On Manin's conjecture for singular del Pezzo surfaces of degree four, I

0.28 MB·English

by  R. de la Breteche

On Manin’s conjecture for singular del Pezzo
surfaces of degree four, I
R. de la Bret`eche1 and T.D. Browning2
6
0 1Universit´e Paris-Sud, Bˆatiment 425, 91405 Orsay Cedex
0 2School of Mathematics, Bristol University, Bristol BS8 1TW
2
[email protected], [email protected]
n
a
J
Abstract
4

This paper contains a proof of the Manin conjecture for the singular
]
T delPezzo surface
N X : x x −x2=x x −x x +x2 =0,
0 1 2 0 4 1 2 3
.
h ofdegreefour. Infact,ifU ⊂X istheopensubsetformedbydeletingthe
t
a unique line from X, and H is the usual height function on P4(Q), then
m theheight zetafunction H(x)−s isanalytically continuedtothe
x∈U(Q)
[ half-plane ℜe(s)>9/10.P
Mathematics Subject Classification (2000): 11G35 (14G05, 14G10)
3
v
6 1 Introduction
8
0
Let Q ,Q Z[x ,...,x ] be a pair of quadratic forms whose common zero
2 1 2 0 4
∈
1 locus defines a geometrically integral surface X P4. Then X is a del Pezzo
⊂
4 surface of degree four. We assume henceforth that the set X(Q) = X P4(Q)
0 of rational points on X is non-empty, so that in particular X(Q) is d∩ense in
/
h X under the Zariski topology. Given a point x = [x0,...,x4] P4(Q), with
∈
t x ,...,x Z such that gcd(x ,...,x ) = 1, we let H(x) = max x .
a 0 4 ∈ 0 4 06i64| i|
m Then H : P4(Q) R>0 is the height attached to the anticanonical divisor
→
K on X, metrized by the choice of norm max x . A finer notion of
: − X 06i64| i|
v density is provided by analysing the asymptotic behaviour of the quantity
i
X
N (B)=# x U(Q):H(x)6B ,
r U,H { ∈ }
a
as B , for appropriate open subsets U X. Since every quartic del Pezzo
→∞ ⊆
surfaceX containsaline, it is naturalto estimateN (B) for the opensubset
U,H
U obtained by deleting the lines from X. The motivation behind this paper is
toconsidertheasymptoticbehaviourofN (B)forsingulardelPezzosurfaces
U,H
of degree four.
A classification of quartic del Pezzo surfaces X P4 can be found in the
⊂
work of Hodge and Pedoe [8, Book IV, XIII.11], showing in particular that
there are only finitely many isomorphism§classes to consider. Let X˜ denote the
minimal desingularisation of X, and let PicX˜ be the Picard group of X˜. Then
Manin has stated a very general conjecture [6] that predicts the asymptotic
1
behaviour of counting functions associated to suitable Fano varieties. In our
setting itleadsus to expectthe existence ofapositive constantc suchthat
X,H
N (B)=c B(logB)ρ−1 1+o(1) , (1.1)
U,H X,H
(cid:0) (cid:1)
as B , where ρ denotes the rank of PicX˜. The constant c has re-
X,H
→ ∞
ceived a conjectural interpretation at the hands of Peyre [9], which in turn has
been generalised to cover certain other cases by Batyrev and Tschinkel [2] and
Salberger [11]. A brief discussion of c will take place in 2.
X,H
§
There has been rather little progress towards the Manin conjecture for del
Pezzo surfaces of degree four. The main successes in this direction are to be
found in work of Batyrev and Tschinkel [1], covering the case of toric varieties,
andinthe workofChambert-LoirandTschinkel[5],coveringthe caseofequiv-
ariant compactifications of vector groups. It is our intention to investigate the
distribution of rational points in the special case that X is defined by the pair
of equations
x x x2 =0,
0 1− 2
x x x x +x2 =0.
0 4− 1 2 3
ThenX P4 is a delPezzosurfaceofdegreefour,with aunique singularpoint
⊂
[0,0,0,0,1] which is of type D . Furthermore, X contains precisely one line
5
x = x = x = 0. It turns out that X is an equivariant compactification of
0 2 3
G2, so that the work of Chambert-Loir and Tschinkel [5, Theorem 0.1] ensures
a
that the asymptotic formula (1.1) holds when U X is taken to be the open
⊂
subsetformedbydeletingtheuniquelinefromX. Nonethelessthereareseveral
reasons why this problem is still worthy of attention. Firstly, in making an
exhaustive study of X it is hoped that a template will be set down for the
treatment of other singular del Pezzo surfaces. In fact no explicit use is made
of the fact that X is an equivariant compactification of G2, and the techniques
a
that we develop in this paper have already been applied to other surfaces [3,
4]. Secondly, in addition to improving upon Chambert-Loir and Tschinkel’s
asymptotic formula for N (B), the results that we obtain lend themselves
U,H
more readily as a bench-test for future refinements of the Manin conjecture,
such as that recently proposed by Swinnerton-Dyer [12] for example.
LetX P4 betheD delPezzosurfacedefinedabove,andletU X bethe
5
⊂ ⊂
corresponding open subset. Our first result concerns the height zeta function
1
Z (s)= , (1.2)
U,H H(x)s
x∈XU(Q)
that is defined when e(s) is sufficiently large. The analytic properties of
ℜ
Z (s) are intimately related to the asymptotic behaviour of the counting
U,H
function N (B). For e(s)>0 we define the functions
U,H
ℜ
E (s+1)=ζ(6s+1)ζ(5s+1)ζ(4s+1)2ζ(3s+1)ζ(2s+1), (1.3)
1
ζ(14s+3)ζ(13s+3)3
E (s+1)= . (1.4)
2 ζ(10s+2)ζ(9s+2)ζ(8s+2)3ζ(7s+2)3ζ(19s+4)
ItiseasilyseenthatE (s)hasameromorphiccontinuationtotheentirecomplex
1
plane witha singlepole ats=1. Similarlyit is clearthatE (s) is holomorphic
2
2
and bounded on the half-plane e(s) > 9/10+ε, for any ε > 0. We are now
ℜ
ready to state our main result.
Theorem 1. Let ε > 0. Then there exists a constant β R, and functions
∈
G (s),G (s) that are holomorphic on the half-plane e(s)>5/6+ε, such that
1 2
ℜ
for e(s)>1 we have
ℜ
12/π2+2β
Z (s)=E (s)E (s)G (s)+ +G (s).
U,H 1 2 1 2
s 1
−
In particular (s 1)6Z (s) has a holomorphic continuation to the half-plane
U,H
−
e(s)>9/10. The function G (s) is bounded for e(s)>5/6+ε and satisfies
1
ℜ ℜ
G (1)=0, and the function G (s) satisfies
1 2
6
G (s) (1+ m(s))6(1−ℜe(s))+ε
2 ε
≪ |ℑ |
on this domain.
An explicit expression for β can be found in (5.24), whereas the formulae
(6.1)–(6.4)canbeusedtodeduceanexplicitexpressionforG . Thereareseveral
1
features of Theorem 1 that are worthy of remark. The first step in the proof of
Theorem 1 is the observation
∞
Z (s)=s t−s−1N (t)dt. (1.5)
U,H U,H
Z
1
Thus we find ourselves in the situation of establishing a preliminary estimate
for N (B) in order to deduce the analytic properties of Z (s) presented in
U,H U,H
Theorem 1, before then using this information to deduce an improvedestimate
for N (B). This will be givenin Theorem2 below. With this orderof events
U,H
in mind we highlight that the term 12(s 1)−1 appearing in Theorem 1 corre-
π2 −
sponds to an isolated conic contained in X. Moreover, whereas the first term
E (s)E (s)G (s)inthe expressionforZ (s) correspondstothe maintermin
1 2 1 U,H
our preliminary estimate for N (B), and arises through the approximation
U,H
of certain arithmetic quantities by real-valued continuous functions, the term
involvingβ hasamorearithmeticinterpretation. Indeed,itwillbeseentoarise
purelyoutoftheerrortermsproducedbyapproximatingthesearithmeticquan-
tities by continuous functions. Finally we make the observation that under the
assumptionoftheRiemannhypothesisE (s)isholomorphicfor e(s)>17/20,
2
ℜ
so that Z (s) has an analytic continuation to this domain.
U,H
We now come to how Theorem 1 can be used to deduce an asymptotic for-
mulaforN (B). Weshallverifyin 2thatthefollowingresultisinaccordance
U,H
§
with the Manin conjecture.
Theorem 2. Let δ (0,1/12). Then there exists a polynomial P of degree 5
∈
such that
N (B)=BP(logB)+O(B1−δ),
U,H
for any B >1. Moreover the leading coefficient of P is equal to
τ 1 6 6 1
∞
1 1+ + ,
28800Yp (cid:16) − p(cid:17) (cid:16) p p2(cid:17)
where
1 1/v
τ = min u3+1,1/v3 max u3 1,0 dudv. (1.6)
∞
Z0 Z−1 (cid:16) {p }−p { − }(cid:17)
3
The deduction of Theorem 2 from Theorem 1 will take place in 7, and
§
amounts to a routine application of Perron’s formula. Although we choose
not to give the details here, it is in fact possible to take δ (0,1/11) in the
∈
statement of Theorem 2 by using more sophisticated estimates for moments
of the Riemann zeta function in the critical strip. By expanding the height
zeta function Z (s) as a power series in (s 1)−1, one may obtain explicit
U,H
−
expressions for the lower order coefficients of the polynomial P in Theorem 2.
It would be interesting to obtain refinements of Manin’s conjecture that admit
conjectural interpretations of the lower order coefficients.
The principal tool in the proof of Theorem 1 is a passage to the universal
torsor above the minimal desingularisation X˜ of X. Although originally in-
troduced to aid in the study of the Hasse principle and Weak approximation,
universal torsors have recently become endemic in the context of counting ra-
tional points of bounded height. In 4 we shall establish a bijection between
§
U(Q) and integer points on the universal torsor, which in this setting has the
natural affine embedding
v y2y v y3y2+v y2 =0.
2 0 4− 0 1 2 3 3
Once we have translated the problem to the universal torsor, Theorem 1 will
beestablishedusingarangeoftechniquesdrawnfromclassicalanalyticnumber
theory. In particular Theorem 1 seems to be the first instance of a height zeta
function that has been calculated via a passage to the universal torsor.
Acknowledgements. Partsofthis paper wereworkedupon while the authors
wereattendingtheconferenceDiophantine geometry atG¨ottingenUniversityin
June 2004, during which time the authors benefited from useful conversations
with several participants. While working on this paper, the first author was at
the E´cole Normale Sup´erieure, and the second author was at Oxford Univer-
sity supported by EPSRC grant number GR/R93155/01. The hospitality and
financial support of these institutions is gratefully acknowledged. Finally, the
authorswouldliketothanktheanonymousrefereeforhiscarefulreadingofthe
manuscript.
2 Conformity with the Manin conjecture
Inthis sectionwe shallshowthatTheorem2agreeswiththe Maninconjecture.
For this we need to calculate the value of c and ρ in (1.1). We therefore
U,H
review some of the geometry of the surface X P4, as defined by the pair of
⊂
quadratic forms
Q (x)=x x x2, Q (x)=x x x x +x2, (2.1)
1 0 1− 2 2 0 4− 1 2 3
where x=(x ,...,x ).
0 4
LetX˜ denotetheminimaldesingularisationofX,andletπ :X˜ X denote
→
the corresponding blow-up map. We let E denote the strict transform of the
6
unique line contained in X, and let E ,...,E denote the exceptional curves
1 5
4
of π. Then the divisors E ,...,E satisfy the Dynkin diagram
1 6
E
3
E E E E E
5 4 1 2 6
andgeneratethePicardgroupofX˜. Inparticularwehaveρ=6in(1.1),which
agrees with Theorem 2.
It remains to discuss the conjectured value of the constant c in (1.1).
X,H
For this we shallfollow the presentationof BatyrevandTschinkel[2, 3.4]. Let
Λ (X˜) PicX˜ R be the cone of effective divisors on X˜, and let §
eff Z
⊂ ⊗
Λ∨ (X˜)= s Pic∨X˜ R: s,t >0 t Λ (X˜)
eff { ∈ ⊗Z h i ∀ ∈ eff }
bethe correspondingdualcone,wherePic∨X˜ denotesthe duallatticetoPicX˜.
Then if dt denotes the Lebesgue measure on Pic∨X˜ R, we define
Z
⊗
α(X˜)= e−h−KX˜,tidt,
ZΛ∨eff(X˜)
where K is the anticanonicaldivisorof X˜. Thus α(X˜) measuresthe volume
− X˜
ofthepolytopeobtainedbyintersectingΛ∨ (X˜)withacertainaffinehyperplane.
eff
Next we discuss the Tamagawa measure on the closure X˜(Q) of X˜(Q) in
X˜(A ),whereA denotestheadelering. WriteL (s,PicX˜)forthelocalfactors
Q Q p
of L(s,PicX˜). Furthermore, let ω denote the archimedean density of points
∞
on X, and let ω denote the usual p-adic density of points on X, for any prime
p
p. Then we may define the Tamagawa measure
ω
τ (X˜)= lim (s 1)ρL(s,PicX˜) ω p , (2.2)
H s→1 − ∞ L (1,PicX˜)
(cid:0) (cid:1) Yp p
where ρ denotes the rank of PicX˜ as above. With these definitions in mind,
the conjectured value of the constant in (1.1) is equal to
c =α(X˜)β(X˜)τ (X˜), (2.3)
X,H H
where β(X˜) = #H1(Gal(Q/Q),PicX˜ Q) = 1, since X˜ is split over the
Q
⊗
ground field Q.
We begin by calculating the value of α(X˜), for which we shall follow the
approachofPeyreandTschinkel[10, 5]. WeneedtodeterminetheconeΛ (X˜)
eff
§
andtheanticanonicaldivisor K . TodeterminethesewemayusetheDynkin
− X˜
diagram to write down the intersection matrix
E E E E E E
1 2 3 4 5 6
E 2 1 1 1 0 0
1
−
E 1 2 0 0 0 1
2
−
E 1 0 2 0 0 0 .
3
−
E 1 0 0 2 1 0
4
−
E 0 0 0 1 2 0
5
−
E 0 1 0 0 0 1
6
−
5
But then it follows that Λ (X˜) is generated by the divisors E ,...,E . Fur-
eff 1 6
thermore,onemployingtheadjunctionformula K .D =2+D2forD PicX˜,
− X˜ ∈
we easily deduce that
K =6E +5E +3E +4E +2E +4E .
− X˜ 1 2 3 4 5 6
It therefore follows that
α(X˜)=Vol (t ,t ,t ,t ,t ,t ) R6 :6t +5t +3t +4t +2t +4t =1
1 2 3 4 5 6 ∈ >0 1 2 3 4 5 6
1 (cid:8) (cid:9)
= Vol (t ,t ,t ,t ,t ) R5 :6t +5t +3t +4t +2t 61 ,
4 1 2 3 4 5 ∈ >0 1 2 3 4 5
(cid:8) (cid:9)
whence in fact
α(X˜)=1/345600. (2.4)
It remains to calculate the value of τ (X˜) in (2.3).
H
Lemma 1. We have τ (X˜)=12τ τ, where τ is given by (1.6) and
H ∞ ∞
1 6 6 1
τ = 1 1+ + . (2.5)
Yp (cid:16) − p(cid:17) (cid:16) p p2(cid:17)
Proof. Recallthedefinition(2.2)ofτ (X˜). Ourstartingpointistheobservation
H
that L(s,PicX˜)=ζ(s)6. Hence it easily follows that
lim (s 1)ρL(s,PicX˜) = lim (s 1)6ζ(s)6 =1.
s→1 − s→1 −
(cid:0) (cid:1) (cid:0) (cid:1)
Furthermore we plainly have
1 6
L (1,PicX˜)−1 = 1 , (2.6)
p
(cid:16) − p(cid:17)
for any prime p.
We proceed by employing the method of Peyre [9] to calculate the value of
the archimedean density ω . It will be convenient to parameterise the points
∞
via the choice of variables x ,x ,x , for which we first observe that the Leray
0 1 4
form ω (X˜) is given by (4x x )−1dx dx dx , since
L 2 3 0 1 4
∂Q1 ∂Q2
∂x2 ∂x2
det = 4x2x3.
−
∂Q1 ∂Q2
∂x3 ∂x3
Now in any real solution to the pair of equations Q (x) = Q (x) = 0, the
1 2
componentsx andx mustnecessarilysharethesamesign. Takingintoaccount
0 1
the fact that x and x represent the same point in P4, we therefore see that
−
ω =2 ω (X˜).
∞ L
Z{x∈R5: Q1(x)=Q2(x)=0, 06x0,x1,x3,|x4|61}
We write ω to denote the contribution to ω from the case x = √x x ,
∞,− ∞ 2 0 1
−
and ω∞,+ for the contribution from the case x2 = √x0x1. But then it follows
that
1 dx dx dx
0 1 4
ω = ,
∞,+
2Z Z Z x2x (x−1/2x3/2 x )
q 0 1 0 1 − 4
6
where the triple integral is over all x ,x ,x R such that
0 1 4
∈
06x ,x , x 61, x3/2 >x1/2x , 1+x x >x3/2x1/2.
0 1 | 4| 1 0 4 0 4 1 0
The change of variables u=x1/2/x1/6, and then v =x1/6, therefore yields
1 0 0
dudvdx
4
ω =6 ,
∞,+
Z Z Z √u3 x
4
−
where the triple integral is now over all u,v,x R such that
4
∈
06v 61, 06u61/v, max 1,u3 1/v6 6x 6min 1,u3 .
4
{− − } { }
On performing the integration over x , a straightforward calculation leads to
4
the equality
1 1/v
ω =12 min u3+1,1/v3 max u3 1,0 dudv.
∞,+
Z0 Z0 (cid:16) {p }−p { − }(cid:17)
The calculation of ω is similar. On following the steps outlined above, one
∞,−
is easily led to the equality
1 0
ω =12 min u3+1,1/v3 dudv.
∞,−
Z Z { }
0 −1 p
Once taken together, these equations combine to show that ω =12τ , where
∞ ∞
τ is given by (1.6).
∞
It remains to calculate the value of ω =lim p−3rN(pr), where we have
p r→∞
written N(pr) = # x (modpr) : Q (x) Q (x) 0 (modpr) . To begin with
1 2
{ ≡ ≡ }
we write
x =pk0x′, x =pk1x′
0 0 1 1
with p ∤ x′x′. Now we have pr x2 if and only if k +k > r, and there are
0 1 | 2 0 1
at most pr/2 square roots of zero modulo pr. When k +k <r, it follows that
0 1
k +k must be even and we may write
0 1
x =p(k0+k1)/2x′,
2 2
with p∤x′ and
2
x′x′ x′2 0(mod pr−k0−k1).
0 1− 2 ≡
The number of possible choices for x′,x′,x′ is therefore
0 1 2
φ(pr−k0)φ(pr−(k0+k1)/2)pk0 if k +k <r,
h (r,k ,k )= 0 1
p 0 1 (cid:26) O(p5r/2−k0−k1) if k +k >r.
0 1
Itremainstodetermine the numberofsolutionsx ,x modulopr suchthat
3 4
pk0x′x pk0/2+3k1/2x′x′ +x2 0(mod pr). (2.7)
0 4− 1 2 3 ≡
In order to do so we distinguish between three basic cases: either k +k < r
0 1
and k 6 3k , or k +k < r and k > 3k , or else k +k > r. For the first
0 1 0 1 0 1 0 1
two ofthese cases we musttake careonly to sum overvalues of k ,k suchthat
0 1
7
k +k is even. We shall denote by N (pr) the contribution to N(pr) from the
0 1 i
ith case, for 16i63, so that
N(pr)=N (pr)+N (pr)+N (pr). (2.8)
1 2 3
We begin by calculating the value of N (pr). For this we write x =pk3x′,
1 3 3
with k = min r/2, k /2 = k /2 . The number of possibilities for x′ is
3 { ⌈ 0 ⌉} ⌈ 0 ⌉ 3
pr−⌈k0/2⌉, each one leading to a congruence of the form
x′x p3k1/2−k0/2x′x′ +p2⌈k0/2⌉−k0x′2 0(mod pr−k0).
0 4− 1 2 3 ≡
Modulo pr−k0, there is one choice for x , and so there are pr+k0−⌈k0/2⌉ =
4
pr+⌊k0/2⌋ possibilities for x and x . On summing these contributions over
3 4
all the relevant values of k ,k , we therefore obtain
0 1
N (pr)= pr+⌊k0/2⌋h (r,k ,k )=p3r−2(p2+4p+1) 1+o(1) ,
1 p 0 1
k0+k1<Xr, k0,k1>0 (cid:0) (cid:1)
k063k1, 2|(k0+k1)
as r .
→∞
Next we calculate N (pr), for which we shall not use the above calculation
2
for h (r,k ,k ). On writing x =pk3x′, with
p 0 1 3 3
k =min r/2, k /4+3k /4 = k /4+3k /4 ,
3 0 1 0 1
{ ⌈ ⌉} ⌈ ⌉
weobservethatk /2+3k /2mustbeevensincep∤x′x′. Thusk =k /4+3k /4
0 1 1 2 3 0 1
and p∤x′. In this way (2.7) becomes
3
pk0/2−3k1/2x′x x′x′ +x′2 0(mod pr−k0/2−3k1/2),
0 4− 1 2 3 ≡
which thereby implies that x′2 x′x′ (mod pk0/2−3k1/2). At this point we
3 ≡ 1 2
recall the auxiliary congruence x′2 x′x′ (mod pr−k0−k1) that is satisfied by
2 ≡ 0 1
x′,x′x′. Weproceedbyfixingvaluesofx′ andx′,forwhichthereareprecisely
0 1 2 2 3
(1 1/p)2pr−k0/2−k1/2pr−k3 choices. But then x′ is fixed modulo pk0/2−3k1/2,
− 1
and so there are pr−k1−(k0/2−3k1/2) possibilities for x′. Finally we deduce from
1
the remaining two congruences that there are pk1 ways of fixing x′, and pk0
0
ways of fixing x . Summing over the relevant values of k and k we therefore
4 0 1
obtain
N (pr)= (1 1/p)2p3r−(k0−k1)/4 =2p3r−1 1+o(1) ,
2
−
k0+k1<Xr, k0,k1>0 (cid:0) (cid:1)
k0>3k1, 4|(k0−k1)
as r .
→∞
Finally we calculate the value of N (pr). In this case we write x =pr/2x′.
3 2 2
ButthenasimilarcalculationultimatelyshowsthatN (pr)=o(p3r)asr .
3
→∞
On combining our estimates for N (pr),N (pr),N (pr) into (2.8), we therefore
1 2 3
deduce that
6 1
N(pr)=p3r 1+ + 1+o(1)
(cid:18) p p2(cid:19)
(cid:0) (cid:1)
as r , whence
→∞
6 1
ω = lim p−3rN(pr)=1+ +
p r→∞ p p2
for any prime p. We combine this with (2.6), in the manner indicated by (2.2),
in order to deduce (2.5).
8
We end this section by combining (2.4) and Lemma 1 in (2.3), in order to
deduce that the conjectured value of the constant in (1.1) is equal to
1
c = τ τ,
X,H ∞
28800
where τ is given by (1.6) and τ is given by (2.5). This agrees with the value
∞
of the leading coefficient obtained in Theorem 2.
3 Congruences
In this section we shall collect together some of the basic facts concerning con-
gruencesthatwillbeneededintheproofofTheorem1. Webeginbydiscussing
the case of quadratic congruences. For any integers a,q such that q > 0, we
definethearithmeticfunctionη(a;q)tobethenumberofpositiveintegersn6q
such that n2 a (mod q). When q is odd it follows that
≡
a
η(a;q)= µ(d) ,
| | d
Xd|q (cid:0) (cid:1)
where (a) is the usual Jacobi symbol. On noting that η(a;2ν) 6 4 for any
d
ν N, it easily follows that
∈
η(a;q)62ω(q)+1 (3.1)
for any q N. Here ω(q) denotes the number of distinct prime factors of q.
∈
Turning to the case of linear congruences, let κ [0,1] and let ϑ be any
∈
arithmetic function such that
∞
(ϑ µ)(d)
| ∗ | < , (3.2)
dκ ∞
Xd=1
where(f g)(d)= f(e)g(d/e)isthe usualDirichletconvolutionofanytwo
∗ e|d
arithmetic functionPs f,g. Then for any coprime integers a,q such that q > 0,
and any t>1, we deduce that
∞
ϑ(n)= (ϑ µ)(d) 1
∗
Xn6t Xd=1 mX6t/d
n≡a(modq) gcd(d,q)=1 md≡a(modq)
∞ ∞
t (ϑ µ)(d) (ϑ µ)(d)
= ∗ +O tκ | ∗ | ,
q d dκ
Xd=1 (cid:16) Xd=1 (cid:17)
gcd(d,q)=1
on using the equality ϑ = (ϑ µ) 1 and the trivial estimate x = x+O(xκ)
∗ ∗ ⌊ ⌋
for any x>0. We summarise this estimate in the following result.
Lemma 2. Letκ [0,1],let ϑbeany arithmetic function suchthat (3.2)holds,
∈
and let a,q Z be such that q >0 and gcd(a,q)=1. Then we have
∈
∞ ∞
t (ϑ µ)(d) (ϑ µ)(d)
ϑ(n)= ∗ +O tκ | ∗ | .
q d dκ
Xn6t Xd=1 (cid:16) Xd=1 (cid:17)
n≡a(modq) gcd(d,q)=1
9
Define the real-valued function ψ(t) = t 1/2, where t denotes the
{ } − { }
fractional part of t R. Then ψ is periodic with period 1. When ϑ(n) =1 for
∈
all n N we are able to refine Lemma 2 considerably.
∈
Lemma 3. Let a,q Z be such that q > 0, and let t ,t R such that
1 2
∈ ∈
t >max 0,t . Then
2 1
{ }
t t
# t <n6t :n a(mod q) = 2− 1 +r(t ,t ;a,q),
1 2 1 2
{ ≡ } q
where
t a t a
1 2
r(t ,t ;a,q)=ψ − ψ − .
1 2
(cid:16) q (cid:17)− (cid:16) q (cid:17)
Proof. Write a=b+qc for some integer 06b<q. Then it is clear that
t b t b
# t <n6t :n b(mod q) = 2− 1− ,
1 2
{ ≡ } j q k−j q k
whence
t t
# t <n6t :n a(mod q) 2− 1 =r(t ,t ;b,q).
1 2 1 2
{ ≡ }− q
We complete the proof of Lemma 3 by noting that r(t ,t ;b,q)=r(t ,t ;a,q),
1 2 1 2
since ψ has period 1.
Weshallalsoneedtoknowsomethingabouttheaverageorderofthefunction
ψ. We proceed by demonstrating the following result.
Lemma 4. Let ε > 0, t > 0 and let X > 1. Then for any b,q Z such that
∈
q >0 and gcd(b,q)=1, we have
t bx2 X
ψ − (qX)ε +q1/2 .
06Xx<X (cid:16) q (cid:17)≪ε (cid:16)q1/2 (cid:17)
Proof. Throughoutthis proofwe shall write e(t)=e2πit and e (t)=e2πit/q. In
q
order to establish Lemma 4 we shall expand the function f(k) = ψ((t k)/q)
−
as a Fourier series. Thus we have
f(k)= a(ℓ)e (kℓ),
q
06Xℓ<q
for any k Z, where the coefficients a(ℓ) are given by
∈
1
a(ℓ)= f(j)e ( jℓ).
q
q −
06Xj<q
Let α denote the distance from α R to the nearest integer. We proceed by
k k ∈
proving the estimates
q−1, ℓ=0,
a(ℓ) (3.3)
≪(cid:26) q−1 ℓ/q −1, ℓ=0.
k k 6
10