Manin's conjecture for a quintic del Pezzo surface with A_2 singularity
aa r X i v : . [ m a t h . N T ] O c t MANIN’S CONJECTURE FOR A QUINTIC DEL PEZZOSURFACE WITH A SINGULARITY
ULRICH DERENTHAL
Abstract.
Manin’s conjecture is proved for a split del Pezzo surface ofdegree 5 with a singularity of type A . Contents
1. Introduction 12. A universal torsor 23. Estimating N b ( B ; A ) 44. Real-valued functions 55. Estimating N a ( B ) and N b ( B ; A ) – first step 66. Estimating N a ( B ) – second step 77. Estimating N b ( B ; A ) – second step 118. The final step 13References 161. Introduction
Let S ⊂ P be the del Pezzo surface of degree 5 defined by x x − x x = x x − x x = x x + x + x x = x x + x x + x = x x + x x + x x = 0 . (1.1)It contains a unique singularity of type A and four lines, all of them definedover Q . Let U ⊂ S be the complement of these lines.We define the height of any rational point x ∈ S ( Q ) that is representedby integral and relatively coprime coordinates ( x , . . . , x ) as H ( x ) := max {| x | , . . . , | x |} . For any B >
1, let N U,H ( B ) := { x ∈ U ( Q ) | H ( x ) B } be the number of rational points in U whose height is at most B .We prove the following result: Theorem.
We have N U,H ( B ) = c S,H B (log B ) + O ( B (log B ) − / ) , Mathematics Subject Classification.
Primary 11G35; Secondary 14G05. where c S,H = 1864 · ω ∞ · Y p (cid:18) − p (cid:19) (cid:18) p + 1 p (cid:19) and ω ∞ = Z | t | , | t | , | t t t + t t | , | t t t | , | t t + t | , | t t + t t t | , t > d t d t d t . Manin’s conjecture [FMT89] predicts that N U,H ( B ) grows as cB (log B ) k − for B → ∞ where k is the rank of the Picard group of the minimal desin-gularization e S of S . As S is a del Pezzo surface of degree 5 whose lines aredefined over Q , we have k = 5, so our result agrees with this conjecture.Peyre [Pey95] predicts that c is a product of a constant α ( e S ) whose valueis 1 /
864 by [Der07] and [DJT07] and a product of local densities. We expectthat (1 − /p ) (1 + 5 /p + 1 /p ) agrees with the density at each prime p , andthat ω ∞ agrees with the real density, but we do not check this here.Note that S is neither toric nor an equivariant compactification of G , soour theorem is not a consequence of [BT98] or [CLT02].For the proof of the theorem, we use the basic strategy of [BB07], [BBD07]and [DT07] together with the techniques introduced in [BD07]. In Section 2,we translate the counting problem to the question of integral points on auniversal torsor and split their counting into three parts. As outlined at theend of Section 2, these parts are handled separately in Sections 3 to 7 andput together again in Section 8 to complete the proof of the theorem.2. A universal torsor
We use the notation η = ( η , . . . , η ) , η ′ = ( η , . . . , η ) , α = ( α , α )and, for ( n , . . . , n ) ∈ Q , η ( n ,n ,n ,n ) := η n η n η n η n . By the method of [DT07] and using the data of [Der06] on the geometry of S and its minimal desingularization e S , we obtain a bijection Ψ : T → U ( Q )with T := { ( η ′ , α ) ∈ Z × Z > × Z | (2.1) and coprimality conditions hold } where(2.1) η η η + η α + η α = 0and the coprimality conditions are described by the extended Dynkin dia-gram of E , . . . , E , A , A in Figure 1, using the rule that two variables arecoprime unless the corresponding divisors in the diagram are connected byan edge. The map Ψ sends ( η ′ , α ) ∈ T to( η (2 , , , η , η (2 , , , α , η α α , η (1 , , , η η α , η (1 , , , α , η (0 , , , η η α )in U ( Q ). ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 3 A AAAAAAAA E CCCCCCCCC E E GFED@ABC E GFED@ABC E A ~~~~~~~~ E {{{{{{{{{ Figure 1.
Configuration of curves on e S .Note that these coprimality conditions imply that the formula above forΨ( η ′ , α ) results in relatively coprime coordinates Ψ( η ′ , α ) i , so H (Ψ( η ′ , α )) = max i {| Ψ( η ′ , α ) i |} . With (2.1), H (Ψ( η ′ , α )) B implies(2.2) η (1 , , , η | η | B, η (0 , , , η | η | B. Using (2.1), the coprimality conditions can be rewritten asgcd( α , η η ) = 1 , (2.3) gcd( α , η η ) = 1 , (2.4) gcd( η , η η η η ) = 1 , (2.5) gcd( η , η η η ) = 1 , (2.6) gcd( η , η ) = 1 , gcd( η , η ) = 1 , gcd( η , η ) = 1 . (2.7)Therefore, the number N U,H ( B ) coincides with the number of ( η ′ , α ) ∈ Z > × Z =0 × Z which satisfy the torsor equation (2.1), the coprimalityconditions (2.3)–(2.7) and the height condition H (Ψ( η ′ , α )) B .Our further strategy is as follows. For fixed η ′ , we estimate the number of α satisfying the torsor equation, the coprimality conditions and the heightcondition. We sum this number over all suitable η ′ afterwards. To get ahold of the error terms in these summations, it will be useful to do thissummations in different orders depending on the relative size of η , . . . , η .We denote the number of ( η ′ , α ) contributing to N U,H ( B ) that fulfill(2.8) | η | > | η | by N a ( B ), and the number of those satisfying(2.9) | η | < | η | . by N b ( B ).We split the elements contributing to N b ( B ) further into two subsets: Forsome A > N b ( B ; A ) be the number of ( η ′ , α )satisfying (2.9) and(2.10) η (2 , , , B (log B ) A , while N b ( B ; A ) is the number of the remaining ones. ULRICH DERENTHAL
We deal with N b ( B ; A ) in the following Section 3. As a first step forboth N a ( B ) and N b ( B ; A ), we estimate the number of α in Section 5. For N a , we sum first over the bigger η and then over η in Section 6, while for N b ( B ; A ), we sum in the reverse order in Section 7. The resulting mainterms are put together and summed over the remaining variables η , . . . , η in Section 8 to complete the proof of the theorem.3. Estimating N b ( B ; A )Our strategy is to estimate the number of ( η ′ , α ) lying in dyadic intervalsfirst, and to sum over all possible intervals in a second step. Lemma 1.
We have N b ( B ; A ) ≪ A B (log B ) (log log B ) .Proof. Let N = N ( N , . . . , N , A , A ) be the number of ( η ′ , α ) subject to N i / < | η i | N i for i ∈ { , . . . , } and A j / < | α j | A j for j ∈ { , } .Because of the height conditions and using the notation N ( n ,n ,n ,n ) := N n N n N n N n , we have, if N > B (log B ) − A ≪ N (2 , , , ≪ B, (3.1) N A A ≪ B, (3.2) N (1 , , , N N A ≪ B, (3.3) N (0 , , , N N A ≪ B, (3.4) N (1 , , , N N ≪ B, (3.5) N ≪ (log B ) A . (3.6)Here, (3.5) follows from (2.2). As in [BD07, Lemma 5, 6], we obtain by esti-mating the number of α , α in two ways first and summing over η , . . . , η afterwards: N ≪ N N N N ( N A ) / ( N A ) / + N N N N N N . Next, we sum this estimate for N ( N , . . . , N , A , A ) over all possibledyadic intervals, with N , . . . , N , A , A subject to (3.1)–(3.6).For the first term, we have using (3.2)–(3.5) X N ,...,N ,A ,A N (1 / , / , , N N A / A / ≪ B / X N ,...,N ,A ,A N (1 / , / , , N N / A / A / ≪ B / X N ,...,N N (1 / , / , / , / N / N / ≪ B X N ,...,N ≪ A B (log B ) (log log B ) . Here we have used that for fixed N , N , N , there are only O A (log log B )possibilities for N and N by (3.1) and (3.6). ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 5
For the second term, we use (3.5) to obtain X N ,...,N ,A ,A N (1 , , , N N ≪ B X N ,...,N ,A ,A N N N ≪ A B (log B ) (log log B ) , which completes the proof. (cid:3) Real-valued functions
Let(4.1) h ( t , t , t , t ) := max ( | t t | , | t t | , | t t t + t t t | , | t t t t | , | t t t + t t | , | t t + t t t t | ) . Defining Y := η (2 , , , B ! / , Y := (cid:18) B η (2 , − , − , − (cid:19) / ,Y := Y − , Y := (cid:18) B η ( − , − , − , (cid:19) / , we note that the height condition H (Ψ( η ′ , α )) B is equivalent to h ( Y , α /Y , η /Y , η /Y ) . Define g ( t , t , t ) := Z h ( t ,t ,t ,t ) t , (4.2) g a ( t , t ; η ; B ) := Z Y t > | Y t | ,t > g ( t , t , t ) d t , (4.3) g b ( t , t ; η ; B ) := Z | Y t | > max { Y t , } g ( t , t , t ) d t , (4.4) g a ( t ; η ; B ) := Z | Y t | > g a ( t , t ; η ; B ) d t , (4.5) g b ( t ; η ; B ) := Z ∞ g b ( t , t ; η ; B ) d t . (4.6)We have g ( t ; η ; B ) := g a ( t ; η ; B ) + g b ( t ; η ; B )= Z h ( t ,t ,t ,t ) , | Y t | > ,t > d t d t d t . (4.7) Lemma 2.
Let η ∈ Z > be given. Then we have: (1) g ( t , t , t ) ≪ t | t | / . (2) g a ( t , t ; η ; B ) ≪ R ∞ g ( t , t , t ) d t ≪ min { t / | t | / , t } . (3) g b ( t , t ; η ; B ) ≪ R ∞−∞ g ( t , t , t ) d t ≪ t t / . ULRICH DERENTHAL
Proof.
Since h ( t , t , t , t ) t t − and t t − , the secondbound of (2) holds.It is not hard to check that given a, b ∈ R \{ } , the condition | at + bt | t whose length is ≪ | a | − / for b | a | , while its lengthis ≪ | b | − ≪ | a | − / for b > | a | .We apply this for a = t t and b = t t , which gives g ( t , t , t ) ≪ ( t | t | ) − / which is (1). Integrating it over t ≪ t / (which holds since | t t t t | | t t + t t t t | | t t |
2) results in (3).For the first bound of (2), we distinguish the case t t t | t | and itsopposite. In the first case, we combine t ≪ t / | t | − / with (1). In thesecond case, we integrate g ( t , t , t ) ≪ t − | t | − over t ≫ t / | t | − / . (cid:3) Finally, we define(4.8) G ( t ) := Z h ( t ,t ,t ,t ) ,t > d t d t d t which is related to ω ∞ defined in the statement of our theorem: Lemma 3.
For any t > , we have G ( t ) = ω ∞ t .Proof. Similar to [BD07, Lemma 7]. (cid:3) Estimating N a ( B ) and N b ( B ; A ) – first step For fixed η ′ subject to the coprimality conditions (2.5)–(2.7), let N bethe number of α , α subject to (2.1), h ( Y , α /Y , η /Y , η /Y ) N = X k | η η µ ( k ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η η η ≡ − η α (mod k η ) ,h ( Y , α /Y , η /Y , η /Y ) , (2.4) holds . The summand vanishes unless gcd( k , η η ) = 1. Since η , η are coprime,we write k = k k uniquely such that k i | η i for i ∈ { , } . We checkthat k | α . We write η = k η ′ , α = k α ′ and obtain N = X k | η ,k | η gcd( k k ,η η )=1 µ ( k ) µ ( k ) N ( k , k )where N ( k , k ) = α ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k η η ′ η ≡ − η α ′ (mod k η ) ,h ( Y , α ′ k /Y , η /Y , η /Y ) , gcd( k α ′ , η η ) = 1 . Note that gcd( k , η η ) = 1 holds automatically, so we may remove thiscondition. We remove the coprimality condition for α ′ by another M¨obiusinversion and obtain, writing α ′ = k α ′′ , N ( k , k ) = X k | η η µ ( k ) ( α ′′ (cid:12)(cid:12)(cid:12) k η η ′ η ≡ − k η α ′′ (mod k η ) ,h ( Y , α ′′ k k /Y , η /Y , η /Y ) , ) . ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 7
Note that the summand vanishes unless gcd( k , k η ) = 1, so we may re-strict the summation over k | η η subject to gcd( k , k η ) = 1. Since thengcd( k η , k η ) = 1, the number of α ′′ is Y k k k η g ( Y , η /Y , η /Y ) + O (1) . Define φ ∗ ( n ) := Q p | n (1 − /p ). Lemma 4.
We have N = Y η g ( Y , η /Y , η /Y ) ϑ ( η ) φ ∗ ( η ) φ ∗ (gcd( η , η )) + O ( R ( η , η , η )) with ϑ ( η ) := X k | η gcd( k ,η η )=1 µ ( k ) φ ∗ ( η η ) k φ ∗ (gcd( η , k η )) and X η ,...,η R ( η , η , η ) ≪ B (log B ) . Proof.
For the main term, note that X k | η ,k | η gcd( k k ,η η )=1 µ ( k ) µ ( k ) k k X k | η η gcd( k ,k η )=1 µ ( k ) k = X k | η gcd( k ,η η )=1 µ ( k ) k · φ ∗ ( η ) φ ∗ (gcd( η , η η )) · φ ∗ ( η η ) φ ∗ (gcd( η η , k η )) . Using gcd( η , η ) = 1 and gcd( η , k η ) = 1, we obtain ϑ .We have R ( η , η , η ) ≪ ω ( η )+ ω ( η )+ ω ( η η ) . We sum this over all suitable η , . . . , η and use (2.2) to obtain X η ,...,η R ( η , η , η ) ≪ X η ,...,η ω ( η )+ ω ( η )+ ω ( η η ) B η (1 , , , η ≪ B (log B ) , completing the proof of this lemma. (cid:3) Estimating N a ( B ) – second step Let N a := N a ( η , η ; B ) be the sum of the main term of Lemma 4 over η subject to (2.6) and (2.8). We sum the main term of N a over η afterwardsto obtain N a ( η ; B ). ULRICH DERENTHAL
Using [BD07, Lemma 2] with α = 0 , q = 1, we obtain (where f a,b ( n ) isdefined to be φ ∗ ( n ) /φ ∗ (gcd( n, a )) if gcd( n, b ) = 1 and to be zero otherwise) N a = Y η ϑ ( η ) X η > | η | f η ,η η η ( η ) g ( Y , η /Y , η /Y ; η ; B )= Y Y η g a ( Y , η /Y ; η ; B ) ϑ ( η ) φ ∗ ( η η η ) ζ (2) Y p | η η η η (cid:18) − p (cid:19) − + O (cid:18) Y η | ϑ ( η ) | (log B )2 ω ( η η η ) sup t g ( Y , t , η /Y ; η ; B ) (cid:19) , where the supremum is taken over t > | η | /Y . Lemma 5.
We have N a = Y Y η g a ( Y , η /Y ; η ; B ) ϑ a ( η ) + O ( R a ( η , η ; B )) with ϑ a ( η ) := ϑ ( η ) φ ∗ ( η η η ) ζ (2) Y p | η η η η (cid:18) − p (cid:19) − and X η ,η R a ( η , η ; B ) ≪ B log B. Proof.
The main term is clear. Define φ † ( n ) := Q p | n (1 + 1 /p ). For theerror term, we use Lemma 2(1) to estimate its sum over η , η as ≪ X η ,η ω ( η η η ) φ † ( η ) Y Y / log BY η | η | / = X η ,η ω ( η η η ) φ † ( η ) B / log Bη / η / | η | / ≪ X η ,η ,η ,η ω ( η η η ) φ † ( η ) B log B η (5 / , / , / , | η | / ≪ B log B. Here, we use η (cid:18) B η (2 , , , | η | (cid:19) / · (cid:18) B η (1 , , , | η | (cid:19) / = B / η (3 / , / , / , | η | which is obtained with (2.8). (cid:3) To sum the main term of N a over η , we remove the coprimality condition(2.5) by a M¨obius inversion and obtain, writing η = k η ′ and applying ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 9 partial summation, N a = Y Y η ϑ a ( η ) X k | η η η η µ ( k ) X | η ′ | > g a ( Y , η ′ k /Y ; η ; B )= Y Y Y η ϑ a ( η ) X k | η η η η µ ( k ) k Z | t | > k /Y g a ( Y , t ; η ; B ) d t + O Y Y η | ϑ a ( η ) | X k | η η η η | µ ( k ) | sup | t | > k /Y g a ( Y , t ; η ; B ) . Lemma 6.
We have N a = Y Y Y η g a ( Y ; η ; B ) ϑ a ( η ) + O ( R a ( η ; B )) with ϑ a ( η ) := ϑ a ( η ) φ ∗ ( η η η η ) and X η R a ( η ; B ) ≪ B (log B ) − / . Proof.
In order to replace the integral over | t | > k /Y in the estimationbefore the statement of the lemma by g a ( Y ; η ; B ), we must add Y Y Y η ϑ a ( η ) X k | η η η η µ ( k ) k Z /Y < | t |
We have N b = Y Y η g b ( Y , η /Y ; η ; B ) ϑ b ( η ) φ ∗ ( η ) φ ∗ (gcd( η , η )) + O ( R b ( η , η ; B )) with ϑ b ( η ) := ϑ ( η ) φ ∗ ( η η η η ) and X η ,η R b ( η , η ; B ) ≪ B log B. Proof.
The main term is clear. We apply Lemma 2(1) to deduce that theerror term can be estimated as X η ,η R b ( η , η ; B ) ≪ X η ,η ω ( η η η η ) φ † ( η ) Y Y / η η / Y = X η ,η ω ( η η η η ) φ † ( η ) B / η / η / η / ≪ X η ,η ,η ,η ω ( η η η ) φ † ( η ) B log B η (5 / , / , / , η / ≪ B log B. In the last step, we have used η (cid:18) B η (2 , , , η (cid:19) / · (cid:18) B η (1 , , , η (cid:19) / = B / η (3 / , / , / , η , which we obtain using (2.2) and (2.9). (cid:3) Next, we sum the main term of Lemma 5 over all suitable η . Apply[BD07, Lemma 2] with α = 0 , q = 1 to obtain N b = Y Y η ϑ b ( η ) X η > f η ,η η η ( η ) g b ( Y , η /Y ; η ; B )= Y Y Y η g b ( Y ; η ; B ) ϑ b ( η ) φ ∗ ( η η η ) ζ (2) Y p | η η η η (cid:18) − p (cid:19) − + O (cid:18) Y Y η | ϑ b ( η ) | (log B )2 ω ( η η η ) sup t g b ( Y , t ; η ; B ) (cid:19) + O Y Y Y η | ϑ b ( η ) | Z t /Y g b ( Y , t ; η ; B ) d t ! , where the supremum is taken over t > /Y . Lemma 8.
We have N b = Y Y Y η g b ( Y ; η ; B ) ϑ b ( η ) + O ( R b ( η ; B )) with ϑ b ( η ) := ϑ b ( η ) φ ∗ ( η η η ) ζ (2) Y p | η η η η (cid:18) − p (cid:19) − and X η R b ( η ; B ) ≪ B (log B ) − A/ , where the sum is taken over η satisfying (2.10) .Proof. The main term is clear from the discussion before the lemma. Thefirst part of the error term makes the contribution ≪ X η ω ( η η η ) φ † ( η ) Y Y log Bη sup g b ( Y , t ; η ; B ) . We use Lemma 2(3) and (2.10) to obtain ≪ X η ω ( η η η ) φ † ( η ) Y Y / Y log Bη Y = X η ω ( η η η ) φ † ( η ) B / log B η (1 / , / , / , / ≪ X η ,η ,η ω ( η η η ) φ † ( η ) B (log B ) − A/ η (1 , , , ≪ B (log B ) − A/ . ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 13
The contribution of the second term is, using Lemma 2(3) and (2.10)again, ≪ X η φ † ( η ) Y Y Y η Z /Y Y t / d t ≪ X η φ † ( η ) Y Y / Y η Y = X η φ † ( η ) B / η (1 / , / , / , / ≪ X η ,η ,η φ † ( η ) B (log B ) − A/ η (1 , , , ≪ B (log B ) − A/ . This completes the proof of the lemma. (cid:3) The final step
By the discussion at the end of Section 2, we have, for any
A > N U,H ( B ) = N a ( B ) + N b ( B ; A ) + N b ( B ; A ) . By Lemma 1, N U,H ( B ) = N a ( B ) + N b ( B ; A ) + O A ( B (log B ) (log log B ) ) . Using Lemmas 4, 5 and 6 and combining their error terms shows that N a ( B ) = X η ∈E ( B ) Y Y Y η g a ( Y ; η ; B ) ϑ a ( η ) + O ( B (log B ) − / ) , where E ( t ) := { η ∈ Z > | (2.7) , η (2 , , , t } for any t >
1, while Lemmas 4, 7 and 8 give, choosing A := 28, N b ( B ; 28) = X η ∈E ( B/ (log B ) ) Y Y Y η g b ( Y ; η ; B ) ϑ b ( η ) + O ( B (log B ) ) . Recall the definition (4.7) of g . Lemma 9.
We have N U,H ( B ) = X η ∈E ( B ) Y Y Y η g ( Y ; η ; B ) ϑ ( η ) + O ( B (log B ) − / ) where ϑ ( η ) := φ ∗ ( η η ) φ ∗ ( η η η ) φ ∗ ( η η η η ) ζ (2) Y p | η η η η (cid:18) − p (cid:19) − × X k | η gcd( k ,η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) if the coprimality conditions (2.7) hold, and ϑ ( η ) := 0 otherwise.Proof. We easily check that ϑ ( η ) agrees with ϑ a ( η ) and ϑ b ( η ) for η satisfying(2.7).In view of the discussion before the lemma, it remains to show that X η ∈E ( B ) \E ( B/ (log B ) ) Y Y Y η g b ( Y ; η ; B ) ϑ b ( η )makes a negligible contribution.Indeed, we estimate this as ≪ X η ∈E ( B ) \E ( B/ (log B ) ) φ † ( η ) Y Y Y η Y = X η ∈E ( B ) \E ( B/ (log B ) ) φ † ( η ) B η (1 , , , ≪ B (log B ) (log log B )since we have g b ( t ; η ; B ) ≪ Z /t g b ( t , t ; η ; B ) d t ≪ Z /t t t / d t ≪ t , using Lemma 2(3) and the fact that g b ( t , t ; η ; B ) = 0 unless t ≪ /t by (4.1). (cid:3) Define E ∗ ( B ) := { η ∈ Z > | η (2 , , , B, η (3 , , , > B } . Lemma 10.
We have N U,H ( B ) = ω ∞ B X η ∈E ∗ ( B ) ϑ ( η ) η (1 , , , + O ( B (log B ) − / ) . Proof.
By Lemma 2(2), we have g ( Y ; η ; B ) ≪ Z | Y t | > Y / | t | / d t ≪ Y / Y / . Therefore, X η ∈E ( B ) \E ∗ ( B ) Y Y Y η g ( Y ; η ; B ) ϑ ( η ) ≪ X η (3 , , , B φ † ( η ) Y Y Y / η Y / ≪ X η (3 , , , B φ † ( η ) B / η (1 / , / , , / ≪ X η ,η ,η φ † ( η ) B η (1 , , , ≪ B (log B ) . ANIN’S CONJECTURE FOR A QUINTIC DEL PEZZO SURFACE 15
This proves that N U,H ( B ) = X η ∈E ∗ ( B ) Y Y Y η g ( Y ; η ; B ) ϑ ( η ) + O ( B (log B ) − / ) . Comparing the definitions (4.7) and (4.8) of the functions g and G to-gether with the estimation X η ∈E ∗ ( B ) Y Y Y η Z h ( Y ,t ,t ,t ) , | Y t | ,t > d t d t d t ≪ X η ∈E ∗ ( B ) φ † ( η ) Y Y Y η Z | Y t | Y d t ≪ X η ∈E ∗ ( B ) φ † ( η ) Y Y η Y = X η ∈E ∗ ( B ) φ † ( η ) B η (4 , , , ≪ X η ,η ,η φ † ( η ) B η (1 , , , ≪ B (log B ) shows that N U,H ( B ) = X η ∈E ∗ ( B ) Y Y Y η G ( Y ) ϑ ( η ) + O ( B (log B ) − / ) . Finally, we note that Y Y Y η G ( Y ) = Y Y Y η Y ω ∞ = B η (1 , , , ω ∞ using Lemma 3. (cid:3) For k = ( k , k , k , k ) ∈ Z > , let∆ k ( n ) := X η ∈ Z > , η ( k ,k ,k ,k = n ϑ ( η ) η (1 , , , . Consider the Dirichlet series F k ( s ) = ∞ X n =1 ∆ k ( n ) n s = X η ∈ Z > ϑ ( η ) η ( k s +1 ,k s +1 ,k s +1 ,k s +1) . It is absolutely convergent for ℜ e ( s ) >
0. We write it as an Euler product F k ( s ) = Q p F k ,p ( s ), where we compute that F k ,p ( s ) is(1 − /p ) · (cid:18) (1 + 1 /p ) + 1 − /pp k s +1 − − /pp k s +1 − − /pp k s +1 −
1+ 1 − /pp k s +1 − (cid:18) (1 − /p ) + 1 − /pp k s +1 − − /pp k s +1 − − /pp k s +1 − (cid:19)(cid:19) . For ε > k ∈ { (2 , , , , (3 , , , } and all s ∈ C lying in the half-plane ℜ e ( s ) > − / ε , we have F k ,p ( s ) Y j =1 (cid:18) − p k j s +1 (cid:19) = 1 + O ε ( p − − ε ) . We define E k ( s ) := Y j =1 ζ ( k j s + 1) , G k ( s ) := F k ( s ) E k ( s )and note that F k ( s ) has a meromorphic continuation to ℜ e ( s ) > − / ε with a pole of order 4 at s = 0.As in [BD07, Lemma 15], we use a Tauberian theorem to show that M k ( t ) := X n t ∆ k ( n )can be estimated as G k (0) P (log t )4! Q j =1 k j + O ( t − δ )for some δ > P a monic polynomial of degree 4.Using Lemma 10 and the definitions of ∆ k and M k , N U,H ( B ) = ω ∞ B X n B (∆ (2 , , , ( n ) − ∆ , , , ( n )) + O ( B (log B ) − / )= ω ∞ G k (0) 14! (cid:18) · − · · (cid:19) B (log B ) + O ( B (log B ) − / )Since α ( e S ) = 1864 = 14! (cid:18) · − · · (cid:19) and G k (0) = Y p (cid:18) − p (cid:19) (cid:18) p + 1 p (cid:19) , this completes the proof of the theorem. References [BB07] R. de la Bret`eche and T. D. Browning,
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