aa r X i v : . [ m a t h . N T ] S e p ON A SENARY QUARTIC FORM
JIANYA LIU, JIE WU & YONGQIANG ZHAO
Abstract.
We count rational points of bounded height on the non-normal senaryquartic hypersurface x = ( y + · · · + y ) z in the spirit of Manin’s conjecture. Introduction
Recently, we [7] proved Manin’s conjecture for singular cubic hypersurfaces(1.1) x = ( y + · · · + y n ) z, where n is a positive multiple of 4. In this short note, we show that our method usedin [7] also works for higher degree forms like(1.2) x m = ( y + · · · + y n ) z m − , where n > m >
4. To illustrate, we establish an asymptotic formula for thenumber of rational points of bounded height on the quartic hypersurface(1.3) Q : x = ( y + y + y + y ) z , in the spirit of Manin’s conjecture.It is easy to see that the subvariety x = z = 0 of Q already contains ≫ B rationalpoints with | x | B, | z | B, and | y j | B with 1 j
4, which is predominant andis much larger than the heuristic prediction that is of order B . One therefore countsrational points on the complement subset U = Q r { x = z = 0 } . Let H be the heightfunction H ( x : y : · · · : y : z ) = max (cid:8) | x | , q y + · · · + y , | z | (cid:9) for ( x, y , . . . , y , z ) = 1. Let B be a large integer, and define N U ( B ) := (cid:12)(cid:12)(cid:8) ( x : y : · · · : y : z ) ∈ U : H ( x : y : · · · : y : z ) B (cid:9)(cid:12)(cid:12) . This counts rational points in U whose height is bounded by B , and the aim of thisnote is obtain an asymptotic formula for it. To this end, we need to understand inadvance a similar quantity N ∗ U ( B ) := X | x | B, y + ··· + y B , | z | Bx =( y + ··· + y ) z . One sees, in N ∗ U ( B ), that the co-prime condition ( x, y , . . . , y , z ) = 1 in N U ( B ) isrelaxed. Our main result is as follows. Date : September 18, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Quartic hypersurface; Manin’s conjecture; rational point; asymptoticformula.
Theorem 1.1. As B → ∞ , we have N U ( B ) = C B log B (cid:26) O (cid:18) √ log B (cid:19)(cid:27) , (1.4) N ∗ U ( B ) = C ∗ B log B (cid:26) O (cid:18) √ log B (cid:19)(cid:27) (1.5) with C := ζ (3) C and C ∗ := C , where C is defined as in (2.6) below, and ζ is theRiemann zeta-function. We note that the exponent of B in the main terms of the above theorem is 3 insteadof 2 as predicted by the usual heuristic. This phenomenon may be explained by thefact that the hypersurface Q is not normal.It is easy to check that Q has an obvious quadric bundle structure given by(1.6) Q [ a : b ] : (cid:26) b x = a ( y + y + y + y ) ,ax − by = 0 , and { Q [ a : b ] } covers Q as long as [ a : b ] goes thorough P ( Q ). From this, it is possibleto interpret Theorem 1.1 in the framework of the generalized Manin’s conjecture byBatyrev and Tschinkel [1], as was done in the work of de la Bret`eche, Browning, andSalberger [3]. However, we will not pursue such an explanation here. The only solepurpose of this short note is to show that our method used in [7] also works for higherdegree forms Q .Finally, we remark that using the method in our joint paper [4] with de la Bret`eche,one can get power-saving error terms in Theorem 1.1, which we will not pursue here.2. Outline of the proof of Theorem 1.1
Denote by r ( d ) the number representations of a positive integer d as the sum of foursquares : d = y + · · · + y with ( y , . . . , y ) ∈ Z . It is well-known (cf. [5, (3.9)]) that(2.1) r ( d ) = 8 r ∗ ( d ) with r ∗ ( d ) := X ℓ | d, ℓ ℓ. Let (cid:3) ( n ) be the characteristic function of squares. In view of the above, we can write(2.2) N ∗ U ( B ) = 32 (cid:26) X n B X d | n d B r ∗ ( d ) (cid:3) (cid:18) n d (cid:19) − X n B X d | n d Theorem 2.1. Let ε > be arbitrary. We have (2.4) S ( x, y ) = xy (cid:0) P ( ψ ) + P ′ ( ψ ) (cid:1) + O ε (cid:0) x y + x + ε y (cid:1) uniformly for x > y > x > , where ψ := log x − log y and P ( t ) is a quadraticpolynomial, defined as in (4.18) below. In particular, for any fixed η ∈ (0 , we have (2.5) S ( x, y ) = 4 C xy (cid:18) log x − 14 log y (cid:19)(cid:26) O (cid:18) x ) η (cid:19)(cid:27) uniformly for x > and x (log x ) − − η ) y x , where (2.6) C := 23150 ζ (5) Y p (cid:18) p + 2 p + 2 p + 1 p + 1 p (cid:19)(cid:18) − p (cid:19) is the leading coefficient of P ( t ) . Now we turn to analyze T ( B ) which is more difficult, since the range of its secondsummation depends on the variable n of the first summation. Thus Theorem 2.1 doesnot apply to T ( B ) directly. In § Theorem 2.2. As B → ∞ , we have (2.7) T ( B ) = 25 C B log B (cid:26) O (cid:18) √ log B (cid:19)(cid:27) , where C is as in (2.6) above. As in [2, 7], we shall firstly establish an asymptotic formula for the quantity(2.8) M ( X, Y ) := Z Y Z X S ( x, y ) d x d y. by applying the method of complex integration. Then we derive the asymptotic formula(2.4) for S ( x, y ) in Theorem 2.1 by the operator D defined below. Let E k be the set ofall functions of k variables. Define the operator D : E → E by(2.9) ( D f )( X, H ; Y, J ) := f ( H, J ) − f ( H, Y ) − f ( X, J ) + f ( X, Y ) . The next lemma summarises all properties of D needed later. Lemma 2.1. (i) Let f ∈ E be a function of class C . Then we have ( D f )( X, H ; Y, J ) = ( J − Y )( H − X ) (cid:26) ∂ f∂x∂y ( X, Y ) + O (cid:0) R ( X, H ; Y, J ) (cid:1)(cid:27) for X H and Y J , where R ( X, H ; Y, J ) := ( H − X ) max X x HY y J (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂x ∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) + ( J − Y ) max X x HY y J (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂x∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) . (ii) Let S ( x, y ) and M ( X, Y ) be defined as in (2.3) and (2.8) . Then ( D M )( X − H, X ; Y − J, Y ) HJ S ( X, Y ) ( D M )( X, X + H ; Y, Y + J ) for H X and J Y . JIANYA LIU, JIE WU & YONGQIANG ZHAO The next elementary estimate ([2, Lemma 6(i)] or [7, Lemma 4.3]) will also be usedseveral times in the paper. Lemma 2.2. Let H X and | σ | . Then for any β ∈ [0 , , we have (2.10) (cid:12)(cid:12) ( X + H ) s − X s (cid:12)(cid:12) ≪ X σ (( | τ | + 1) H/X ) β , where the implied constant is absolute. Dirichlet series associated with S ( x, y )In view of the definition of S ( x, y ) in (2.3), we define the double Dirichlet series(3.1) F ( s, w ) := X n > n − s X d | n d − w r ∗ ( d ) (cid:3) (cid:18) n d (cid:19) for ℜ e s > ℜ e w > 0. The next lemma states that the function F ( s, w ) enjoys anice factorization formula. Lemma 3.1. For min j ℜ e ( s + 2 jw − j ) > , we have (3.2) F ( s, w ) = Y j ζ ( s + 2 jw − j ) G ( s, w ) , where G ( s, w ) is an Euler product, given by (3.8) , (3.10) and (3.11) below. Further, forany ε > and for min j ℜ e ( s + 2 jw − j ) > + ε , G ( s, w ) converges absolutely and (3.3) G ( s, w ) ≪ ε . Proof. Since the functions r ∗ ( d ) and n − s P d | n d − w r ∗ ( d ) (cid:3) ( n /d ) are multiplicative, for ℜ e s > ℜ e w > F ( s, w ) = Y p X ν > p − νs X µ ν p − µw r ∗ ( p µ ) = Y p F p ( s, w ) . In the above computations, speacial attention should be paid to the effect of the func-tion (cid:3) . The next is to simplify each F p ( s, w ). To this end, we recall (2.1) so that(3.4) r ∗ ( p µ ) = 1 − p µ +1 − p ( p > , r ∗ (2 µ ) = 3for all integers µ > 1. On the other hand, a simple formal calculation shows(3.5) X ν > x ν X µ ν y µ − z µ +1 − z = 11 − z X ν > x ν (cid:18) − y ν +2 − y − z − ( yz ) ν +2 − y z (cid:19) = 1 + xy (1 + z + z ) + xy ( z + z + z ) + x y z (1 − x )(1 − xy )(1 − xy z )and(3.6) 1 + X ν > x ν (cid:18) a X µ ν y µ (cid:19) = 1 + X ν > x ν (cid:18) a y − y ν +2 − y (cid:19) = 1 + axy + ( a − xy (1 − x )(1 − xy ) · N A SENARY QUARTIC FORM 5 When p > 2, in view of (3.4), we can apply (3.5) with ( x, y, z ) = ( p − s , p − w , p ) to write(3.7) F p ( s, w ) = Y j (cid:0) − p − ( s +2 jw − j ) (cid:1) − G p ( s, w ) , where(3.8) G p ( s, w ):= (cid:18) p + p + 1 p s +2 w + p + p + pp s +4 w + p p s +6 w (cid:19)(cid:18) − p p s +2 w (cid:19)(cid:18) − p s +4 w (cid:19) − . While for p = 2, the formula (3.6) with ( x, y, z, a ) = (2 − s , − w , , 3) gives(3.9) F ( s, w ) = Y j (cid:0) − − ( s +2 jw − j ) (cid:1) − G ( s, w ) , where(3.10) G ( s, w ) := 1 + 3 · − s − w + 2 − s − w +1 − − s − w Y j (1 − − ( s +2 jw − j ) ) . Combining (3.7)–(3.10), we get (3.2) with(3.11) G ( s, w ) := Y p G p ( s, w ) ( ℜ e s > , ℜ e w > . It is easy to verify that for min j ( σ + 2 ju − j ) > + ε , we have | G p ( s, w ) | =1 + O ( p − − ε ). This shows that under the same condition, the Euler product G ( s, w )converges absolutely and (3.3) holds. By analytic continuation, (3.2) is also true in thesame domain. This completes the proof. (cid:3) Proof of Theorem 2.1 In the sequel, we suppose(4.1) 10 X Y X , ( XY ) T U X , H X, J Y, and for brevity we fix the following notation:(4.2) s := σ + i τ, w := u + i v, L := log X, κ := 1 + L − , λ := 1 + 4 L − . The following proposition is an immediate consequence of Lemmas 4.2-4.5 below. Proposition 4.1. Under the previous notation, we have M ( X, Y ) = X Y P (log X − log Y ) + R ( X, Y ) + R ( X, Y ) + R ( X, Y ) + O (1) uniformly for ( X, Y, T, U, H, J ) satisfying (4.1) , where R , R , R and P ( t ) are definedas in (4.8) , (4.13) , (4.16) and (4.18) below, respectively. The proof is divided into several subsections. JIANYA LIU, JIE WU & YONGQIANG ZHAO Application of Perron’s formula. The first step is to apply Perron’s formulatwice to transform M ( X, Y ) into a form that is ready for future treatment. Lemma 4.2. Under the previous notation, we have (4.3) M ( X, Y ) = M ( X, Y ; T, U ) + O (1) uniformly for ( X, Y, T, U ) satisfying (4.1) , where the implied constant is absolute and (4.4) M ( X, Y ; T, U ) := 1(2 π i) Z κ +i Tκ − i T (cid:18) Z λ +i Uλ − i U F ( s, w ) Y w +1 w ( w + 1) d w (cid:19) X s +1 s ( s + 1) d s. The proof is the same as that of [7, Lemma 6.2].4.2. Application of Cauchy’s theorem. In this subsection, we shall apply Cauchy’stheorem to evaluate the integral over w in M ( X, Y ; T, U ). We write(4.5) w j = w j ( s ) := (2 j + 1 − s ) / (2 j ) (1 j F ∗ ( s ) := ζ ( s ) ζ (2 − s ) G ( s, w ( s )) , F ∗ ( s ) := ζ ( s ) ζ ( s +12 ) G ( s, w ( s )) . Lemma 4.3. Under the previous notation, for any ε > we have (4.7) M ( X, Y ; T, U ) = I + I + R ( X, Y ) + O ε (1) uniformly for ( X, Y, T, U ) satisfying (4.1) , where I := 42 π i Z κ +i Tκ − i T F ∗ ( s ) X s +1 Y (5 − s ) / (3 − s )(5 − s ) s ( s + 1) d s,I := 162 π i Z κ +i Tκ − i T F ∗ ( s ) X s +1 Y (9 − s ) / (5 − s )(9 − s ) s ( s + 1) d s, and (4.8) R ( X, Y ) := 1(2 π i) Z κ +i Tκ − i T (cid:18) Z + ε +i U + ε − i U F ( s, w ) Y w +1 w ( w + 1) d w (cid:19) X s +1 s ( s + 1) d s. Furthermore we have (4.9) ( D R )( X, X + H ; Y, Y + J )( D R )( X − H, X ; Y − J, Y ) (cid:27) ≪ ε X + ε Y + ε H J + X ε Y + ε HJ uniformly for ( X, Y, T, U, H, J ) satisfying (4.1) .Proof. We want to calculate the integral12 π i Z λ +i Uλ − i U F ( s, w ) Y w +1 w ( w + 1) d w for any individual s = σ + i τ with σ = κ and | τ | T . We move the line of integration ℜ e w = λ to ℜ e w = + ε . By Lemma 3.1, for σ = κ and | τ | T , the points w j ( s ) ( j = 1 , + ε u λ and | v | U . The residues of F ( s,w ) w ( w +1) Y w +1 at the poles w j ( s ) are(4.10) 4 F ∗ ( s ) Y (5 − s ) / (3 − s )(5 − s ) , F ∗ ( s ) Y (9 − s ) / (5 − s )(9 − s ) , N A SENARY QUARTIC FORM 7 respectively, where F ∗ j ( s )( j = 1 , 2) are defined as in (4.6).It is well-known that (cf. e.g. [8, page 146, Theorem II.3.7])(4.11) ζ ( s ) ≪ | τ | max { (1 − σ ) / , } log | τ | ( σ > , | τ | > c > σ = κ and + ε u λ , it is easily checked thatmin j ( σ + 2 ju − j ) > + ε − 1) = + 3 ε > + ε. It follows from (4.11) and (3.3) that F ( s, w ) ≪ ε U − u ) L for σ = κ, | τ | T, + ε u λ and v = ± U . This implies that Z λ ± i U + ε ± i U F ( s, w ) Y w +1 w ( w + 1) d w ≪ ε Y L Z λ (cid:18) YU (cid:19) u d u ≪ ε Y L U ≪ ε . Cauchy’s theorem then gives12 π i Z λ +i Uλ − i U F ( s, w ) Y w +1 w ( w + 1) d w = 4 F ∗ ( s ) Y (5 − s ) / (3 − s )(5 − s ) + 16 F ∗ ( s ) Y (9 − s ) / (5 − s )(9 − s )+ 12 π i Z + ε +i U + ε − i U F ( s, w ) Y w +1 w ( w + 1) d w + O ε (1) . Inserting the last formula into (4.4), we obtain (4.7).Finally we prove (4.9). For σ = κ , | τ | T , u = + ε and | v | U , we apply (4.11)and (3.3) as before, to get F ( s, w ) ≪ ( | τ | + | v | + 1) L ≪ (cid:8) ( | τ | + 1) + ( | v | + 1) (cid:9) L . Also, for σ, τ, u, v as above, we have r s,w ( X, H ; Y, J ) := (cid:0) ( X + H ) s +1 − X s +1 (cid:1)(cid:0) ( Y + J ) w +1 − Y w +1 (cid:1) ≪ X (( | τ | + 1) H/X )) − ε Y + ε (( | v | + 1) J/Y ) − ε ≪ X + ε Y + ε H J ( | τ | + 1) − ε ( | v | + 1) − ε by (2.10) of Lemma 2.2 with β = − ε and with β = 1 − ε . Similarly, r s,w ( X, H ; Y, J ) = (cid:0) ( X + H ) s +1 − X s +1 (cid:1)(cid:0) ( Y + J ) w +1 − Y w +1 (cid:1) ≪ X (( | τ | + 1) H/X )) − ε Y + ε (( | v | + 1) J/Y ) − ε ≪ X ε Y + ε HJ ( | τ | + 1) − ε ( | v | + 1) − ε by Lemma 2.2 with β = 1 − ε and with β = − ε . These and Lemma 2.1(i) imply( D R )( X, X + H ; Y, Y + J ) = Z κ +i Tκ − i T Z + ε +i U + ε − i U F ( s, w )(2 π i) r s,w ( X, H ; Y, J ) s ( s + 1) w ( w + 1) d w d s ≪ ε X + ε Y + ε H J + X ε Y + ε HJ . This completes the proof. (cid:3) JIANYA LIU, JIE WU & YONGQIANG ZHAO Evaluation of I .Lemma 4.4. Under the previous notation, we have (4.12) I = R ( X, Y ) + O (1) uniformly for ( X, Y, T ) satisfying (4.1) , where (4.13) R ( X, Y ) := 42 π i Z +i T − i T F ∗ ( s ) X s +1 Y (5 − s ) / (3 − s )(5 − s ) s ( s + 1) d s. Further we have (4.14) ( D R )( X, X + H ; Y, Y + J )( D R )( X − H, X ; Y − J, Y ) (cid:27) ≪ X Y HJ uniformly for ( X, Y, T, H, J ) satisfying (4.1) .Proof. We shall prove (4.12) by moving the contour ℜ e s = κ to ℜ e s = . When κ σ , it is easy to check thatmin j ( σ + 2 jw ( σ ) − j ) = min j ( j + (1 − j ) σ ) > · By Lemma 3.1 the integrand is holomorphic in the rectangle κ σ and | τ | T ;and we can apply (4.11) and (3.3) to get F ∗ ( s ) ≪ T ( σ − / L in this rectangle, whichimplies that Z ± i Tκ ± i T F ∗ ( s ) X s +1 Y (5 − s ) / (3 − s )(5 − s ) s ( s + 1) d s ≪ X Y L T Z κ (cid:18) XT / Y / (cid:19) σ − d σ ≪ X Y L T + X Y L T ≪ . This proves (4.12).To establish (4.14), we note that F ∗ ( s ) ≪ ( | τ | + 1) for σ = and | τ | T . By(2.10) of Lemma 2.2 with β = 1, r s,w ( s ) ( X, H ; Y, J ) := (cid:0) ( X + H ) s +1 − X s +1 (cid:1)(cid:0) ( Y + J ) (5 − s ) / − Y (5 − s ) / (cid:1) ≪ X Y HJ ( | τ | + 1) . Combining these with Lemma 2.1(ii), we deduce that( D R )( X, X + H ; Y, Y + J ) = 42 π i Z +i T − i T F ∗ ( s ) r s,w ( s ) ( X, H ; Y, J )(3 − s )(5 − s ) s ( s + 1) d s ≪ X Y HJ, from which the desired result follows. (cid:3) N A SENARY QUARTIC FORM 9 Evaluation of I .Lemma 4.5. Under the previous notation, for any ε > we have (4.15) I = X Y P (log X − log Y ) + R ( X, Y ) + O ε (1) uniformly for ( X, Y, T ) satisfying (4.1) , where P ( t ) is defined as in (4.18) below and (4.16) R ( X, Y ) := 162 π i Z + ε +i T + ε − i T F ∗ ( s ) X s +1 Y (9 − s ) / (5 − s )(9 − s ) s ( s + 1) d s. Further we have (4.17) ( D R )( X, X + H ; Y, Y + J )( D R )( X − H, X ; Y − J, Y ) (cid:27) ≪ ε X + ε Y HJ uniformly for ( X, Y, T, H, J ) satisfying (4.1) .Proof. We move the line of integration ℜ e s = κ to ℜ e s = + ε . Obviously s = 1 isthe unique pole of order 2 of the integrand in the rectangle + ε σ κ and | τ | T ,and the residue is X Y P (log X − log Y ) with(4.18) P ( t ) := (cid:18) s − F ∗ ( s )e t ( s − (5 − s )(9 − s ) s ( s + 1) (cid:19) ′ (cid:12)(cid:12)(cid:12)(cid:12) s =1 . Here P ( t ) is a linear polynomial with the leading coefficient C given by (2.6) above.When + ε σ κ , we check thatmin j ( σ + 2 jw ( σ ) − j ) = min j ( j + (2 − j ) σ ) > + ε. Hence when + ε σ κ and | τ | T , (4.11) and (3.3) yields F ∗ ( s ) ≪ T (1 − σ ) / L . Itfollows that Z κ ± i T + ε ± i T F ∗ ( s ) X s +1 Y (9 − s ) / (5 − s )(9 − s ) s ( s + 1) d s ≪ X Y L T Z κ (cid:18) Y T X (cid:19) (1 − σ ) / d s ≪ X Y L T + X Y L T ≪ . These establish (4.15). To prove (4.17), we note that for σ = + ε and | τ | T , wehave F ∗ ( s ) ≪ ε ( | τ | + 1) / thanks to (4.11) and (3.3), and r s,w ( s ) ( X, H ; Y, J ) := (cid:0) ( X + H ) s +1 − X s +1 (cid:1)(cid:0) ( Y + J ) (9 − s ) / − Y (9 − s ) / (cid:1) ≪ ε X + ε Y HJ ( | τ | + 1) by Lemma 2.2 with β = 1. Combining these with Lemma 2.1(i), we deduce that( D R )( X, X + H ; Y, Y + J ) = 162 π i Z + ε +i T + ε − i T F ∗ ( s ) r s,w ( s ) ( X, H ; Y, J )(5 − s )(9 − s ) s ( s + 1) d s ≪ ε X + ε Y HJ. This proves the lemma. (cid:3) Completion of proof of Theorem 2.1. Denote by M ( X, Y ) the main term inthe asymptotic formula of M ( x, y ) in Proposition 4.1, that is M ( X, Y ) := X Y P ( ψ )and ψ := log( X/Y / ). Then Lemma 2.1(i) gives( D M )( X, X + H ; Y, Y + J ) = (cid:8) XY (cid:0) P ( ψ ) + P ′ ( ψ ) (cid:1) + O ( XJ L + Y H L ) (cid:9) HJ. Since D is a linear operator, this together with Proposition 4.1 implies that( D M )( X, X + H ; Y, Y + J ) = (cid:8) XY (cid:0) P ( ψ ) + P ′ ( ψ ) (cid:1) + O ε ( R ) (cid:9) HJ with R := X + ε Y H − + X ε Y J − + X Y + X + ε Y + XJ L + Y H L . The same formula also holds for ( D M )( X − H, X ; Y − J, Y ). Now Lemma 2.1(ii) with H = XY − and J = Y allows us to deduce S ( X, Y ) = XY (cid:0) P ( ψ ) + P ′ ( ψ ) (cid:1) + O ε (cid:0) X Y + X + ε Y (cid:1) , where we have used the following facts( X Y ) − ε ( X + ε Y ) ε = X − − ε ) ε Y + ε > X ε Y + ε , ( X Y ) ( X + ε Y ) = X ε Y > X ε Y . This finally completes the proof of Theorem 2.1.5. Proof of Theorems 2.2 and 1.1 Proof of Theorems 2.2. The idea is to apply Theorems 2.1 in a delicate way. Triviallywe have r ∗ ( d ) dτ ( d ) (here τ ( n ) is the divisor function), and therefore(5.1) S ( x, y ) y X n x X d | n τ ( d ) ≪ xy (log x ) for all x > y > 2, where the implied constant is absolute.Let δ := 1 − (log B ) − and let k be a positive integer such that δ k < (log B ) − δ k − . Note that k ≍ (log B ) log log B . In view of (5.1), we can write(5.2) T ( B ) = X k k X δ k B By noticing that(1 − δ ) 1 − δ k − δ = 1 − δ k δ + δ + δ = 15 + O (cid:18) √ log B (cid:19) , ( δ − − δ − δ k +1) − δ = δ − δ k +4 δ + δ + δ + δ = 15 + O (cid:18) √ log B (cid:19) . The desired asymptotic formula (2.7) follows from (5.4) and (5.5). (cid:3) Proof of Theorem 1.1. Applying (2.4) of Theorem 2.1 with ( x, y ) = ( B, B ), we have(5.6) X n B X d | n d B r ∗ ( d ) (cid:3) (cid:18) n d (cid:19) = 2 C B log B (cid:26) O (cid:18) √ log B (cid:19)(cid:27) . Inserting this and (2.7) into (2.2), we obtain (1.5) with C ∗ = C .Finally (1.4) follows from (1.5) via the inversion formula of M¨obius. (cid:3) Acknowledgements. This work was supported by National Natural Science Foun-dation of China (Grant Nos. 11531008 and 11771121) and the Program PRC 1457-AuForDiP (CNRS-NSFC). References [1] V. Batyrev and Y. Tschinkel, Tamagawa numbers of polarized algebraic varieties , Ast´erisque (1998), 299–340.[2] R. de la Bret`eche, Sur le nombre de points de hauteur born´ee d’une certaine surface cubiquesinguli`ere , Ast´erisque (1998), 51–77.[3] R. de la Bret `eche, T. Browning and P. Salberger, Counting rational points on the Cayley ruledcubic , European Journal of Mathematics (2016), 55–72.[4] R. de la Bret`eche, J. Liu, J. Wu and Y. Zhao. On a certain non-split cubic surface , arXiv:1709.09476.[5] E. Grosswald, Representations of integers as sums of squares , Springer-Verlag, New York, 1985.xi+251 pp. ISBN: 0-387-96126-7.[6] H. Iwaniec, Topics in classical automorphic forms , Graduate Studies in Mathematics, vol. ,American Mathematical Society, Providence, Rhode Island, 1997.[7] J. Liu, J. Wu and Y. Zhao, Manin’s conjecture for a class of singular cubic hypersurfaces . IMRN,Vol. 2017, No. 00, pp 1–36, doi: 10.1093/imrn/rnx179.[8] G. Tenenbaum, Introduction to analytic and probabilistic number theory , Translated from thesecond French edition (1995) by C. B. Thomas, Cambridge Studies in Advanced Mathematics , Cambridge University Press, Cambridge, 1995. xvi+448 pp. Jianya Liu, School of Mathematics, Shandong University, Jinan, Shandong 250100,China E-mail address : [email protected] Jie Wu, CNRS UMR 8050, Laboratoire d’analyse et de math´ematiques appliqu´ees,Universit´e Paris-Est Cr´eteil, 94010 Cr´eteil cedex, France E-mail address : [email protected] Yongqiang Zhao, Westlake University, School of Science, Shilongshan Road, CloudTown, Xihu District, Hangzhou, Zhejiang 310024, China E-mail address ::