Applications of the asymptotic large sieve

by

BrianConrey

Abstrat. InthisnotewedesribesomeresultsofConrey,Iwanie,andSoundararajan. The

detailswillappearelsewhere.

Résumé. Dansettenote,nousprésentonsdesrésultats deConrey,IwanieetSoundara-

rajan.Lesdétailsserontpubliésultérieurement.

1. Introdution

Thelarge sieve inequality, initsmultipliative form[H-B1℄,asserts thatfor any sequeneof

omplex numbers

a _n

^,

S (~a, Q) := X

q ≤ Q

q φ(q)

X ∗

χ mod q

X ^N

n=1

a _n χ(n) ² ≤ (Q ² + N ) X N

n=1

| a _n | ² .

Many appliations have followed from this basi tool, espeially, in this form, appliations

having to do with primes in arithmeti progressions, or to do with statistial averages of

Dirihlet L-funtions. Generalizationsofthis basitoolhave been givenwheretheharaters

χ

^{arereplaedbytheoeientsofL-funtionsfromafamily. Thus,thereareversionswhih}

involve only the real haraters (see [H-B4℄, as well as versions with Fourier oeients of

modular forms (see [DI℄, [Iwa1 ℄, and [Iwa2℄) and many others. The large sieves reveal an

almost orthogonality amongthesets of oeients.

Forertainappliations,itisdesirabletohaveanasymptotiformulainplaeoftheinequality

above. Wehave developedsuhan asymptoti formula, for ertainsequenes,and even have

formulas for

S (~a, Q) := X

q ≤ Q

q φ(q)

X ∗

χ mod q

| L(1/2, χ) | ² X ^N

n=1

a _n χ(n) ²

2000 MathematisSubjetClassiation. 11M06,11M26.

Keywordsandphrases. Largesieve,DirihletL-funtions,GeneralizedRiemannHypothesis,moments

ofL-funtions.

forL-funtions of degrees 1,2,and3 and for

S (~a, Q) := X

q≤Q

q φ(q)

X ∗

χ mod q

| L(1/2, χ) | ^2k X N

n=1

a _n χ(n) ²

provided that

k

^{times the degreeof Lis not toolarge.}

Asa sample appliation, we an show that, assuming the generalized Riemann Hypothesis,

thereare

L

^{-funtions whihhave zeros spaedloser than0.366 times theaverage spaing.}

Anotherappliationisto ritialzerosofafamilyof twists ofaxedL-funtion. Let

L(s)

^be

anL-funtion. We onsiderthe olletion

{ L(s, χ) }

where

χ

^{ranges over all primitive haraters}

χ

^modulo

q

^with

q ≤ Q

^{. We onsider all of the}

zerosofall ofthese

L

^{-funtionsup toa height}

log Q

^{. If}

L

^{isadegree oneL-funtion, thenat}

least55%of all of these zeros areon the 1/2-line. If

L

^{isdegree 2, thenat least35% areon}

theritialline,and if

L

^{is degree3 thenat least one-half ofone perent of thezeros areon}

theritial line.

A third appliation is to moments of Dirihlet L-funtions. Here the best result is for the

sixth moment of Dirihlet L-funtions. We prove a formula, whih ontains the full ninth

degreepolynomial polynomial, and whihagrees perfetly withtheonjeturesmotivatedby

randommatrix theory.

2. The basi idea

Theesseneof

S(~a, Q)

^is

∆

^{whih isdened through}

S(~a, Q) = X

m,n ≤ X

a _m b _n ∆(m, n);

thus

∆(m, n) := X

q

W (q/Q) φ(q)

X _∗

χ mod q

χ(m)χ(n).

Herearesome basi lemmasto getus started analyzingthis

∆

^symbol.

Lemma 2.1. If

(mn, q) = 1

^,then

X _∗

χ mod q

χ(m)χ(n) = X

d | q d | (m − n)

φ(d)µ(q/d).

ApplyingLemma 2.1we nd that

∆(m, n) = X

(cd,mn)=1 d | m − n

W (cd/Q)µ(c)φ(d)

φ(cd) .

Lemma 2.2. We have

φ(d) φ(cd) = 1

φ(c) X

a | c a|d

µ(a) a .

Thus,

∆(m, n) = X

(acd,mn)=1 ad | (m − n)

W (a ² cd/Q)µ(a)µ(ac) aφ(ac) .

Nowwe separatethe diagonal terms fromthenon-diagonal ones.

Proposition 2.3. We have

∆(m, m) = ˆ W (1)Q φ(m) m

Y

p

1 − 1

p ² − 1 p ³

Y

p | m

1 − 1

p ² − 1 p ³

₋₁

+ O _ǫ ((Qm) ^ǫ )

Proof. We have

∆(m, m) = X

(acd,m)=1

µ(a)µ(ac) aφ(ac) W

a ² cd Q

= 1 2πi

Z

(2)

Q ^s W ˆ (s)ζ(s) Y

p | m

(1 − _p ^{1 s} ) X

(ac,m)=1

µ(a)µ(ac) a ^1+2s c ^s φ(ac) ds.

Thesums over

a

^and

c

^{areabsolutelyonvergent for}

σ > 0

^and

W ˆ (s)

^{isofrapid deayinthe}

vertial diretion. Let

ǫ > 0

^{. We shift the path of} integration to the

ǫ

^{-line and pik up the}

residuefrom the poleof

ζ(s)

^at

s = 1

^{. Thus}

∆(m, m) = ˆ W (1)Q φ(m) m

X

(ac,m)=1

µ(a)µ(ac)

a ³ cφ(ac) + O (Qm) ^ǫ .

Thesum over

a

^and

c

^{inthe main termis}

= Y

p ∤ m

1 + 1

p ³ (p − 1) − 1 p(p − 1)

= Y

p ∤ m

1 − 1

p ² − 1 p ³

.

Nowwe shallassume that

m 6 = n

^{. We introdue a parameter}

C

^{and split thesum over}

c

ⁱⁿ

∆

^{sothatwe have}

∆(m, n) = L(m, n) + U (m, n)

^where

L(m, n) = X

(acd,mn)=1 ad | m − n,c ≤ C

W (a ² cd/Q)µ(a)µ(ac) aφ(ac)

and

U(m, n) = X

(acd,mn)=1 ad|m−n,c>C

W (a ² cd/Q)µ(a)µ(ac)

aφ(ac) .

The term

U (m, n)

^{is relatively easy to analyze. Basially we detet the ongruene}

m ≡ n mod ad

^{bygoingtoharatersmodulo}

ad

^{. Sine}

c

^islargeand

acd ≈ Q

^{itmustbethease}

that

ad

^{is smalland soitis nottoo expensiveto sum overall of theharaters modulo}

ad

^.

For theterm

L(m, n)

^withsmaller

c

^{and larger}

ad

^{we write}

| m − n | = ade

^where

e

^{is alled}

the omplementary modulus and we then, after elimination of

d

^{from the rest of the sum,}

we have

m ≡ n mod ae

^{and detet this ongruene by haraters modulo}

ae

^{. The diulty}

in doing this is largely tehnial; to eliminate the variable

d

^{everywhere by replaing it by}

| m − n | /ae

^{is ompliated! In arryingout this proedurefor spei sequenes}

a _n

^{one often}

enounters main terms arising from the prinipal haraters modulo

ad

ⁱⁿ

U

^{and from the}

prinipalharaters modulo

ae

ⁱⁿ

L

^{. Thetwolargestofthese maintermsanel;therealo-}

diagonal ontribution omes from a seondarymain term intheprinipal haraters modulo

ae

^of

L

^{. Thisterm ombines withthe diagonal to formthenalmain term.}

Inthenexttwo setions we givefurther examplesof preisetheorems we an prove.

3. The analogue of the Balasubramanian, Conrey, Heath-Brown onjeture

Theorem 3.1. Suppose that

W

^{issmooth and supported on}

[1, 2]

^{. Then}

X

q

W q Q

X ^♭

χ mod q

| L(1/2, χ) | ² χ(h)χ(k) = Res

s=0



  W ˜ (s + 2)π ^−s

Γ _s+1/2

2

2

Γ ¹ ₄ 2 Q ^2+s ζ (1 + 2s) s

× φ(hk) hk

(h, k) ^1+2s (hk) ^1/2+s

Y

p ∤ hk

1 − 1 p ^2+2s − 2

p ² + 2 p ^3+2s + 1

p ⁴ − 1 p ^5+2s



 + E h,k

where

X

h,k ≤ N

λ _h λ _k E h,k ≪ (Q ^{2 − ǫ} + (QN ) ^1+ǫ ) X

h ≤ N

| (λ ∗ λ) _h | ² .

Here

λ ∗ λ

^{istheDirihletonvolutionofthe sequene}

λ

^{withitself. Notethatfora sequene}

λ _h ≪ h ^{ǫ − 1/2}

^{themaintermaboveislikelytobe}

≫ Q ²

^{sothatwehave anasymptotiformula}

provided that

N ≪ Q ^1−ǫ

^.

The above is the analogue of a onjeture of Balasubramanian, Conrey, and Heath-Brown

[BCH-B℄who proved that

Z T

0 | ζ(1/2 + it) | ²

X N

n=1

λ n n ^{− it}

2

dt = T X

h,k ≤ N

λ _h λ _k

(h, k)

√ hk

log T (h, k) ²

2πhk + 2γ − 1

+ E (h, k)

where

X

h,k≤N

λ _h λ _k E h,k ≪ (T ^{1 − ǫ} + T ^1/2 N ) X

n≤N

| λ n | ² .

Thisformulagivesanasymptotiformulaprovided that

N ≪ T ^1/2−ǫ

^{. Theauthorsonjeture}

thatthe asymptoti formula atually holds provided that

N ≪ T ^{1 − ǫ}

^{(a onjeturewhih is}

equivalent to theLindelöf hypothesis).

4. The sixth moment

Understanding moments of families of

L

^{-funtions has long been an important subjetwith}

manynumbertheoretiappliations. It isonlyinthelast tenyearsthatanunderstanding of

thenerstruture ofmomentshasbegunto emerge. Thenewvision began withthework of

KeatingandSnaith,whorealizedthatthedistributionofthevaluesofanL-funtion,orfamily

of L-funtions,an be modeled by harateristi polynomials fromlassial ompat groups,

andtoKatzandSarnakfortheirrealizationthatfamiliesofL-funtionshavesymmetrytypes

assoiatedwiththemthatrevealwhihofthelassialgroupstousetomodelthefamily. See

[KaSa ℄,[KS1℄, [KS2℄,and [CF℄.

Prior to theseworks, Conrey andGhosh predited, onnumber theoreti grounds, that

Z _T

0 | ζ (1/2 + it) | ⁶ dt ∼ 42 Y

p

1 − 1

p 4

1 + 4 p + 1

p ²

T log ⁹ T 9! .

TheonjetureofKeatingandSnaithagreeswiththis. Numerially,however,thisonjeture

isuntestable. For example,

Z 2350000

0 | ζ(1/2 + it) | ⁶ dt = 3317496016044.9 = 3.3 × 10 ¹²

whereas

42 Y

p

1 − 1

p 4

1 + 4 p + 1

p ²

× 2350000 × (log 2350000) ⁹

9! = 4.22 × 10 ¹¹

is nowhere near the predition. This situation has been retied by the onjetures of

[CFKRS℄ whihassert, for example,thatfor any

ǫ > 0

^,

Z T

0 | ζ(1/2 + it) | ⁶ dt = Z T

0

P ₃

log t 2π

dt + O(T ^1/2+ǫ )

where

P ₃

^{is a polynomial of degree 9 whose exat oeients are speied as ompliated}

inniteproduts andseries over primes, butwhose approximate oeients are

P 3 (x) = 0.000005708 x ⁹ + 0.0004050 x ⁸ + 0.01107 x ⁷ + 0.1484 x ⁶

+1.0459 x ⁵ + 3.9843 x ⁴ + 8.6073 x ³ + 10.2743 x ² + 6.5939 x + 0.9165.

For this polynomial, we have

Z 2350000

0

P 3

log t

2π

dt = 3317437762612.4

whih agreeswell withthenumeris.

Aproofofanasymptotiformulaforthesixthmomentof

ζ(s)

^{ontheitiallineisompletely}

outofreahoftoday'stehnology. However,wehaveprovedananalogousformulaforDirihlet

L

^{-funtions suitablyaveraged. Ourformulaagrees exatlywiththe onjetureof [CFKRS℄}

and soprovides,wehope,a newglimpse into themehanis ofmoments.

Thepreisestatement of ourtheorem onthe sixthmoment ofDirihlet L-funtionsis alittle

ompliatedtostate andrequiressomesetup. We invitethereader tolookat theend ofthe

paperforaversionofthetheoremwhihshowstherstmaintermintheasymptotiformula.

Let

χ mod q

^{be aneven,primitive Dirihlet harater andlet}

L(s, χ) =

X ∞

n=1

χ(n) n ^s

be its assoiated

L

^{-funtion. Suh an}

L

^{-funtion has an Euler produt:}

L(s, χ) = Q

p (1 − χ(p)/p ^s ) ^{− 1}

^{and a funtionalequation}

Λ(s, χ) := (q/π) ^{(s − 1/2)/2} Γ(s/2)L(s, χ) = ǫ _χ Λ(1 − s, χ)

where

ǫ _χ

^{is a omplex numberof absolute value 1. We prove an asymptoti formula, witha}

powersavings,forasuitableaverageofthesixthpoweroftheabsolutevalueoftheseprimitive

Dirihlet

L

^{-funtionsnearthe ritial point 1/2.}

Anupperboundforsuhanaveragefollowsfromthelargesieveinequality. Huxleyusedthis

toprove that

X

q≤Q

X ∗

χ mod q

| L(1/2, χ) | ⁶ ≪ Q ² log ⁹ Q

and

X

q≤Q

X ∗

χ mod q

| L(1/2, χ) | ⁸ ≪ Q ² log ¹⁶ Q.

Hereisa statement of thoseonjetures andthe theoremof thispaper. Let

A

^and

B

^besets

ofomplexnumberswithequalardinality

| A | = | B | = K

^{. Supposethat}

|ℜ α | , |ℜ β | ≤ 1/4

^for

α ∈ A

^,

β ∈ B

^{. Thesearethe shifts. Let}

Λ _A,B (χ) := Y

α ∈ A

Λ(1/2 + α, χ) Y

β ∈ B

Λ(1/2 + β, χ)

= q π

δ A,B G _A,B L A,B (χ)

where

L A,B (χ) := Y

α ∈ A

L(1/2 + α, χ) Y

β ∈ B

L(1/2 + β, χ),

G _A,B := Y

α ∈ A

Γ

1/2 + α 2

Y

β ∈ B

Γ

1/2 + β 2

and

δ _A,B = 1 2



 X

α∈A

α + X

β ∈ B

β



 .

Further, let

Z (A, B) := Y

α∈A β∈B

ζ(1 + α + β)

and

A (A, B) := Y

p

B p (A, B) Z p (A, B) ^{− 1}

with

Z p (A, B) := Y

α ∈ A β ∈ B

ζ p (1 + α + β)

and

ζ _p (x) =

1 − 1 p ^x

₋ 1

;

also

B p (A, B) :=

Z ₁

0

Y

α ∈ A

z _p,θ (1/2 + α) Y

β ∈ B

z _{p, − θ} (1/2 + β ) dθ

with

z _p,θ (x) = 1/(1 − e(θ)/p ^x ).

^{Theonditionsontherealpartsofelementsof}

A

^and

B

^ensure

thattheEuler produtfor

A

^onverges absolutely. Let

B q = Q

p|q B p

^{. Let}

Q A,B (q) := X

S ⊂ A T ⊂ B

| S | = | T |

Q (S ∪ ( − T ), T ∪ ( − S); q)

where

S

^{denotes the omplement of}

S

ⁱⁿ

A

^{, and bythe set}

− S

^{we mean}

{− s : s ∈ S }

^and,

for any sets

X

^and

Y

^,

Q (X, Y ; q) = q π

δ X,Y

G _X,Y AZ B q

(X, Y ).

For example,if

A = { α ₁ , α ₂ , α ₃ }

^and

B = { β ₁ , β ₂ , β ₃ }

^,then

Q A,B (q) = Q ( { α ₁ , α ₂ , α ₃ } , { β ₁ , β ₂ , β ₃ } , q) + Q ( {− β ₁ , α ₂ , α ₃ } , {− α ₁ , β ₂ , β ₃ } , q) + · · · + Q ( {− β ₁ , − β ₂ , − β ₃ } , {− α ₁ , − α ₂ , − α ₃ } , q),

isasumof8terms;the rstsummandorrespondsto

S = T = φ

^sothat

X = A

^and

Y = B

^;

theseondsummandorrespondsto

S = { α 1 }

^and

T = { β 1 }

^{,andsoon. Ingeneral,}

Q A,B (q)

will have

2K K

summands.

Nowwe state the onjeture.

Conjeture 4.1. [CFKRS℄Assumingthatthe shifts

α ∈ A

^,

β ∈ B

^satisfy

|ℜ α | , |ℜ β | ≤ 1/4,

^and

ℑ α, ℑ β ≪ q ^{1 − ǫ} ,

^{we onjeture that}

X ♭

χ mod q

Λ _A,B (χ) = X ♭ χ mod q

Q A,B (q)(1 + O(q ^−1/2+ǫ ))

where

X ♭

denotes a sumover even primitive haraters.

When

| A | = | B | = 3

^{and allof theshiftsare0,theonjetureimplies that,as}

q → ∞

^with

q

notongruent to 2modulo4,we have

Conjeture 4.2. [CFKRS℄

1 φ ^♭ (q)

X ♭

χ mod q

| L( 1

2 , χ) | ⁶ ∼ 42a ₃ Y

p | q

1 − ¹ _p 5

1 + ⁴ _p + _p ¹ 2

log ⁹ q 9! .

where

φ ^♭ (q)

^{is the number of even primitiveharaters modulo}

q

^and

a ₃ = Y

p

1 − 1

p 4

1 + 4 p + 1

p ²

.

Notethat

a ₃

^{is theonstant that appears inthe onjeture for the sixthmoment of}

ζ

^{. The}

`42' here played an important role in the disovery by Keating and Snaith that moments

of

L

^{-funtions ouldbe modeled bymoments of} harateristi polynomials, see Beineke and Hughes[BeHu℄for an aount ofthis story.

For aset

A

^{it isonvenient to let}

A _t

^{be the setof translates}

A _t = { α + t : α ∈ A } .

Notethat

(A s 1 ) s 2 = A s 1 +s 2

^.

Theorem 4.3. Suppose that

| A | = | B | = 3

^{and that}

α, β ≪ 1/ log Q,

^for

α ∈ A, β ∈ B

^.

Suppose that

Ψ

^{is smooth on}

[1, 2]

^and

Φ(t)

^{is an entire funtionof}

t

^{whihdeays rapidly as}

| t | → ∞

^{in any xed horizontal strip. Then}

X

q

Ψ q

Q Z _∞

−∞

Φ(t) X

χ

♭ Λ _{A it ,B − it} (χ) dt

= X

q

Ψ q

Q Z _∞

−∞

Φ(t)φ ^♭ (q) Q A it ,B _−it (q) dt + O(Q ^7/4+ǫ ).

We ould equally well prove a theorem for odd primitive haraters. The answer would be

similarwithjusttheGamma-fatorshangedslightlytoreetthediereneinthefuntional

equation for odd primitive Dirihlet

L

^{-funtions. When the shifts are all 0, this dierene}

disappears,intheleading order mainterm.

Corollary 4.4. We have

X

q

X _∗

χ mod q

Ψ(q/Q) Z _∞

−∞

Φ(t) | L(1/2 + it, χ) | ⁶ dt

∼ 42a ₃ X

q

Ψ q

Q Y

p | q

1 − ¹ _p 5

1 + ⁴ _p + _p ¹ ₂ φ ^♭ (q) log ⁹ q 9!

Z _∞

−∞

Φ(t) | Γ((1/2 + it)/2) | ⁶ dt

∼ 42a 3 ( L )Q ² log ⁹ Q 9!

Z _∞

0

Ψ(x)x dx Z _∞

−∞

Φ(t) | Γ((1/2 + iy)/2) | ⁶ dt

where

a ₃ ( L ) = Y

p

1 − 1 p

5 1 + 5

p − 5 p ² + 14

p ³ − 15 p ⁴ + 5

p ⁵ + 4 p ⁶ − 4

p ⁷ + 1 p ⁸

.

Thus, our theorem asserts that Conjetures 4.1 and 4.2 are true (but with a weaker error

term)on averageover

q

^{and amildaverage over}

y

^.

5. Conlusion

The asymptoti large sieve is potentially a very useful tool with many appliations. In its

initial inarnations, the proofsare fairly diult, espeially when omplex main terms have

to be unearthed. The authors areseeking good ways to present these new ideas, ways that

will simplifysome of theseompliations.

Referenes

[BCH-B℄ Balasubramanian,R.; Conrey,J. B.; Heath-Brown, D. R. Asymptotimean square ofthe

produtoftheRiemannzeta-funtionandaDirihletpolynomial.J.ReineAngew.Math.357(1985),

161181.

[BeHu℄ Beineke,J.andHughes,C.GreatmomentsoftheRiemannzetafuntion,inBisuitsofnumber

theory.EditedbyArthurT.BenjaminandEzraBrown.TheDolianiMathematialExpositions,34.

MathematialAssoiationofAmeria,Washington,DC,2009.xiv+311pp.

[BD℄ Bombieri, E.; Davenport, H. Some inequalities involving trigonometrial polynomials. Ann.

SuolaNorm. Sup.Pisa(3)23(1969),223241.

[Con℄ Conrey,J.B.Themean-squareofDirihlet

L

^-funtions,arXiv:0708.2699.

[CF℄ Conrey,J. B.;Farmer,D. W. Meanvaluesof

L

^{-funtions and symmetry. Internat.Math. Res.}

Noties2000,no.17,883908.

[CFKRS℄ Conrey, J. B.; Farmer, D. W.; Keating, J. P.; Rubinstein, M. O.; Snaith, N. C. Integral

momentsof

L

^{-funtions.Pro.LondonMath. So.(3)91(2005),no.1,33104.}

[CG℄ Conrey,J.B.;Ghosh,A.AonjetureforthesixthpowermomentoftheRiemannzeta-funtion.

Internat.Math. Res.Noties1998,no.15,775780.

[CGo℄ Conrey,J.B.;Gonek,S.M.HighmomentsoftheRiemannzeta-funtion.DukeMath.J.107

(2001),no.3,577604.

[CIS℄ Conrey,J.B.,Iwanie, H.,Soundararajan,K.Theasymptotilargesieve.Inpreparation.

[DI℄ Deshouillers,J.-M.;Iwanie,H.KloostermansumsandFourieroeientsofuspforms.Invent.

Math.70(1982/83),no.2,219288.

[H-B1℄ Heath-Brown, D. R.Thefourth powermomentof theRiemannzetafuntion.Pro.London

Math.So.(3)38(1979),no.3,385422.

[H-B2℄ Heath-Brown, D. R.An asymptotiseries forthemeanvalueof Dirihlet

L

^{-funtions.Com-}

ment.Math.Helv.56(1981),no.1,148161.

[H-B3℄ Heath-Brown,D.R.ThefourthpowermeanofDirihlet's

L

^{-funtions.Analysis1(1981),no.}

1,2532.

[H-B4℄ Heath-Brown, D. R. A mean valueestimate for realharater sums.Ata Arith. 72 (1995),

no.3,235275.

[Hux℄ Huxley,M.N. Thelargesieveinequalityforalgebrainumberelds. II.Meansof momentsof

Hekezeta-funtions.Pro.London Math.So.(3)21(1970),108-128.

[Iwa1℄ Iwanie, Henryk Introdution to the spetraltheory of automorphi forms. Bibliotea de la

RevistaMatematiaIberoameriana.RevistaMatematiaIberoameriana,Madrid,1995.

[Iwa2℄ Iwanie,HenrykTopisinlassialautomorphiforms.GraduateStudiesinMathematis,17.

AmerianMathematialSoiety,Providene,RI,1997.

[KaSa℄ Katz,Niholas M.;Sarnak,Peter Randommatries,Frobeniuseigenvalues,andmonodromy.

AmerianMathematialSoietyColloquiumPubliations,45.AmerianMathematialSoiety,Prov-

idene,RI,1999.

[KS1℄ Keating,J.P.; Snaith,N. C.Randommatrixtheoryand

ζ(1/2 + it)

^{.Comm. Math.Phys.214}

(2000),no.1,5789.

[KS2℄ Keating,J.P.;Snaith,N.C.Randommatrixtheoryand

L

^-funtionsat

s = 1/2

^.Comm.Math.

Phys.214(2000),no.1,91110.

[Sou℄ Soundararajan,K. Thefourthmomentof Dirihlet

L

^{-funtions,Analytinumbertheory,239-}

246,ClayMath.Pro.,7,Amer.Math. So.,Providene,RI,2007;arxivmath.NT/0507150.

[Sou1℄ Soundararajan,K.MomentsoftheRiemannzeta-funtion,Ann.ofMath. (2)170(2009),no.

2,981-993.;arxiv math.NT/0612106.

[You℄ Young,Matthew.ThefourthmomentofDirihlet

L

^{-funtions,toappear,Ann.of}Math.;arxiv math.NT/0610335.

Lundi31 janvier2011

BrianConrey, AmerianInstitute ofMathematis, 360Portage Ave Palo Alto, CA 94306-2244

Department of Mathematis University Walk, Clifton, Bristol BS8 1TW, U.K.

E-mail:onreyaimath.org