New A Symplectic Test of the L-Functions Ratios Conjecture · 2008. 5. 13. · Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs Identifying the Symmetry

Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs

A Symplectic Test of the L-FunctionsRatios Conjecture

Steven J MillerBrown University

[email protected]@aya.yale.edu

http://www.math.brown.edu/∼sjmiller

Cornell University, June 5th, 2008

1


L-functions

Riemann zeta function:

ζ(s) =

∞∑

n=1

1ns =

∏

p

(1 − 1

ps

)−1

.

Dirichlet L-functions:

L(s, χd) =

∞∑

n=1

χd(n)

ns =∏

p

(1 − χd(p)

ps

)−1

.

Elliptic curve L-functions: build up with data related tonumber of solutions modulo p.

2


Properties of zeros of L-functions

infinitude of primes, primes in arithmetic progression.

Chebyshev’s bias: π3,4(x) ≥ π1,4(x) ‘most’ of the time.

Birch and Swinnerton-Dyer conjecture.

Goldfeld, Gross-Zagier: bound for h(D) fromL-functions with many central point zeros.

Even better estimates for h(D) if a positivepercentage of zeros of ζ(s) are at most 1/2 − ε of theaverage spacing to the next zero.

3


Distribution of zeros

ζ(s) 6= 0 for Re(s) = 1: π(x), πa,q(x).

GRH: error terms.

GSH: Chebyshev’s bias.

Analytic rank, adjacent spacings: h(D).

4


Measures of Spacings: n-Level Correlations2 NICHOLAS M. KATZ AND PETER SARNAK

Figure 1. Nearest neighbor spacings among 70 million zeroes be-yond the 1020-th zero of zeta, verses µ1(GUE).

RH (needless to say, in the numerical experiments reported on below all zeroesfound were on the line Re(s) = 1

2 ) and order the ordinates γ:

. . . . . . γ−1 ≤ 0 ≤ γ1 ≤ γ2 . . . .(4)

Then γj = −γ−j, j = 1, 2, . . . , and in fact γ1 is rather large, being equal to14.1347 . . . . It is known (apparently already to Riemann) that

#{j : 0 ≤ γj ≤ T } ∼ T log T

2π, as T → ∞.(5)

In particular, the mean spacing between the γ′js tends to zero as j → ∞. In order

to examine the (statistical) law of the local spacings between these numbers were-normalize (or “unfold” as it is sometimes called) as follows:Set

γj =γj log γj

2πfor j ≥ 1.(6)

The consecutive spacings δj are defined to be

δj = γj+1 − γj , j = 1, 2, . . . .(7)

More generally, the k − th consecutive spacings are

δ(k)j = γj+k − γj , j = 1, 2, . . . .(8)

What laws (i.e. distributions), if any, do these numbers obey?During the years 1980-present, Odlyzko [OD] has made an extensive and pro-

found numerical study of the zeroes and in particular their local spacings. Hefinds that they obey the laws for the (scaled) spacings between the eigenvalues of

n-level correlation: {αj} an increasing sequence ofnumbers, B ⊂ Rn−1 a compact box:

limN→∞

#

{(αj1 − αj2, . . . , αjn−1 − αjn

)∈ B, ji 6= jk ≤ N

}

N

Odlyzko: Normalized spacings of ζ(s) starting at 1020

Montgomery, Hejhal: Pair, triple correlations of ζ(s)Rudnick-Sarnak: n-level correlations for allautomorphic cupsidal L-fnsKatz-Sarnak: n-level correlations for the classicalcompact groupsinsensitive to any finite set of zeros

5


Measures of Spacings: n-Level Density and Families

Let gi be even Schwartz functions whose FourierTransform is compactly supported, L(s, f ) an L-functionwith zeros 1

2 + iγf and conductor Qf :

Dn,f (g) =∑

j1,...,jnji 6=±jk

g1

(γf ,j1

log Qf

2π

)· · ·gn

(γf ,jn

log Qf

2π

)

Properties of n-level density:� Individual zeros contribute in limit� Most of contribution is from low zeros� Average over similar L-functions (family)To any geometric family, Katz-Sarnak predict then-level density depends only on a symmetry group (aclassical compact group) attached to the family.

6


n-Level Density

n-level density: F = ∪FN a family of L-functions orderedby conductors, gk an even Schwartz function:

Dn,F(g) = limN→∞

1|FN|

∑

f∈FN

∑

j1,...,jnji 6=±jk

g1

(log Qf

2πγj1;f

)· · ·gn

(log Qf

2πγjn;f

)

As N → ∞, 1-level density converges to∫

g(x)W1,G(F)(x)dx =

∫g(u)W1,G(F)(u)du.

Conjecture (Katz-Sarnak)(In the limit) Distribution of zeros near central point agreeswith distribution of eigenvalues near 1 of a classicalcompact group.

7


1-Level Densities

Let G be one of the classical compact groups: Unitary,Symplectic, Orthogonal (or SO(even), SO(odd)).If supp(g) ⊂ (−1, 1), 1-level density of G is

g(0) − cGg(0)

2,

where

cG =

0 G is Unitary1 G is Symplectic

−1 G is Orthogonal.

8


Some Results

Orthogonal:� Iwaniec-Luo-Sarnak: 1-level density for H±

k (N), Nsquare-free.� Miller, Young: families of elliptic curves.� Güloglu: 1-level for {Symr f : f ∈ Hk(1)}, r odd.Symplectic:� Rubinstein: n-level densities for L(s, χd).� Güloglu: 1-level for {Symr f : f ∈ Hk(1)}, r even.Unitary:� Hughes-Rudnick, Miller: families of primitiveDirichlet characters.

9


Identifying the Symmetry Groups

Often an analysis of the monodromy group in thefunction field case suggests the answer.All simple families studied to date are built from GL1or GL2 L-functions.Tools: Explicit Formula, Orthogonality of Characters /Petersson Formula.How to identify symmetry group in general? Onepossibility is by the signs of the functional equation:Folklore Conjecture: If all signs are even and nocorresponding family with odd signs, Symplecticsymmetry; otherwise SO(even). (False!)

10


Explicit Formula

π: cuspidal automorphic representation on GLn.Qπ > 0: analytic conductor of L(s, π) =

∑λπ(n)/ns.

By GRH the non-trivial zeros are 12 + iγπ,j.

Satake parameters {απ,i(p)}ni=1;

λπ(pν) =∑n

i=1 απ,i(p)ν.

L(s, π) =∑

nλπ(n)

ns =∏

p

∏ni=1 (1 − απ,i(p)p−s)

−1.

∑

j

g(

γπ,jlog Qπ

2π

)= g(0)−2

∑

p,ν

g(

ν log plog Qπ

)λπ(pν) log ppν/2 log Qπ

11


Some Results: Rankin-Selberg Convolution of Families

Symmetry constant: cL = 0 (resp, 1 or -1) if family L hasunitary (resp, symplectic or orthogonal) symmetry.

Rankin-Selberg convolution: Satake parameters forπ1,p × π2,p are

{απ1×π2(k)}nmk=1 = {απ1(i) · απ2(j)} 1≤i≤n

1≤j≤m.

Theorem (Dueñez-Miller)If F and G are nice families of L-functions, thencF×G = cF · cG.

12


Ratios Conjecture

13


History

Farmer (1993): Considered∫ T

0

ζ(s + α)ζ(1 − s + β)

ζ(s + γ)ζ(1 − s + δ)dt ,

conjectured (for appropriate values)

T(α + δ)(β + γ)

(α + β)(γ + δ)− T 1−α−β (δ − β)(γ − α)

(α + β)(γ + δ).

Conrey-Farmer-Zirnbauer (2007): conjectureformulas for averages of products of L-functions overfamilies:

RF =∑

f∈F

ωfL(

12 + α, f

)

L(1

2 + γ, f) .

14


Uses of the Ratios Conjecture

Applications:� n-level correlations and densities;� mollifiers;� moments;� vanishing at the central point;

Advantages:� RMT models often add arithmetic ad hoc;� predicts lower order terms, often to square-rootlevel.

15


Inputs for 1-level density

Approximate Functional Equation:

L(s, f ) =∑

m≤x

am

ms + εXL(s)∑

n≤y

an

n1−s ;

� ε sign of the functional equation,� XL(s) ratio of Γ-factors from functional equation.

Explicit Formula: g Schwartz test function,

∑

f∈F

ωf

∑

γ

g(

γlog Nf

2π

)=

12πi

∫

(c)

−∫

(1−c)

R′F(· · · )g (· · · )

� R′F(r) = ∂

∂αRF (α, γ)

∣∣∣α=γ=r

.

16


Procedure

Use approximate functional equation to expandnumerator.Expand denominator by generalized Mobius function:cusp form

1L(s, f )

=∑

h

µf (h)

hs ,

where µf (h) is the multiplicative function equaling 1for h = 1, −λf (p) if n = p, χ0(p) if h = p2 and 0otherwise.Execute the sum over F , keeping only main(diagonal) terms.Extend the m and n sums to infinity (complete theproducts).Differentiate with respect to the parameters.

17


Main Results

18


Symplectic Families

Fundamental discriminants: d square-free and 1modulo 4, or d/4 square-free and 2 or 3 modulo 4.Associated character χd :� χd(−1) = 1 say d even;� χd(−1) = −1 say d odd.� even (resp., odd) if d > 0 (resp., d < 0).

Will study following families:� even fundamental discriminants at most X ;� {8d : 0 < d ≤ X , d an odd, positive square-freefundamental discriminant}.

19


Prediction from Ratios Conjecture

1

X∗

∑

d≤X

∑

γd

g(

γdlog X

2π

)=

1

X∗ log X

∫∞

−∞g(τ)

∑

d≤X

[log

d

π+

1

2

Γ′

Γ

(1

4±

iπτ

log X

)]dτ

+2

X∗ log X

∑

d≤X

∫∞

−∞g(τ)

[ζ′

ζ

(1 +

4πiτ

log X

)+ A′

D

(2πiτ

log X;

2πiτ

log X

)

− e−2πiτ log(d/π)/ log XΓ(

14 − πiτ

log X

)

Γ(

14 + πiτ

log X

) ζ

(1 −

4πiτ

log X

)AD

(−

2πiτ

log X;

2πiτ

log X

)]dτ + O(X− 1

2 +ε),

with

AD(−r , r) =∏

p

(1 − 1

(p + 1)p1−2r −1

p + 1

)·(

1 − 1p

)−1

A′D(r ; r) =

∑

p

log p(p + 1)(p1+2r − 1)

.

20



Main term is

1X ∗

∑

d≤X

∑

γd

g(

γdlog X

2π

)=

∫ ∞

−∞

g(x)

(1 − sin(2πx)

2πx

)dx

+ O(

1log X

),

which is the 1-level density for the scaling limit ofUSp(2N). If supp(g) ⊂ (−1, 1), then the integral of g(x)against − sin(2πx)/2πx is −g(0)/2.

21



Assuming RH for ζ(s), for supp(g) ⊂ (−σ, σ) ⊂ (−1, 1):

−2

X∗ log X

∑

d≤X

∫∞

−∞g(τ) e

−2πiτ log(d/π)log X

Γ(

14 − πiτ

log X

)

Γ(

14 + πiτ

log X

) ζ

(1 −

4πiτ

log X

)AD

(−

2πiτ

log X;

2πiτ

log X

)dτ

= −g(0)

2+ O(X− 3

4 (1−σ)+ε);

the error term may be absorbed into the O(X−1/2+ε) errorif σ < 1/3.

22


Main Results

Theorem (M– ’07)

Let supp(g) ⊂ (−σ, σ), assume RH for ζ(s). 1-LevelDensity agrees with prediction from Ratios Conjecture

up to O(X−(1−σ)/2+ε) for the family of quadraticDirichlet characters with even fundamentaldiscriminants at most X;up to O(X−1/2 + X−(1− 3

2 σ)+ε + X− 34 (1−σ)+ε) for our

sub-family. If σ < 1/3 then agrees up to O(X−1/2+ε).

23


Numerics (J. Stopple): 1,003,083 negative fundamentaldiscriminants −d ∈ [1012, 1012 + 3.3 · 106]

Histogram of normalized zeros (γ ≤ 1, about 4 million).� Red: main term. � Blue: includes O(1/ log X ) terms.

� Green: all lower order terms.24


Sketch of Proofs

25


Ratios Calculation

Hardest piece to analyze is

R(g; X ) = − 2X ∗ log X

∑

d≤X

∫ ∞

−∞

g(τ)e−2πiτ log(d/π)log X

Γ(

14 − πiτ

log X

)

Γ(

14 + πiτ

log X

)

· ζ

(1 − 4πiτ

log X

)AD

(− 2πiτ

log X;

2πiτlog X

)dτ,

AD(−r , r) =∏

p

(1 − 1

(p + 1)p1−2r −1

p + 1

)·(

1 − 1p

)−1

.

Proof: shift contours, keep track of poles of ratios of Γ andzeta functions.

26


Ratios Calculation: Weaker result for supp(g) ⊂ (−1, 1).

d -sum is X ∗e−2πi(1− log πlog X )τ

(1 − 2πiτ

log X

)−1+ O(X 1/2);

decay of g restricts τ -sum to |τ | ≤ log X , Taylorexpand everything but g: small error term and

∫

|τ |≤log Xg(τ)

N∑

n=−1

an

logn X(2πiτ)ne−2πi(1− log π

log X )τdτ

=N∑

n=−1

an

logn X

∫

|τ |≤log X(2πiτ)ng(τ)e−2πi(1− log π

log X )τdτ ;

from decay of g can extend the τ -integral to R

(essential that N is fixed and finite!), for n ≥ 0 get theFourier transform of g(n) (the nth derivative of g) at1 − π

log X , vanishes if supp(g) ⊂ (−1, 1).27


Number Theory Sums

Seven = − 2X ∗

∑

d≤X

∞∑

`=1

∑

p

χd(p)2 log pp` log X

g(

2log p`

log X

)

Sodd = − 2X ∗

∑

d≤X

∞∑

`=0

∑

p

χd(p) log pp(2`+1)/2 log X

g(

log p2`+1

log X

).

28


Number Theory Sums

LemmaLet supp(g) ⊂ (−σ, σ) ⊂ (−1, 1). Then

Seven = −g(0)

2+

2log X

∫ ∞

−∞

g(τ)ζ ′

ζ

(1 +

4πiτlog X

)dτ

+2

log X

∫ ∞

−∞

g(τ)A′D

(2πiτlog X

;2πiτlog X

)+ O(X− 1

2+ε)

Sodd = O(X− 1−σ2 log6 X ).

If instead we consider the family of characters χ8d for odd,positive square-free d ∈ (0, X ) (d a fundamentaldiscriminant), then

Sodd = O(X−1/2+ε + X−(1− 32 σ)+ε).

29


Analysis of Seven

χd(p)2 = 1 except when p|d . Replace χd(p)2 with 1, andsubtract off the contribution from when p|d :

Seven = −2∞∑

`=1

∑

p

log pp` log X

g(

2log p`

log X

)

+2

X ∗

∑

d≤X

∞∑

`=1

∑

p|d

log pp` log X

g(

2log p`

log X

)

= Seven;1 + Seven;2.

Lemma (Perron’s Formula)

Seven;1 = −g(0)

2+

2log X

∫ ∞

−∞

g(τ)ζ ′

ζ

(1 +

4πiτlog X

)dτ.

30


Analysis of Seven: Seven;2

This piece gives us∫

g(τ)A′D(− · · · , · · · ).

Main ideas:� Restrict to p ≤ X 1/2.� For p < X 1/2:

∑d≤X ,p|d 1 = X∗

p+1 + O(X 1/2).� Use Fourier Transform to expand g.

31


Analysis of Sodd

Sodd = − 2X ∗

∞∑

`=0

∑

p

log pp(2`+1)/2 log X

g(

log p2`+1

log X

)∑

d≤X

χd(p).

Jutila’s bound

∑

1<n≤Nn non−square

∣∣∣∣∣∣∑

0<d≤Xd fund. disc.

χd(n)

∣∣∣∣∣∣

2

� NX log10 N.

Proof: Cauchy-Schwarz and Jutila: p2`+1 non-square:

∞∑

`=0

∑

p(2`+1)/2≤Xσ

∣∣∣∣∣∑

d≤X

χd(p)

∣∣∣∣∣

2

1/2

� X1+σ

2 log5 X .

32


Analysis of Sodd: Extending Support

More technical, replace Jutila’s bound by applyingPoisson Summation to character sums.

LemmaLet supp(g) ⊂ (−σ, σ) ⊂ (−1, 1). For family{8d : 0 < d ≤ X , d an odd, positive square-freefundamental discriminant}, Sodd = O(X− 1

2+ε + X−(1− 32 σ)+ε).

In particular, if σ < 1/3 then Sodd = O(X−1/2+ε).

33


Conclusions

34


Conclusions

Ratios Conjecture gives detailed predictions (up toX 1/2+ε).

Number Theory agrees with predictions for suitablyrestricted test functions.

Numerics quite good.

35


Appendix

36


RH and the Prime Number Theorem

From ζ(s) =∑

n−s =∏

p (1 − p−s)−1, logarithmic derivative is

−ζ ′(s)

ζ(s)=∑ Λ(n)

ns ,

where Λ(n) = log p if n = pk and is 0 otherwise.

Take Mellin transform, integrate and shift contour. Find∑

n≤x

Λ(n) = x −∑

ρ

xρ

ρ,

where ρ = 1/2 + iγ runs over non-trivial zeros of ζ(s).

Partial summation gives Prime Number Theorem (to first order, thereare x/ log x primes at most x) if Reρ < 1.

The smaller max Re(ρ) is, the better the error term in the PrimeNumber Theorem. The Riemann Hypothesis (RH) says Re(ρ) = 1/2.

37


Primes in Arithmetic Progression

To study number primes p ≡ a mod q, use

L(s, χ) =∑ χ(n)

ns =∏

p

(1 − χ(p)

ps

)−1

.

Key sum: 1φ(q)

∑χ mod q χ(n) is 1 if n ≡ 1 mod q and 0 otherwise.

Similar arguments give

∑

p≡a mod q

log pps = − 1

φ(q)

∑

χ mod q

L′(s, χ)

L(s, χ)χ(a) + Good(s).

Note: To understand {p ≡ a mod q} need to understand all L(s, χ);see benefit of studying a family.

38


GSH and Chebyshev’s Bias

π3,4(x) ≥ π1,4(x) and π2,3(x) ≥ π1,3(x) ‘most’ of the time. Useanalytic density:

Denan(S) = lim sup1

log T

∫

S∩[2,T ]

dtt

.

Have π3,4(x) ≥ π1,4(x) with analytic density .9959 (first flip at 26861);π2,3(x) ≥ π1,3(x) with analytic density .9990 (first flip ≈ 6 · 1011).

Non-residues beat residues. Key ingredient Generalized SimplicityHypothesis (GSH): the zeros of L(s, χ) are linearly independent overQ.

Structure of zeros important: GSH used to show a flow on a torus isfull (becomes equidistributed).

39


Class Number

Class number: measures failure of unique factorization (order of idealclass group).

Imaginary quadratic field Q(√

D), fundamental discriminant D < 0, Igroup of non-zero fractional ideals, P subgroup of principal ideals,H = I/P class group, h(D) = #H the class number. Dirichlet proved

L(1, χD) =2πh(D)

wD√

D,

where χD the quadratic character and wD = 2 if D < −4, 4 if D = −4and 6 if D = −3.

Theorem: h(D) = 1 ⇔ −D ∈ {3, 4, 7, 8, 11, 19, 43, 67, 163}.

40


Class Number and Distribution of Zeros I

Expect√

|D|

log log |D| � h(D) �√|D| log log |D|. Siegel proved

h(D) > c(ε)|D|1/2−ε (but ineffective).

Goldfeld, Gross-Zagier: f primitive cusp form of weight k , level N,trivial central character, suppose m = ords=1/2L(s, f )L(s, χD) ≥ 3,g = m − 1 or m − 2 so that (−1)g = ω(f )ω(fχD ) (signs of fnal eqs).Then have effective bound

h(D) � (log |D|)g−1∏

p|D

(1 +

1p

)−3(1 +

λ(p)√

pp + 1

)−1

.

Good result from using an elliptic curve that vanishes to order 3 ats = 1/2, application of many zeros at central point.

41


Class Number and Distribution of Zeros II

Assume a positive percent of zeros (or cT log T/(log |D|)A) of zeroswith γ ≤ T ) of ζ(s) are at most 1/2 − ε of the average spacing fromthe next zero ζ(s). Then h(D) �

√|D|/(log |D|)B , all constants

computable.

See actual spacings between zeros are tied to number theory (havepositive percent are less than half the average spacing if GUEConjecture holds for adjacent spacings).

Instead of 1/2 − ε, under RH have: .68 (Montgomery), .5179(Montgomery-Odlyzko), .5171 (Conrey-Ghosh-Gonek), .5169(Conrey-Iwaniec) (Montgomery says led to pair correlation conjectureby looking at gaps between zeros of ζ(s) and h(D)).

42


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New A Symplectic Test of the L-Functions Ratios Conjecture · 2008. 5. 13. · Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs Identifying the Symmetry