼ N , then, the condition A ( B, κ )) is satisfied by β = ( β n ) n ∼ N , but with a function A κ ( A ) independentof N . Then we are in an interesting situation, on letting N tend to infinity, andchoosing A very large, but fixed.We shall also frequently suppose that, on average, the sequences are less than apower of log 2 n by introducing, for B > 0, the assumption( A ( B )) | β n | B τ B ( n ) for all n ∼ N. Sometimes it will be asked that β n = 0 when n has a small prime divisor in thefollowing sense: let x > A ( x )) be the hypothesis( A ( x )) β n = 0 ⇒ (cid:8) p | n ⇒ p > exp(log x/ (log log x ) ) (cid:9) . In other words, we ask the support of β to be included in the set of quasi primes.We shall also sometimes work with very particular λ = ( λ ℓ ) ℓ ∼ L satisfying( A ( z )) There exists an interval L ⊂ [ L, L [ and z > λ = z L . First recall a classical consequence of the large sieve inequality, which, after com-binatorial preparations, leads to Proposition 1 (Bombieri–Vinogradov Theorem). Proposition 4. Let ǫ , x , B , M and N be real numbers such that ǫ > , B > , x = M N and M, N > max(2 , x ǫ ) . Let κ : R → R be a real function. Let α =( α m ) m ∼ M , β = ( β n ) n ∼ N be two complex sequences such that β satisfies ( A ( B, κ )) .Then, for every C > , there exists A , depending only on B and C such that thefollowing inequality holds X q x L − A max ( a,q )=1 (cid:12)(cid:12) ∆( α ∗ β ; q, a ) (cid:12)(cid:12) ≪ k α k k β k x L − C , where the constant implied in the ≪ –symbol depends at most on ǫ , κ , B and C . However, Proposition 4 says nothing when Q ≍ x . We now recall severalsituations, when Q ( level of distribution ) can be taken greater than x . The relativesizes of the factors of the convolution are crucial to allow to go beyond x L − A ,which is the natural limit of the large sieve.The first situation is Proposition 5. Let a = 0 be an integer. Let ǫ , x , B , M and N be real numberssuch that ǫ > , B > , x = M N and M, N > max(2 , x ǫ ) . Let κ : R → R bea real function. Let α = ( α m ) m ∼ M , β = ( β n ) n ∼ N be two complex sequences suchthat β satisfies ( A ( B, κ )) , ( A ( B )) and ( A ( x )) .Then for every C > , we have X q ∼ Q ( q,a )=1 | ∆( α ∗ β ; q, a ) | ≪ k α k k β k x L − C , uniformly for x ǫ − Q < N < x − ǫ Q − , where the constant implied in the ≪ –symbol depends at most on ǫ , κ , a , B and C . The first and stronger version of Proposition 5 can be found in [11, Th´eor`eme1] (without the restriction ( A ( x ))). A new proof is given in [3, Theorem 3] and itappears again as [4, Theorem 1]. It is obvious that we can take Q ≍ x as soon as N satisfies x ǫ < N < x − ǫ . This result is quite convenient for applications.We shall also use the following result which is one of the key ingredient in theproof of Proposition 2. Proposition 6. ( [4, Theorem 2] ) Let a = 0 be an integer. Let ǫ , x , y , y , B , C , N and Q be real numbers such that ǫ > , B > , C > , y > y > , x > , x ǫ N x − ǫ and Q x − ǫ . Let κ : R → R be a real function. Let β = ( β n ) n ∼ N be a complex sequence such that β satisfies ( A ( B, κ )) and ( A ( x )) .Then, for every double sequence ξ = ξ ( ℓ, m ) of complex numbers, we have theinequality X q ∼ Q ( q,a )=1 (cid:16) X X X ℓmn ∼ x, n ∼ Ny The sec-ond type of results concerns the convolution of three sequences(37) η = ( η k ) k ∼ K , λ = ( λ ℓ ) ℓ ∼ L , α = ( α m ) m ∼ M ,x = KLM, L = log 2 x, with K, L, M > . We have Proposition 7. Let ǫ , B and C be given positive real numbers. Let a = 0 be aninteger. Let κ : R → R be a real function. Let x , K , L , M be real numbers and η , λ and α be three sequences as in (37). Furthermore, suppose that the followingconditions are satisfied • K , L , M > x ǫ , • η satisfies ( A ( B )) and ( A ( x )) , • λ satisfies ( A ( B, κ )) , ( A ( B )) and ( A ( x )) , • α satisfies ( A ( B )) .Then, there exists A , depending only on B and C , such that the following in-equality (38) X q ∼ Q ( q,a )=1 (cid:12)(cid:12) ∆( η ∗ λ ∗ α ; q, a ) (cid:12)(cid:12) ≪ x L − C . holds as soon as one of two sets of inequalities is verified ( S Q L A < KL, K L < Qx L − A and K L ( K + L ) < x − ǫ , or ( S Q L A < KL, KL Q < x L − A and K x ǫ < Q. The constant implied in the ≪ –symbol of (38) depends at most on ǫ , κ , a , B and C . The conditions ( S 1) correspond to [4, Theorem 3], and the set ( S 2) to [4, The-orem 4]. Note that in the original statement of [4, Theorems 3 & 4], the sequence α is supposed to satisfy ( A ( x )). Actually, this restriction is unnecessary, since theproof of [4, Formula (4.3)], based on Cauchy–Schwarz inequality does not requiresuch a condition.Note that if in ( S 1) or ( S 2) the factor L A was replaced by the larger factor x ǫ ,Proposition 7 would be too weak for the proof of Proposition 3 and Theorem 4.This is the reason why we cannot appeal to [3, Theorem 4], which also deals withthe convolution of three sequences.3.4. Other types of results on the convolution of three sequences. Thecondition ( A ( x )) that must satisfy λ in Proposition 7 is rather annoying in theapplication that we have in mind. It could certainly be removed by writing withgreat care the original proof of Theorems 3 & 4 of [4]. We prefer to modify theproof of the following result of Fouvry [10, Th´eor`eme 2]. Proposition 8. Let a be an integer. Let κ : R → R be a real function. Let ǫ , x , C , L , M , N be real numbers such that : ǫ and C > , L , M and N > , x = LM N , < | a | x and such that ( S L N M − ǫ , L N M − ǫ and log N > ǫ log M. Let α , β and λ be the characteristic functions of three sets of integers respectivelyincluded in [ M, M [ , [ N, N [ and [ L, L [ . Suppose that β satisfies (cid:12)(cid:12)(cid:12) X n ≡ b mod q β n − ϕ ( q ) X ( n,q )=1 β n (cid:12)(cid:12)(cid:12) κ ( A ) (cid:16)X n | β n | (cid:17) (log 2 N ) − A , for every real A and for every integers b and q such that ( b, q ) = 1 . Then we havethe inequality X q ( LN )1 − ǫ ( q,a )=1 (cid:12)(cid:12)(cid:12) ∆( α ∗ β ∗ λ ; q, a ) (cid:12)(cid:12)(cid:12) ≪ x L − C , where the constant implied in the ≪ –symbol depends at most on ǫ , κ and C . It is worth to notice the large uniformity over a compared with the results con-tained in Propositions 5–7. This is due to the use of Weil’s classical bound forKloosterman sums instead of kloostermania. However we shall not use this unifor-mity here. Nevertheless the condition ( A ( x )) is now absent from the hypothesis,but the range of summation for q is not satisfactory for our application. As inProposition 7, we would like to go up to q ( LN ) L − A . We now give the improvement of Proposition 8 necessary for our application. Proposition 9. Let a be an integer. Let κ : R → R be a real function. Let ǫ , x , C , L , M , N be real numbers such that : ǫ and C > , L , M and N > x ǫ , x = LM N , such that ( S ) is satisfied. Let a be an integer such that < | a | x .Let α = ( α m ) m ∼ M , β = ( β n ) n ∼ N and λ = ( λ ℓ ) ℓ ∼ L be three sequences such that • α , β , and λ satisfy ( A ( B )) , • β satisfies ( A ( B, κ )) .Then there exists A depending only on B and C such that we have X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) ∆( α ∗ β ∗ λ ; q, a ) (cid:12)(cid:12)(cid:12) ≪ x L − C , for Q ( LN ) L − A . The constant implied in the ≪ –symbol depends at most on ǫ , κ , B and C .Proof. When Q ( LN ) − ǫ , the extension from Proposition 8 to Proposition 9 isstraightforward by following the proof of [10, Th´eor`eme 2].Hence we are left with the case(39) ( LN ) − ǫ < Q ( LN ) L − A . We shall follow the notations of [10] the most possible, even if they are differentsometimes from [4]. Let γ = β ∗ λ . Hence γ = ( γ k ) k has its support included in [ K, K [, with K := LN . Note that(40) | γ k | B τ B +1 ( k ) , by ( A ( B )). In the following proof, we shall denote by B ∗ a constant dependingonly on the constant B appearing in the assumptions ( A ( B, κ )) and ( A ( B )). Thevalue of B ∗ may change at each time it appears. N THE GREATEST PRIME FACTOR OF ab + 1 17 Let E ( Q ) be the sum E ( Q ) := X q ∼ Q ( q,a )=1 X ( m,q )=1 | α m | (cid:12)(cid:12)(cid:12) X k ≡ am mod q γ k − ϕ ( q ) X ( k,q )=1 γ k (cid:12)(cid:12)(cid:12) . (here m is the multiplicative inverse of m mod q .) Obviously, E ( Q ) satisfies theinequality X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) ∆( α ∗ β ∗ λ ; q, a ) (cid:12)(cid:12)(cid:12) E ( Q ) . By the Cauchy–Schwarz inequality, by the assumption ( A ( B )) for α and by inver-sion of summation, we get the inequality (see [10, p.365])(41) E ( Q ) k α k Q D ( Q ) ≪ M Q L B ∗ D ( Q ) , where the dispersion D ( Q ) is(42) D ( Q ) := W ( Q ) − V ( Q ) + U ( Q ) , with U ( Q ) := X q ∼ Q ( q,a )=1 X m ∼ M ( m,q )=1 (cid:16) ϕ ( q ) X ( k,q )=1 γ k (cid:17) ,V ( Q ) := X q ∼ Q ( q,a )=1 X m ∼ M ( m,q )=1 (cid:16) X k ≡ am mod q γ k (cid:17)(cid:16) ϕ ( q ) X ( k ,q )=1 γ k (cid:17) , and W ( Q ) := X q ∼ Q ( q,a )=1 X m ∼ M ( m,q )=1 (cid:16) X k ≡ am mod q γ k (cid:17) . Let also A ( Q ) := X q ∼ Q ( q,a )=1 q ϕ ( q ) (cid:16) X ( k,q )=1 γ k (cid:17) . Following the proof of [10, Form.(6)] and appealing to Lemma 1, we prove theequality(43) U ( Q ) = M A ( Q ) + O (cid:0) K Q − L B ∗ (cid:1) . By the proof of [10, Form.(12)], we also have(44) V ( Q ) = M A ( Q ) + O ǫ (cid:0) KQ − x − ǫ + K Q − x ǫ (cid:1) , where ǫ appears in (39).The study of W ( Q ) is more delicate. Firstly we take some care to get rid of thecommon divisors. Let(45) ∆ := 3 KQ − , and, by (39), we can suppose that(46) 3 L A ∆ < K ǫ . Then we notice that if k and k are two distinct integers of the interval [ K, K [,satisfying ( k k , q ) = 1 and k − k = q q , for some q ∼ Q and some positive integer q we then have(47) ( k , k ) = ( k , k − k ) = ( k , q q ) = ( k , q ) q ∆ . Following [10, § VI], we write the equality(48) W ( Q ) = X q ∼ Q ( q,a )=1 X k ≡ k q ( k k ,q )=1 γ k γ k X m ∼ Mm ≡ ak q . We first notice that the contribution, say W = ( Q ), to W ( Q ) of the ( k , k ) with k = k satisfies (cid:12)(cid:12) W = ( Q ) (cid:12)(cid:12) ≪ X k K τ B +2 ( k ) X m ∼ Mkm = a τ ( | km − a | ) + Qx ǫ . Writing t = km , we deduce that(49) (cid:12)(cid:12) W = ( Q ) (cid:12)(cid:12) ≪ X t xt = a τ B +3 ( t ) τ ( | t − a | ) + Qx ǫ ≪ x L B ∗ , by Cauchy–Schwarz inequality and by Lemma 1. We will see that the bound (49)is acceptable in view of (39) & (41) by choosing A sufficiently large.Let W = ( Q ) be the contribution to W ( Q ) of the pairs ( k , k ) with k = k (see(48)). By (47), we know that d := ( k , k ) is less than ∆. Decomposing W = ( Q )according to the the value of d and writing k i = dk ′ i ( i = 1, 2) we have the equality(compare with [10, Form.(13)])(50) W = ( Q ) = X q ∼ Q ( q,a )=1 X d ∆( d,q )=1 X k ′ ≡ k ′ q, k ′ = k ′ k ′ ,k ′ k ′ k ′ ,q )=1 γ dk ′ γ dk ′ X m ∼ Mm ≡ adk ′ q . We continue to prepare the variable by extracting from k ′ all the prime factorsappearing also in d . So we write k ′ = d k ′′ with d | d ∞ and ( k ′′ , d ) = 1 and weuse the following crude estimate Lemma 5. Uniformly for d integer > and y > , we have the inequality X d | d ∞ d > y d ≪ τ ( d ) y . Proof. We may restrict to the case where d = p · · · p r is squarefree. FollowingRankin’s method, we write, for every κ ∈ ]0 , X d | d ∞ d > y d X d | d ∞ d · (cid:16) d y (cid:17) κ = 1 y κ r Y i =1 (cid:16) − p κ − i (cid:17) − ≪ y κ exp (cid:0) r X i =1 p κ − i (cid:1) . Fixing κ = 1 / 2, we get the desired upper bound. (cid:3) Inspired by [10, p.368], we see that the contribution to the right part of (50) of d > y is= X q ∼ Q ( q,a )=1) X d ∆( d,a )=1 X d | d ∞ d >y X d k ′′ ≡ k ′ mod qd k ′′ = k ′ ( k ′′ ,dk ′ )=( k ′′ k ′ ,d q )=1 γ dd k ′′ γ dk ′ X m ∼ Mm ≡ add k ′′ q . Using (40) and the inequality τ ( n ) ≪ X ǫ B +1) (0 < n X ) several times, andseparating the cases dd k ′′ m − a = 0 from the case dd k ′′ m − a = 0, we see, by N THE GREATEST PRIME FACTOR OF ab + 1 19 (45), that the above contribution is ≪ M x ǫ X d ∆( d,a )=1 X d | d ∞ d >y X k ′′ K/ ( dd ) KdQ + Kx ε ≪ K M Q − x ǫ y − + Kx ε ≪ K M Q − x − ǫ , (51)by choosing(52) y = x ǫ , and applying Lemma 5. Note that (51) is acceptable in view of (39) & (41). Gath-ering (48), (49), (50) & (51) and slightly changing the notations, we write theequality W ( Q ) = X q ∼ Q ( q,a )=1 X d ∆( d,q )=1 X d | d ∞ d y X d k ′ ≡ k q, d k ′ = k k ′ ,dk k ′ k ,d q )=1 γ dd k ′ γ dk X m ∼ Mm ≡ add k ′ q O (cid:0) x L B ∗ + x M − Q − x − ǫ (cid:1) . (Compare with [10, Form.(14)]). The main term in (53) certainly comes in replacingthe last sum by its approximation M/q . When this replacement is done we canforget the conditions d k = k and d y . We introduce an error which is in ≪ x L B ∗ + K M Q − x − ǫ (same computations as for (49) & (51)). Let(54) B ( Q ) := X q ∼ Q ( q,a )=1 q X k ≡ k q ( k k ,q )=1 γ k γ k = X q ∼ Q ( q,a )=1 q X κ mod q ( κ,q )=1 (cid:16) X k ≡ κ mod q γ k (cid:17) . The above discussion transforms (53) into the following equality, which has to becompared with [10, Form.(15)](55) W ( Q ) = M B ( Q ) + W ( M, Q ) − W (2 M, Q ) + O (cid:0) x L B ∗ + x M − Q − x − ǫ (cid:1) , with W ( Y, Q ) = X q ∼ Q ( q,a )=1 X d ∆( d,q )=1 X d | d ∞ d y X d k ′ ≡ k q, d k ′ = k k ′ ,dk k ′ k ,d q )=1 γ dd k ′ γ dk ψ (cid:16) Y − add k ′ q (cid:17) , where ψ ( t ) + 1 / t . Our present formula of W ( Y, Q )coincides with the corresponding formula of W ( Y, Q ) given in [10, p.369], withthe tiny difference that the sum is over d x ǫ instead of d ∆. In [10], theproblem of bounding W ( Y, Q ) (with Y = M or 2 M ) is accomplished by appealingto Weil’s bound for Kloosterman sums. It is easy to check, that, in this paper, thesummation over d is always made on the norms of the corresponding sums. Hence,since by (46), we have ∆ x ǫ , we can apply [10, Form.(26)], in our case, givingthe bound(56) W ( Y, Q ) ≪ L M N Q − x − ǫ + L N Q − x ǫ + L N Q − x ǫ , for Y = M or 2 M . By the orthogonality of characters, the large sieve inequalityand the assumption ( A ( B, κ )) for β , we get (compare with [10, Form.(37) & (40)]) the inequality0 M B ( Q ) − M A ( Q ) X q ∼ Q ( q,a )=1 q ϕ ( q ) X χ mod qχ = χ (cid:12)(cid:12)(cid:12) X k χ ( k ) γ k (cid:12)(cid:12)(cid:12) ≪ x M − Q − L B ∗ − C + x L B ∗ , (57)which is true for any C > 0. Gathering (41), (42), (43), (44), (55), (56) & (57), wecan write E ( Q ) ≪ M Q L B ∗ n L N Q − + LN Q − x − ǫ + L N Q − x ǫ + L M N Q − x − ǫ + L N Q − x ǫ + L N Q − x ǫ + L M N Q − L − C + x o , which simplifies into E ( Q ) ≪ x L B ∗ − C + M Qx L B ∗ + L M N x ǫ + L M N x ǫ . This gives bound claimed in Proposition 9, under the assumptions ( S 3) and (39)after changing the value of ǫ and C . (cid:3) Particular cases of equidistribution. We now finish with some particularcases where λ is the characteristic function of quasi primes. The first result is [4,Theorem 5*]. Proposition 10. Let a = 0 be an integer. Let ǫ , z , B and C be positive numbers.Let x , K , L , M , η , λ and α as in (37), and satisfying the extra conditions • z exp(log 2 x/ (log log 2 x )) , • K , L , M > x ǫ , • α and η satisfies ( A ( B )) , • λ satisfies ( A ( z )) .Then there exists A depending only on B and C , such that the following in-equality holds (58) X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) ∆( η ∗ λ ∗ α ; q, a ) (cid:12)(cid:12)(cid:12) ≪ x L − A , as soon as Q satisfies ( S Q L A < KL, M K Q < x − ǫ and M K Q < x − ǫ . The constant implied in the ≪ –symbol in (58) depends at most on ǫ , a , B , C . Finally we recall a consequence of bounds of exponential sums (coming eitherfrom Weil’s or Deligne’s work) and of the fundamental lemma in sieve theory. Wehave (see [4, Lemma 2*]) Proposition 11. Let ǫ > . Let K , L and M > and x > KLM , such that either K = 1 or K > x ǫ and similarly either L = 1 or L > x ǫ and either M = 1 or M > x ǫ . Let K , L and M be three intervals respectively included in [ K, K [ , [ L, L [ and [ M, M [ . Then there exists an absolute positive constant δ such that ∆ (cid:0) z ( K ∗ L ∗ M ); q, a ) ≪ xϕ ( q ) exp (cid:16) − ǫ log x log z (cid:17) N THE GREATEST PRIME FACTOR OF ab + 1 21 uniformly for ( q, a ) = 1 and q x + δ . The constant implied in the ≪ –symboldepends at most on ǫ . Proof of Theorem 4 We arrive now at the central part of our work. Of course, our proof highlyimitates the proof given in [4]. The combinatorics is heavy and we were unable tofind shortcuts to simplify the technique of [4].4.1. Notations and first reductions of the proof of Theorem 4. As in [4, § X n ∗ means that we are summing over integers n , with z ( n ) = 1 , and z now has the value(59) z := exp (cid:16) log x (log log x ) (cid:17) . To prove Theorem 4, we consider S ( x, Q, P , P ) := X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) X X P Q xy with(61) L A y exp (cid:16) log x log log x (cid:17) := y , since y > y , (60) is trivial, by (13). In (61), A is a constant the definition of whichwill be given in § § Q > x − ǫ , otherwise, (60) is a direct consequence of Proposition 4. Of course Theorem 4 istrivial also when P is too large ( P > x , since the sum is empty). Using theclassical formulas X m xpm ≡ b mod q xpq + O (1) , and(62) X m xp ( m,q )=1 ϕ ( q ) q · xp + O ( τ ( q )) , we deduce the inequality S ( x, Q, P , P ) ≪ X q ∼ Q τ ( q ) X P x y − . Furthermore, if P There exists an absolute positive c , such that, uniformly for v > u > and x > we have the inequality Θ( x ; u, v ) := (cid:12)(cid:12) { n x ; Y pν k np u p ν > v } (cid:12)(cid:12) ≪ x exp (cid:16) − c log v log u (cid:17) . Let W be a number which satisfies(66) W ∼ exp (cid:16) log x √ log log x (cid:17) . By Lemma 6, the contribution of the triples ( q, n, t ) such that t † > W to the rightpart of (64) is(67) ≪ L (cid:16)X ∗ X x/ To transform the function Λinto bilinear forms, we appeal to the identity of Heath–Brown (see [18, Prop. 13.3]for instance) Lemma 7. Let J > and n < x . We then have the equality Λ( n ) = J X j =1 ( − j (cid:18) Jj (cid:19) X m ,...,m j x /J µ ( m ) · · · µ ( m j ) X m ...m j n ...n j = n log n . We apply this lemma to Λ( n ) inside (68) with J = 7. It gives the inequality(70) ˜ E ( x, Q, P , P ) (cid:18) (cid:19) X j =1 ˜ E j ( x, Q, P , P ) , with˜ E j ( x, Q, P , P ) := X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) X ∗ · · · X ∗ X † m ··· m j n ··· n j uw ≡ a mod q µ ( m ) · · · µ ( m j ) log n (71) − ϕ ( q ) X ∗ · · · X ∗ X † ( m ··· m j n ··· n j uw,q )=1 µ ( m ) · · · µ ( m j ) log n (cid:12)(cid:12)(cid:12) , where the variables of summation satisfy the inequalities x/ < m · · · m j n · · · n j uw x, P < m · · · m j n · · · n j P (72) and m , . . . , m j D, where D is any number > x . and where w satisfies (69). Dissection of the set of summation. In (72), the variables m i , n i , u and w have to satisfy several multiplicative inequalities. To make these variables inde-pendent we process as usual in such problems, see [4, p.388] for instance. We definea parameter δ satisfying x − ǫ < δ < 1, and introduce the notation g ≃ G to mean that the integer variable g satisfies G g < (1 + δ ) G . Let D := (cid:8) (1 + δ ) ν ; ν = 0 , , , . . . (cid:9) . To transform (72), we precise (66) and the choice of D by imposing D, W ∈ D and D ≃ x . The conditions (72) are equivalent to(73) m i ≃ M i , n i ≃ N i (1 i j ) , u ≃ U and w ≃ W, for some numbers M ,..., M j , N ,..., N j , U and W from D satisfying x/ < M · · · M j N · · · N j U W x,P < M · · · M j N · · · N j P ,M , . . . , M j D/ (1 + δ ) ,W W / (1 + δ ) , (74)unless the 2 j + 2–uple ( m , · · · , m j , n , · · · , n j , u, w ) is too near from some edge ofthe dissection, that means satisfies at least one of the following four conditions x < m · · · m j n · · · n j uw x (1 + δ ) j +2 ,x/ < m · · · m j n · · · n j uw ( x/ δ ) j +2 ,P < m · · · m j n · · · n j P (1 + δ ) j and m · · · m j n · · · n j uw x,P < m · · · m j n · · · n j P (1 + δ ) j and m · · · m j n · · · n j uw x. (75)Write r := m · · · m j n · · · n j and t := uw (note that each t can be written inan unique way in that form, since u (resp. w ) has all its prime factors greater(resp. less) than z ). It is easy to see that the contribution (denoted by C ,j ) to˜ E j ( x, Q, P , P ) of the (2 j + 2)–uples ( m , · · · , m j , n , · · · , n j , u, w ) which satisfy atleast one of the four conditions of (75) is C ,j L n X x