Publications Mathématiques de Besançon

Philippe Lebacque

Alexey Zykin

Asymptotic methods in number theory and algebraic geometry

Christophe Ritzenthaler

Optimal curves of genus $1$, $2$ and $3$

Brian Conrey

APPLICATIONS OF THE ASYMPTOTIC LARGE SIEVEbyBrian Conrey
Abstrat.  In this note we desribe some results of Conrey, Iwanie, and Soundararajan. Thedetails will appear elsewhere.Résumé.  Dans ette note, nous présentons des résultats de Conrey, Iwanie et Soundara-rajan. Les détails seront publiés ultérieurement.
1. IntrodutionThe large sieve inequality, in its multipliative form [H-B1℄, asserts that for any sequene ofomplex numbers an,
S(~a,Q) :=
∑
q≤Q
q
φ(q)
∑∗
χ mod q
∣∣
N∑
n=1
anχ(n)
∣∣2 ≤ (Q2 +N)
N∑
n=1
|an|2.Many appliations have followed from this basi tool, espeially, in this form, appliationshaving to do with primes in arithmeti progressions, or to do with statistial averages ofDirihlet L-funtions. Generalizations of this basi tool have been given where the haraters
χ are replaed by the oeients of L-funtions from a family. Thus, there are versions whihinvolve only the real haraters (see [H-B4℄, as well as versions with Fourier oeients ofmodular forms (see [DI℄, [Iwa1℄, and [Iwa2℄) and many others. The large sieves reveal analmost orthogonality among the sets of oeients.For ertain appliations, it is desirable to have an asymptoti formula in plae of the inequalityabove. We have developed suh an asymptoti formula, for ertain sequenes, and even haveformulas for
S(~a,Q) :=
∑
q≤Q
q
φ(q)
∑∗
χ mod q
|L(1/2, χ)|2
∣∣
N∑
n=1
anχ(n)
∣∣22000 Mathematis Subjet Classiation.  11M06, 11M26.Key words and phrases.  Large sieve, Dirihlet L-funtions, Generalized Riemann Hypothesis, momentsof L-funtions.
38 Appliations of the asymptoti large sievefor L-funtions of degrees 1, 2, and 3 and for
S(~a,Q) :=
∑
q≤Q
q
φ(q)
∑∗
χ mod q
|L(1/2, χ)|2k
∣∣
N∑
n=1
anχ(n)
∣∣2provided that k times the degree of L is not too large.As a sample appliation, we an show that, assuming the generalized Riemann Hypothesis,there are L-funtions whih have zeros spaed loser than 0.366 times the average spaing.Another appliation is to ritial zeros of a family of twists of a xed L-funtion. Let L(s) bean L-funtion. We onsider the olletion
{L(s, χ)}where χ ranges over all primitive haraters χ modulo q with q ≤ Q. We onsider all of thezeros of all of these L-funtions up to a height logQ. If L is a degree one L-funtion, then atleast 55% of all of these zeros are on the 1/2-line. If L is degree 2, then at least 35% are onthe ritial line, and if L is degree 3 then at least one-half of one perent of the zeros are onthe ritial line.A third appliation is to moments of Dirihlet L-funtions. Here the best result is for thesixth moment of Dirihlet L-funtions. We prove a formula, whih ontains the full ninthdegree polynomial polynomial, and whih agrees perfetly with the onjetures motivated byrandom matrix theory. 2. The basi ideaThe essene of S(~a,Q) is ∆ whih is dened through
S(~a,Q) =
∑
m,n≤X
ambn∆(m,n);thus
∆(m,n) :=
∑
q
W (q/Q)
φ(q)
∑∗
χ mod q
χ(m)χ(n).Here are some basi lemmas to get us started analyzing this ∆symbol.Lemma 2.1.  If (mn, q) = 1, then
∑∗
χ mod q
χ(m)χ(n) =
∑
d|q
d|(m−n)
φ(d)µ(q/d).Applying Lemma 2.1 we nd that
∆(m,n) =
∑
(cd,mn)=1
d|m−n
W (cd/Q)µ(c)φ(d)
φ(cd)
.
Publiations mathématiques de Besançon - 2011
Brian Conrey 39Lemma 2.2.  We have
φ(d)
φ(cd)
=
1
φ(c)
∑
a|c
a|d
µ(a)
a
.Thus,
∆(m,n) =
∑
(acd,mn)=1
ad|(m−n)
W (a2cd/Q)µ(a)µ(ac)
aφ(ac)
.Now we separate the diagonal terms from the non-diagonal ones.Proposition 2.3.  We have
∆(m,m) = Ŵ (1)Q
φ(m)
m
∏
p
(
1− 1
p2
− 1
p3
)∏
p|m
(
1− 1
p2
− 1
p3
)−1
+Oǫ((Qm)
ǫ)Proof.  We have
∆(m,m) =
∑
(acd,m)=1
µ(a)µ(ac)
aφ(ac)
W
(
a2cd
Q
)
=
1
2πi
∫
(2)
QsŴ (s)ζ(s)
∏
p|m
(1− 1ps )
∑
(ac,m)=1
µ(a)µ(ac)
a1+2scsφ(ac)
ds.The sums over a and c are absolutely onvergent for σ > 0 and Ŵ (s) is of rapid deay in thevertial diretion. Let ǫ > 0. We shift the path of integration to the ǫ-line and pik up theresidue from the pole of ζ(s) at s = 1. Thus
∆(m,m) = Ŵ (1)Q
φ(m)
m
∑
(ac,m)=1
µ(a)µ(ac)
a3cφ(ac)
+O
(
(Qm)ǫ
)
.The sum over a and c in the main term is
=
∏
p∤m
(
1 +
1
p3(p − 1) −
1
p(p− 1)
)
=
∏
p∤m
(
1− 1
p2
− 1
p3
)
.Now we shall assume that m 6= n. We introdue a parameter C and split the sum over c in
∆ so that we have ∆(m,n) = L(m,n) + U(m,n) where
L(m,n) =
∑
(acd,mn)=1
ad|m−n,c≤C
W (a2cd/Q)µ(a)µ(ac)
aφ(ac)and
U(m,n) =
∑
(acd,mn)=1
ad|m−n,c>C
W (a2cd/Q)µ(a)µ(ac)
aφ(ac)
.Publiations mathématiques de Besançon - 2011
40 Appliations of the asymptoti large sieveThe term U(m,n) is relatively easy to analyze. Basially we detet the ongruene m ≡
n mod ad by going to haraters modulo ad. Sine c is large and acd ≈ Q it must be the asethat ad is small and so it is not too expensive to sum over all of the haraters modulo ad.For the term L(m,n) with smaller c and larger ad we write |m − n| = ade where e is alledthe omplementary modulus and we then, after elimination of d from the rest of the sum,we have m ≡ n mod ae and detet this ongruene by haraters modulo ae. The diultyin doing this is largely tehnial; to eliminate the variable d everywhere by replaing it by
|m− n|/ae is ompliated! In arrying out this proedure for spei sequenes an one oftenenounters main terms arising from the prinipal haraters modulo ad in U and from theprinipal haraters modulo ae in L. The two largest of these main terms anel; the real o-diagonal ontribution omes from a seondary main term in the prinipal haraters modulo
ae of L. This term ombines with the diagonal to form the nal main term.In the next two setions we give further examples of preise theorems we an prove.3. The analogue of the Balasubramanian, Conrey, Heath-Brown onjetureTheorem 3.1.  Suppose that W is smooth and supported on [1, 2]. Then
∑
q
W
( q
Q
) ∑♭
χ mod q
|L(1/2, χ)|2χ(h)χ(k) = Res
s=0

W̃ (s+ 2)π−s
Γ
(
s+1/2
2
)2
Γ
(
1
4
)2 Q2+s
ζ(1 + 2s)
s
×φ(hk)
hk
(h, k)1+2s
(hk)1/2+s
∏
p∤hk
(
1− 1
p2+2s
− 2
p2
+
2
p3+2s
+
1
p4
− 1
p5+2s
)
+ Eh,kwhere ∑
h,k≤N
λhλkEh,k ≪ (Q2−ǫ + (QN)1+ǫ)
∑
h≤N
|(λ ∗ λ)h|2.Here λ ∗ λ is the Dirihlet onvolution of the sequene λ with itself. Note that for a sequene
λh ≪ hǫ−1/2 the main term above is likely to be ≫ Q2 so that we have an asymptoti formulaprovided that N ≪ Q1−ǫ.The above is the analogue of a onjeture of Balasubramanian, Conrey, and Heath-Brown[BCH-B℄ who proved that
∫ T
0
|ζ(1/2 + it)|2
∣∣∣∣∣
N∑
n=1
λnn
−it
∣∣∣∣∣
2
dt = T
∑
h,k≤N
λhλk
(
(h, k)√
hk
(
log
T (h, k)2
2πhk
+ 2γ − 1
)
+ E(h, k)
)where ∑
h,k≤N
λhλkEh,k ≪ (T 1−ǫ + T 1/2N)
∑
n≤N
|λn|2.This formula gives an asymptoti formula provided that N ≪ T 1/2−ǫ. The authors onjeturethat the asymptoti formula atually holds provided that N ≪ T 1−ǫ (a onjeture whih isequivalent to the Lindelöf hypothesis).Publiations mathématiques de Besançon - 2011
Brian Conrey 414. The sixth momentUnderstanding moments of families of L-funtions has long been an important subjet withmany number theoreti appliations. It is only in the last ten years that an understanding ofthe ner struture of moments has begun to emerge. The new vision began with the work ofKeating and Snaith, who realized that the distribution of the values of an L-funtion, or familyof L-funtions, an be modeled by harateristi polynomials from lassial ompat groups,and to Katz and Sarnak for their realization that families of L-funtions have symmetry typesassoiated with them that reveal whih of the lassial groups to use to model the family. See[KaSa℄, [KS1℄, [KS2℄, and [CF℄.Prior to these works, Conrey and Ghosh predited, on number theoreti grounds, that
∫ T
0
|ζ(1/2 + it)|6 dt ∼ 42
∏
p
(
1− 1
p
)4(
1 +
4
p
+
1
p2
)
T
log9 T
9!
.The onjeture of Keating and Snaith agrees with this. Numerially, however, this onjetureis untestable. For example,
∫ 2350000
0
|ζ(1/2 + it)|6 dt = 3317496016044.9 = 3.3 × 1012whereas
42
∏
p
(
1− 1
p
)4(
1 +
4
p
+
1
p2
)
× 2350000 × (log 2350000)
9
9!
= 4.22 × 1011is nowhere near the predition. This situation has been retied by the onjetures of[CFKRS℄ whih assert, for example, that for any ǫ > 0,
∫ T
0
|ζ(1/2 + it)|6 dt =
∫ T
0
P3
(
log
t
2π
)
dt+O(T 1/2+ǫ)where P3 is a polynomial of degree 9 whose exat oeients are speied as ompliatedinnite produts and series over primes, but whose approximate oeients are
P3(x) = 0.000005708x
9 + 0.0004050x8 + 0.01107x7 + 0.1484x6
+1.0459x5 + 3.9843x4 + 8.6073x3 + 10.2743x2 + 6.5939x + 0.9165.For this polynomial, we have
∫ 2350000
0
P3
(
log
t
2π
)
dt = 3317437762612.4whih agrees well with the numeris.A proof of an asymptoti formula for the sixth moment of ζ(s) on the itial line is ompletelyout of reah of today's tehnology. However, we have proved an analogous formula for Dirihlet
L-funtions suitably averaged. Our formula agrees exatly with the onjeture of [CFKRS℄and so provides, we hope, a new glimpse into the mehanis of moments.Publiations mathématiques de Besançon - 2011
42 Appliations of the asymptoti large sieveThe preise statement of our theorem on the sixth moment of Dirihlet L-funtions is a littleompliated to state and requires some set up. We invite the reader to look at the end of thepaper for a version of the theorem whih shows the rst main term in the asymptoti formula.Let χ mod q be an even, primitive Dirihlet harater and let
L(s, χ) =
∞∑
n=1
χ(n)
nsbe its assoiated L-funtion. Suh an L-funtion has an Euler produt: L(s, χ) = ∏p(1 −
χ(p)/ps)−1 and a funtional equation
Λ(s, χ) := (q/π)(s−1/2)/2Γ(s/2)L(s, χ) = ǫχΛ(1− s, χ)where ǫχ is a omplex number of absolute value 1. We prove an asymptoti formula, with apower savings, for a suitable average of the sixth power of the absolute value of these primitiveDirihlet L-funtions near the ritial point 1/2.An upper bound for suh an average follows from the large sieve inequality. Huxley used thisto prove that
∑
q≤Q
∑∗
χ mod q
|L(1/2, χ)|6 ≪ Q2 log9 Qand
∑
q≤Q
∑∗
χ mod q
|L(1/2, χ)|8 ≪ Q2 log16 Q.Here is a statement of those onjetures and the theorem of this paper. Let A and B be setsof omplex numbers with equal ardinality |A| = |B| = K. Suppose that |ℜα|, |ℜβ| ≤ 1/4 for
α ∈ A, β ∈ B. These are the shifts. Let
ΛA,B(χ) :=
∏
α∈A
Λ(1/2 + α,χ)
∏
β∈B
Λ(1/2 + β, χ)
=
( q
π
)δA,BGA,BLA,B(χ)where
LA,B(χ) :=
∏
α∈A
L(1/2 + α,χ)
∏
β∈B
L(1/2 + β, χ),
GA,B :=
∏
α∈A
Γ
(
1/2 + α
2
) ∏
β∈B
Γ
(
1/2 + β
2
)and
δA,B =
1
2

∑
α∈A
α+
∑
β∈B
β

 .Publiations mathématiques de Besançon - 2011
Brian Conrey 43Further, let
Z(A,B) :=
∏
α∈A
β∈B
ζ(1 + α+ β)and
A(A,B) :=
∏
p
Bp(A,B)Zp(A,B)−1with
Zp(A,B) :=
∏
α∈A
β∈B
ζp(1 + α+ β)and
ζp(x) =
(
1− 1
px
)−1
;also
Bp(A,B) :=
∫ 1
0
∏
α∈A
zp,θ(1/2 + α)
∏
β∈B
zp,−θ(1/2 + β) dθwith zp,θ(x) = 1/(1−e(θ)/px). The onditions on the real parts of elements of A and B ensurethat the Euler produt for A onverges absolutely. Let Bq = ∏p|q Bp. Let
QA,B(q) :=
∑
S⊂A
T⊂B
|S|=|T |
Q(S ∪ (−T ), T ∪ (−S); q)where S denotes the omplement of S in A, and by the set −S we mean {−s : s ∈ S} and,for any sets X and Y ,
Q(X,Y ; q) =
( q
π
)δX,Y GX,Y
AZ
Bq
(X,Y ).For example, if A = {α1, α2, α3} and B = {β1, β2, β3}, then
QA,B(q) = Q({α1, α2, α3}, {β1, β2, β3}, q) +Q({−β1, α2, α3}, {−α1, β2, β3}, q)
+ · · ·+Q({−β1,−β2,−β3}, {−α1,−α2,−α3}, q),is a sum of 8 terms; the rst summand orresponds to S = T = φ so that X = A and Y = B;the seond summand orresponds to S = {α1} and T = {β1}, and so on. In general, QA,B(q)will have (2KK ) summands.Now we state the onjeture.Conjeture 4.1.  [CFKRS℄ Assuming that the shifts α ∈ A, β ∈ B satisfy |ℜα|, |ℜβ| ≤
1/4, and ℑα,ℑβ ≪ q1−ǫ, we onjeture that
∑♭
χ mod q
ΛA,B(χ) =
∑♭
χ mod q
QA,B(q)(1 +O(q−1/2+ǫ))Publiations mathématiques de Besançon - 2011
44 Appliations of the asymptoti large sievewhere ∑♭ denotes a sum over even primitive haraters.When |A| = |B| = 3 and all of the shifts are 0, the onjeture implies that, as q → ∞ with qnot ongruent to 2 modulo 4, we haveConjeture 4.2.  [CFKRS℄
1
φ♭(q)
∑♭
χ mod q
|L(1
2
, χ)|6 ∼ 42a3
∏
p|q
(
1− 1p
)5
(
1 + 4p +
1
p2
) log
9 q
9!
.where φ♭(q) is the number of even primitive haraters modulo q and
a3 =
∏
p
(
1− 1
p
)4(
1 +
4
p
+
1
p2
)
.Note that a3 is the onstant that appears in the onjeture for the sixth moment of ζ. The`42' here played an important role in the disovery by Keating and Snaith that momentsof L-funtions ould be modeled by moments of harateristi polynomials, see Beineke andHughes [BeHu℄ for an aount of this story.For a set A it is onvenient to let At be the set of translates
At = {α+ t : α ∈ A}.Note that (As1)s2 = As1+s2 .Theorem 4.3.  Suppose that |A| = |B| = 3 and that α, β ≪ 1/ logQ, for α ∈ A, β ∈ B.Suppose that Ψ is smooth on [1, 2] and Φ(t) is an entire funtion of t whih deays rapidly as
|t| → ∞ in any xed horizontal strip. Then
∑
q
Ψ
(
q
Q
)∫ ∞
−∞
Φ(t)
∑
χ
♭ΛAit,B−it(χ) dt
=
∑
q
Ψ
(
q
Q
)∫ ∞
−∞
Φ(t)φ♭(q)QAit,B−it(q) dt+O(Q7/4+ǫ).We ould equally well prove a theorem for odd primitive haraters. The answer would besimilar with just the Gamma-fators hanged slightly to reet the dierene in the funtionalequation for odd primitive Dirihlet L-funtions. When the shifts are all 0, this dierenedisappears, in the leading order main term.Publiations mathématiques de Besançon - 2011
Brian Conrey 45Corollary 4.4.  We have
∑
q
∑∗
χ mod q
Ψ(q/Q)
∫ ∞
−∞
Φ(t)|L(1/2 + it, χ)|6 dt
∼ 42a3
∑
q
Ψ
(
q
Q
)∏
p|q
(
1− 1p
)5
(
1 + 4p +
1
p2
)φ♭(q) log
9 q
9!
∫ ∞
−∞
Φ(t)|Γ((1/2 + it)/2)|6 dt
∼ 42a3(L)Q2
log9 Q
9!
∫ ∞
0
Ψ(x)x dx
∫ ∞
−∞
Φ(t)|Γ((1/2 + iy)/2)|6 dtwhere
a3(L) =
∏
p
(
1− 1
p
)5(
1 +
5
p
− 5
p2
+
14
p3
− 15
p4
+
5
p5
+
4
p6
− 4
p7
+
1
p8
)
.Thus, our theorem asserts that Conjetures 4.1 and 4.2 are true (but with a weaker errorterm) on average over q and a mild average over y.5. ConlusionThe asymptoti large sieve is potentially a very useful tool with many appliations. In itsinitial inarnations, the proofs are fairly diult, espeially when omplex main terms haveto be unearthed. The authors are seeking good ways to present these new ideas, ways thatwill simplify some of these ompliations.Referenes[BCH-B℄ Balasubramanian, R.; Conrey, J. B.; Heath-Brown, D. R. Asymptoti mean square of theprodut of the Riemann zeta-funtion and a Dirihlet polynomial. J. Reine Angew. Math. 357 (1985),161181.[BeHu℄ Beineke, J. and Hughes, C. Great moments of the Riemann zeta funtion, in Bisuits of numbertheory. Edited by Arthur T. Benjamin and Ezra Brown. The Doliani Mathematial Expositions, 34.Mathematial Assoiation of Ameria, Washington, DC, 2009. xiv+311 pp.[BD℄ Bombieri, E.; Davenport, H. Some inequalities involving trigonometrial polynomials. Ann.Suola Norm. Sup. Pisa (3) 23 (1969), 223241.[Con℄ Conrey, J. B. The mean-square of Dirihlet L-funtions, arXiv:0708.2699.[CF℄ Conrey, J. B.; Farmer, D. W. Mean values of L-funtions and symmetry. Internat. Math. Res.Noties 2000, no. 17, 883908.[CFKRS℄ Conrey, J. B.; Farmer, D. W.; Keating, J. P.; Rubinstein, M. O.; Snaith, N. C. Integralmoments of L-funtions. Pro. London Math. So. (3) 91 (2005), no. 1, 33104.[CG℄ Conrey, J. B.; Ghosh, A. A onjeture for the sixth power moment of the Riemann zeta-funtion.Internat. Math. Res. Noties 1998, no. 15, 775780.[CGo℄ Conrey, J. B. ; Gonek, S. M. High moments of the Riemann zeta-funtion. Duke Math. J. 107(2001), no. 3, 577604.[CIS℄ Conrey, J. B., Iwanie, H., Soundararajan, K. The asymptoti large sieve. In preparation.Publiations mathématiques de Besançon - 2011
46 Appliations of the asymptoti large sieve[DI℄ Deshouillers, J.-M.; Iwanie, H. Kloosterman sums and Fourier oeients of usp forms. Invent.Math. 70 (1982/83), no. 2, 219288.[H-B1℄ Heath-Brown, D. R. The fourth power moment of the Riemann zeta funtion. Pro. LondonMath. So. (3) 38 (1979), no. 3, 385422.[H-B2℄ Heath-Brown, D. R. An asymptoti series for the mean value of Dirihlet L-funtions. Com-ment. Math. Helv. 56 (1981), no. 1, 148161.[H-B3℄ Heath-Brown, D. R. The fourth power mean of Dirihlet's L-funtions. Analysis 1 (1981), no.1, 2532.[H-B4℄ Heath-Brown, D. R. A mean value estimate for real harater sums. Ata Arith. 72 (1995),no. 3, 235275.[Hux℄ Huxley, M. N. The large sieve inequality for algebrai number elds. II. Means of moments ofHeke zeta-funtions. Pro. London Math. So. (3) 21 (1970), 108-128.[Iwa1℄ Iwanie, Henryk Introdution to the spetral theory of automorphi forms. Bibliotea de laRevista Matematia Iberoameriana. Revista Matematia Iberoameriana, Madrid, 1995.[Iwa2℄ Iwanie, Henryk Topis in lassial automorphi forms. Graduate Studies in Mathematis, 17.Amerian Mathematial Soiety, Providene, RI, 1997.[KaSa℄ Katz, Niholas M.; Sarnak, Peter Random matries, Frobenius eigenvalues, and monodromy.Amerian Mathematial Soiety Colloquium Publiations, 45. Amerian Mathematial Soiety, Prov-idene, RI, 1999.[KS1℄ Keating, J. P.; Snaith, N. C. Random matrix theory and ζ(1/2 + it). Comm. Math. Phys. 214(2000), no. 1, 5789.[KS2℄ Keating, J. P.; Snaith, N. C. Random matrix theory and L-funtions at s = 1/2. Comm. Math.Phys. 214 (2000), no. 1, 91110.[Sou℄ Soundararajan, K. The fourth moment of Dirihlet L-funtions, Analyti number theory, 239-246, Clay Math. Pro., 7, Amer. Math. So., Providene, RI, 2007; arxiv math.NT/0507150.[Sou1℄ Soundararajan, K. Moments of the Riemann zeta-funtion, Ann. of Math. (2) 170 (2009), no.2, 981-993.; arxiv math.NT/0612106.[You℄ Young, Matthew. The fourth moment of Dirihlet L-funtions, to appear, Ann. of Math.;arxivmath.NT/0610335.Lundi 31 janvier 2011Brian Conrey, Amerian Institute of Mathematis, 360 Portage Ave Palo Alto, CA 94306-2244Department of Mathematis University Walk, Clifton, Bristol BS8 1TW, U.K.E-mail : onreyaimath.org
Publiations mathématiques de Besançon - 2011


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