Main terms in moments Brian Conrey AIM and Bristol Cetraro July - - PowerPoint PPT Presentation

Main terms in moments

Brian Conrey AIM and Bristol

July 8, 2019

Cetraro

At the Amalfi conference in 1989:

Theorem : (Conrey and Ghosh)

1 T Z T |ζ(1/2 + it)|6 dt ≥ 10.13 a3 log9 T 9!

ak = Y

p

✓ 1 − 1 p ◆(k−1)2 k−1 X

j=0

k−1

j

2 pj

Conrey and Ghosh conjecture: 1992 Conrey and Gonek conjecture: 1998

Keating and Snaith formula:

g1 = 1 g2 = 2 g3 = 42 g4 = 24024

g5 = 701149020

1 T Z T |ζ(1/2 + it)|2k dt ∼ gkak logk2 T k2!

Conjecture (KS):

Z

U(N)

| det(I − U)|2k dU = (N + 1)(N + 2)2 . . . (N + k)k(N + k + 1)k−1 . . . (N + 2k − 1) 1 · 22 · · · · · kk · (k + 1)k−1 · · · · · (2k − 1) ∼ gk N k2 k2!

gk = k2! 1 · 22 · · · · · kk · (k + 1)k−1 · · · · · (2k − 1)

where Theorem:

BA,B(s) =

∞

X

n=1

τA(m)τB(m) ms

Y

α∈A

ζ(s + α) =

∞

X

n=1

τA(n) ns

Let A and B be sets of small complex numbers and Then, with s=1/2+it

Conjecture (CFKRS) The Recipe

where

Z ψ ✓ t T ◆ Y

α∈A

ζ(s + α) Y

β∈B

ζ(1 − s + β) dt

= Z ψ ✓ t T ◆ X

U⊂A,V ⊂B |U|=|V |

✓ t 2π ◆−(U+V ) BU∪V −,V ∪U −(1) dt + O(T 1−δ)

and ψ ∈ C∞[1, 2]

Note that each term has a total of |A| |B| singularities; but the sum is analytic .

Conjecture (C, Farmer, Keating, Rubinstein, Snaith) where

RMT analogue

This matches perfectly with the recipe!

Theorem (CFKRS). Let Z(A, B) = Y

α∈A β∈B

z(α + β) where z(x) = (1 − e−x)−1. Then Z

U(N)

Y

α∈A

ΛX(e−α) Y

β∈B

ΛX∗(e−β) dX = X

S⊂A T ⊂B |S|=|T |

e−N(S+T )Z(S ∪ T −, T ∪ S−)

Conrey, Iwaniec, Soundararajan proved a sixth moment estimate for Dirichlet L-functions which found the full 9th degree polynomial above. Nathan Ng proved that the sixth moment of zeta with all lower order terms can be obtained from precise information about the shifted divisor problem. Chandee and Li obtained the leading order term (the 24024) assuming RH for the 8th moment of this family.

There is interest in how new main terms enter the picture in moment problems as the order of the moment grows. Classically we can do the second moment of zeta by diagonal analysis; but the fourth moment requires shifted convolution sums. For averages of quadratic L-functions one uses diagonal analysis for the first and second moment and then Soundararajan’s Poisson formula for quadratic characters for the third moment. New main terms arise from the “square” values of k after using Poisson. For averaging cusp form L-functions at the center via the Peterson formula one initially uses only the diagonal terms at the start; then Kowalski, Michel and Vanderkaam show how to use parts of the Kloosterman sum to obtain some off-diagonal contributions; further investigation leads to off-off-diagonal contributions to the main term. In the asymptotic large sieve applications of Conrey, Iwaniec and Soundararajan (eg for the sixth moment of Dirichlet L-functions) the final main terms seem to be located in a remote part of the complex plane, far from other contributing singularities. In Zhang and Diaconu using multiple Dirichlet series found a polar term at 3/4!

Combinatorics of main terms

Moments of long Dirichlet polynomials

+T X

m6=n

ambn √mn ˆ ψ ✓ T 2π log n m ◆ = T ˆ ψ(0) X

n≤X

anbn n Z ∞ ψ ✓ t T ◆ X

m≤X

am m1/2+it X

n≤X

bn n1/2−it dt

We can often rewrite this as The off-diagonal piece is

X

h6=0 0<m+hX

ambm+h p m(m + h) ˆ ψ ✓ T 2π log ✓ 1 + h m ◆◆ ∼ X

h6=0 0<mX

ambm+h m ˆ ψ ✓ Th 2πm ◆

The case of t-aspect for Ramanujan tau L-function

S(X, h) := X

n≤X

τ ∗(n)τ ∗(n + h)

τ ∗(n) = τ(n)n−11/2 S(X, h) ⌧h X2/3

1 p X X

h≤ √ X

|S(X, h)| ⌧ X1/2+✏

Good (1983) How does one average the moments of a cusp form L-function in t-aspect? The shifted convolutions play a role.

X

n≤X

τ ∗(n)τ ∗(n + h) ⌧ X1−δ

uniformly for h ⌧ X2−η

Blomer (2005)

S(X,1) for 1< X < 10000

uniformly for H ⌧ X1/2

T(X; H) := X

n≤X

• X

h≤H

τ ∗(n + h)

• 2

= CXH + O(X2/3+✏H5/3)

Good:

T(X;H) for X = 5000; 1< H < 100

Z ψ ✓ t T ◆

• X

n≤X

τ ∗(n) n1/2+it

• 2

dt

⇠ ( T ˆ ψ(0) P

n≤X τ ∗(n)2 n

X ⌧ T 2 R ψ t

T

• |Lτ ∗(1/2 + it)|2 dt

X T 2

Z ψ ✓ t T ◆ |Lτ ∗(1/2 + it)|2 dt ∼ T ˆ ψ(0) X

n≤T 2

τ ∗(n)2 n

But Averaging a long tau polynomial

= T ˆ ψ(0) X

n≤X

τ ∗(n) n + 2T X

h6=0 T mX

τ ∗(m)τ ∗(m + h) m ˆ ψ ✓ Th 2πm ◆ + o(T)

So, it must be the case that

for X T 2

2 X

h6=0 T mX2

τ ∗(m)τ ∗(m + h) m ˆ ψ ✓ Th 2πm ◆ ∼ C ˆ ψ(0) log T 2 X

X

n≤X

τ ∗(n)2 n ∼ C log X

where C is such that

The case of zeta

M2(α) =    α4 if 0 < α < 1 −α4 + 8α3 − 24α2 + 32α − 14 if 1 < α < 2 2 if α > 2

Zeta-polynomials

1 T Z T

• X

n≤X

dk(n) ns − Resw=1−s ζ(s + w)kXw w

• 2

dt ∼ Mk(α) ak logk2 T k2!

where s = 1/2 + it and X = T α. For example

Let and

ΛU(s) =

N

X

j=0

Scj(U)sj ζ(s)k =

∞

X

n=1

dk(n) ns

To what extent do the coefficients

• f ΛU(s)k behave like dk(n)nit?

Diaconis - Gamburd formula

Z

U(N)

Scj1(U) . . . Scjk(U)Sch1(U) . . . Sch`(U) du

If j1 + · · · + jk = h1 + · · · + h` ≤ N, then

is the number of k × ` matrices with non-negative integer entries and row sums j1, . . . , jk and column sums h1, . . . h`.

For example, Z

U(N)

Scj(U)Sck(U) du = 0 Z

U(N)

|Scj(U)|2 du = 1 and Z

U(N)

|Scj(U)Sck(U)|2 dU = 1 + min{j, k} if j 6= k and j + k  N.

∼ ak N k2 Z

U(N)

• X

j1+···+jk≤αN ji≤N

Scj1 . . . Scjk

• 2

dU

Conjecture

Sandro Bettin assumes the recipe and proves this.

1 T(log T)k2 Z T

• X

m≤T α

dk(m) m1/2+it − Ress=1 ζ(s)kT αs s

• 2

dt

Z

U(N)

• X

j1+···+jk≤αN

Scj1 . . . Scjk

• 2

du = X

m≤αN

Ik(m, N)

By orthogonality

Ik(m; N) := Z

U(N)

• X

j1+···+jk=m ji≤N

Scj1(U) . . . Scjk(U)

• 2

dU

where

Keating, Rodgers, Roddity-Gershon, Rudnick formula

If m < N then By the functional equation this also works for (k-1)N , m < kN. It’s not so clear what the formula looks like when N < m < 2n.

Ik(m; N) = ✓k2 − 1 + m m ◆

Keating, Rodgers, Roddity-Gershon, Rudnick formula

Ik(m; N) is equal to the number of k × k matrices with non-negative integer entries at most N in size whose rows are weakly increasing, columns are weakly decreasing and whose anti-diagonal sums to kN − m

(Gelfond-Tsetlin patterns)

Zeros of R

U(25) ΛU(x)3ΛU ∗(x)3 dU = P25 m=0 Ik(m, 25)x2m

Ik(m; N) = γk(c)N k2−1 + O(N k2−2) c = m/N

γk(c) = 1 k!G(1 + k2) Z

[0,1]k δc(w1 + · · · + wk)∆(w1, . . . , wk)2dw

Keating, Rodgers, Roddity-Gershon, Rudnick where G is the Barnes function.

γk(c) = M 0

k(c)

k2!

1 X Z 2X

X

• X

x≤n≤x+H

dk(n)

• 2

dx − @ 1 X Z 2X

X

X

x≤n≤x+H

dk(n) dx 1 A

2

Conjecture: Keating, Rodgers, Roddity-Gershon, Rudnick; Rodgers, Soundararajan Lester has made progress on the divisor problem in short intervals. Keating, Rodgers, Roddity-Gershon, Rudnick prove the function field analogue of this.

log X log X

H

→ c ∈ (0, k)

with

∼ ak(q)γk(c)H ✓ log X H ◆k2−1

X

1≤a≤q (a,q)=1

• X

n≡a mod q n≤X

dk(n) − 1 φ(q) X

(n,q)=1 n≤X

dk(n)

• 2

Conjecture: Keating, Rodgers, Roddity-Gershon, Rudnick; Rodgers, Soundararajan Rodgers and Soundararajan prove this for delta<c<2-delta (assuming GRH). Keating, Rodgers, Roddity-Gershon, Rudnick prove a function field analogue of this.

log X log q → c ∈ (0, k)

with

∼ ak(q)γk(c)H(log X)k2−1

γk(c) = 1 k!G(1 + k2) Z

[0,1]k δc(w1 + · · · + wk)∆(w1, . . . , wk)2dw

g(u) = Z 1 exp(−tx) dx

γk(c) = 1 G(1 + k)2 Z ∞

−∞

exp(2πiuc)Dk(2πiu) du

Dk(t) = det

k×k

⇣ g(i+j−2)(u) ⌘

Basor, Ge, Rubinstein

Dk(t) = Dk(0) exp ∞ X

m=1

cm(k) m tm !

cM(k) = 1 (M − 1)(M − 2)

M−3

X

m=0

(m + 2)cm+2(k) (cM−m−2(k − 1) + cM−m−2(k + 1) − 2cM−m−2(k)) c1(k) = −k 2

c2(k) = k2 4(4k2 − 1)

D satisfies a Painleve equation which leads to a recursion formula:

Back to moments

• f zeta

X

n≤X

τA(n)τB(n + h)

We need information about The delta method of Duke, Friedlander and Iwaniec (1993) can provide the needed conjecture. Bill Duke John Friedlander Henryk Iwaniec

Divisor correlations

Delta method conjecture

The poles of this Dirichlet series can be determined by replacing the exponential by Dirichlet characters and finding the coefficient of the trivial character (i.e. zeta).

hτA(m)τB(m + h)im=u ⇠

∞

X

q=1

rq(h)hτA(m)e(m/q)im=uhτB(n)e(n/q)in=u

hτA(m)e(m/q)im=u = 1 2πi Z

|w−1|=✏

DA(w, e(1/q))uw−1 dw where DA(w, e(1/q)) =

∞

X

n=1

τA(n)e(n/q) nw rq(h) is the Ramanujan sum and DA is the Estermann function

Z ∞ ψ( t T ) X

m≤T r n≤T r

τA(m)τB(n) √mn ⇣m n ⌘it dt

Assuming delta-conjecture

+O(T 1−δ) where 1 ≤ r < 2 and

= Z 1 ψ( t T ) ✓ BA,B(1; T r) + X

α2A β2B

✓ t 2π ◆αβ BA0,B0(1; T r) ◆ A0 = A − {α} ∪ {−β}, B0 = B − {β} ∪ {−α}

This relies on the identity

where and

∞

X

j=0

τA0∪{− ˆ

β}(pj)τB0∪{−ˆ α}(pj)

pj Equating Euler products, the identity is:

Suppose ˆ α ∈ A and ˆ β ∈ B. Let A0 = A − {ˆ α} and B0 = B − {ˆ β}. Then

Identifying `geometric’ factors and the location of the poles of both sides of the identity is a matter of equating the `linear’ term of the Euler products; i.e. the value at p of the arithmetic function. So, in some sense the identification of the whole p-factor is more complex.

=

∞

X

h=0

1 ph(1−ˆ

α− ˆ β) ∞

X

j=0

τA0(pj) pj(1−ˆ

α) ∞

X

k=0

τB0(pk) pk(1− ˆ

β) ∞

X

d,q=0

µ(pq) pd+q(2−ˆ

α− ˆ β) ∞

X

m=0

τA0(pm+d+q) pm(1−ˆ

α) ∞

X

n=0

τB0(pn+d+q) pn(1− ˆ

β)

Thus, we see the `one-swap’ terms arise from the standard shifted divisor problem.

What is the RMT analogue of the input from the (averaged over h) delta method?

Where are the rest of the terms from the recipe???

What if r > 2? Say ` < r < ` + 1

To find the higher-swap terms we will need convolutions of shifted divisors …

Conrey - Keating approach

is related to

X

m,n<T r

τA(m)τB(n) √mn ˆ ψ T 2π log m n

• X

A=A!∪···∪A` B=B1∪···∪B`

X

M1...M`=N1...N` (Mi,Ni)=1

`

Y

j=1

@ X

mj,nj

τAj(mj)τBj(nj) √mn 1 A ˆ ψ T 2π log m1 . . . m` n1 . . . n`

• ∗

   M1m1 = N1n1 + h1 . . . M`m` = N`n` + h`   

Note that

ˆ ψ T 2π log m1 . . . m` n1 . . . n`

• ∼ ˆ

ψ( T 2π X hi niNi )

Split A and B up into A = A1 ∪ · · · ∪ A` and B = B1 ∪ · · · ∪ B`. Then

subject to Q mi, Q ni ≤ T r and

which controls the ranges of the sums.

hτA(m)τB(n)i(∗)

m=u

⇠ 1 M

∞

X

q=1

rq(h)hτA(m)e(mN/q)im=uhτB(n)e(nM/q)in= uN

M

Delta method conjecture with general linear constraint

hτA(m)e(mN/q)im=u = 1 2πi Z

|w−1|=✏

DA(w, e N q

• )uw−1 dw

where DA(w, e N q

• ) =

∞

X

n=1

τA(n)e(nN/q) nw .

(∗) : mM − nN = h

where The poles of this Dirichlet series can be determined by replacing the exponential by Dirichlet characters and finding the coefficient of the trivial character (i.e. zeta).

1 2⇡i Z

(2)

Xs T 2⇡ −`s X

(M1,N1)=···=(M`,N`)=1 N1...N`=M1...M` ✏j ∈{−1,+1}

Z

0<v1,...,v`<∞

ˆ (✏1v1 + · · · + ✏`v`)

`

Y

j=1

 1 (2⇡i)2 ZZ

|wj −1|=✏ |zj −1|=✏

M −zj

j

N s+1−wj

j

X

hj,qj

rqj(hj) hs+2−wj−zj

j

vs+1−wj−zj

j

DAj(wj, e Nj qj

• )DBj(zj, e

Mj qj

• )

T 2⇡ wj+zj−2dwjdzj dvj ds s

1 2⇡i Z

(2)

Xs Z ∞ (t) X

U(`)⊂A V (`)⊂B

✓Tt 2⇡ ◆− P

ˆ ↵∈U(`) ˆ ∈V (`)

(ˆ α+ ˆ β+s)

×B(As − U(`)s + V (`)−, B − V (`) + U(`)s

−, 1) dt ds

s

= Connecting divisor correlations and the recipe where U(`) denotes a set of cardinality ` with precisely one element from each

• f A1, . . . , A` and similarly V (`) denotes a set of cardinality ` with precisely one

element from each of B1, . . . , B` .

If we sum this over all the ways to split up A and B we get what the recipe predicts times a factor But this is the number of automorphisms of the *-system.

Automorphisms

`!2`2k−2`

50

We can use this approach to discover a formula for γk(c), c > 1.

Z

z,w

(η + X (zj + wj))k2 Q

j

⇥ (1 − zj)k(1 − wj)k⇤ ∆(z)2∆(w)2 Q

j

⇥ zk

j wk j

⇤ Q

i,j [(1 − wi − zj)(1 + wi + zj)] dz dw

It is a linear combination of the functions

where η is the fractional part of c and the products are for 1 ≤ i, j ≤ c

Wooley has pointed out the connection with counting points on varieties and Manin’s idea of counting points on certain varieties by counting points on a stratified set of subvarieties; this idea may be relevant here. Trevor Wooley

• Y. I.Manin

Manin stratification

We can see an example already from the case k=2. We expect that is a good approximation to zeta(s)^2 when X >>T^2.

X

m≤X |h|≥1

d(m)d(m + h) ˆ ψ(hT/m)

But the analysis of averaged over h fails to reveal a large main term of size X/T^2 as well as a secondary main term that reflects the change in behavior as X passes T^2.

X

n≤X

d(n) ns − Resw=1−s ζ(s + w)2Xw w

There is a closer analogy between moments of zeta and averages

• f characteristic polynomials than just that the main terms agree once

we insert the arithmetic factor. In a shifted moment of zeta we let the arithmetic factors go to 1 and we replace zeta(1+x) by z(x). The shift alpha for zeta becomes exp(alpha) in RMT. Finally N becomes t/(2 pi). At that point ALL of the main terms agree. It stands to reason that we can learn something by carefully analyzing all of the pieces from both points of view. And, if we regard the matrix size N in RMT as the infinite prime it stands to reason that we should investigate carefully what happens with the finite primes as well.

The same circle of ideas works for averages of ratios

• f zeta-functions and characteristic polynomials.

In particular, using ratios, we can revisit the Bogomolny-Keating papers on `Hardy-Littlewood implies GUE’

• n a similar footing as this work.

GUE

RMT problems

• 1. Find (simple) exact formulas for for m>N?
• 2. Do the satisfy a Painleve?
• 3. What is the RMT version of CK V?
• 4. Does the answer to 3 lead to a recursion formula for
• 5. Can we find the analogue of for O(N) and USp(2N)?
• 6. Does 5 lead to a formulation of CK V for moments in other families?

Ik(m; N) Ik(m; N)

Ik(m; N) ? Ik(m; N)

Function field problems

• 1. Can we do pair correlation over function fields?
• 2. Can we reproduce any of KRRR for fixed q?
• 3. Analogue of CK I-V for function fields?
• 4. Analogue of divisors in short intervals and in arithmetic progressions

for other families?

Number field problems

• 1. Do Rodgers-Soundararajan with shifts and with power savings.
• 2. What is the precise connection between the answer to 1 and

the moment polynomials? Work out the sizes (asymptotics?) and the relationship between these averages for various lambda and ranges of T and X. When X > T^2 is it best to average

• ver h first, possibly with Voronoi (see Jutila, Ivic)?

Connections with Manin stratification.)

Mλ(X, T; W) := X

m≤X h≥1

λ(m)λ(m + h) m W ✓hT m ◆

• 3. Let
• 4. Extend Nathan Ng’s work to rigorously obtain averages
• f long Dirichlet polynomials with general divisor coefficients.
• 5. Asymptotics of 10th moment of L-functions in cusp form families
• 6. Rigorously derive n-correlation conjecture in a range [0,2]

from H-L conjectures

Sλ(X; T) := X

m≤X

• H

X

h=1

λ(m + h)

• 2

and

FRG Project

A newly funded NSF project with PIs Conrey, Iwaniec, Keating, Soundararajan, Wooley and senior scientists Brad Rodgers and Caroline Turnage-Butterbaugh will be devoted to understanding these questions. If you are interested in this project send me email at conrey@aimath.org with subject line FRG

The End