Zeros, moments and determinants Integrability and Randomness in - - PowerPoint PPT Presentation

Zeros, moments and determinants

Nina Snaith

Integrability and Randomness in Mathematical Physics and Geometry, April 8th, 2019

det B B B B B B @

• 1

K−1

• 3

K−1

• 5

K−1

• · · ·

2K−1

K−1

• 2

K−1

• 4

K−1

• 6

K−1

• · · ·

2K

K−1

• 3

K−1

• 5

K−1

• 7

K−1

• · · ·

2K−1

K−1

• .

. . . . . . . . ... . . . K

K−1

• K+2

K−1

• K+4

K−1

• · · ·

3K−2

K−1

• 1

C C C C C C A = (−1)K+12(

K 2)

Emilia Alvarez (thesis)

det

K×K

       

1 Γ(2K) 1 Γ(2K−2) 1 Γ(2K−4)

· · ·

1 Γ(2) 1 Γ(2K−1) 1 Γ(2K−3) 1 Γ(2K−5)

· · ·

1 Γ(1)

. . . . . . . . . ... . . .

1 Γ(K+2) 1 Γ(K) 1 Γ(K−2)

· · ·

1 Γ(−K+4) 1 Γ(K) 1 Γ(K−2) 1 Γ(K−4)

· · ·

1 Γ(−K+2)

        = K(K − 1) 2 det

K×K

     

1 Γ(2K−1) 1 Γ(2K−3) 1 Γ(2K−5)

· · ·

1 Γ(1) 1 Γ(2K−2) 1 Γ(2K−4) 1 Γ(2K−6)

· · ·

1 Γ(0)

. . . . . . . . . ... . . .

1 Γ(K) 1 Γ(K−2) 1 Γ(K−4)

· · ·

1 Γ(−K+2)

     

Ian Cooper (thesis)

• 1

Γ(2k) 1 Γ(2k−1)

· · ·

1 Γ(k+1) 1 Γ(2k) −1 Γ(2k−1)

· · ·

(−1)k−1 Γ(k+1) 1 Γ(2k−1) 1 Γ(2k−2)

· · ·

1 Γ(k) −1 Γ(2k−1) 1 Γ(2k−2)

· · ·

(−1)k Γ(k)

. . . . . . ... . . . . . . . . . ... . . .

1 Γ(1) 1 Γ(0)

· · ·

1 Γ(2−k) −1 Γ(1) 1 Γ(0)

· · ·

(−1)3k−2 Γ(2−k)

• = (−1)k2k2 k−1

Y

`=1

`! (k + `)!

Conrey, Farmer, Keating, Rubinstein, Snaith. Proc. London.

• Math. Soc. (3) 91 (2005)

The Riemann Zeta Function

ζ(s) =

∞

X

n=1

1 ns , Res > 1 = Y

p

(1 − 1/ps)−1

Speiser: RH is equivalent to all complex zeros

• f ζ0(s) having real part at least 1/2

Picture By Sarah Froelich

Green: zeros of zeta Blue: zeros of zeta’

Conrey (1989): More than two fifths of the zeros

• f the Riemann zeta function are on the critical line

Levinson (1974): More than one third of the zeros

• f the Riemann zeta function are on the critical line

Bui, Conrey, Young (2011): More than 41% of the zeros of the Riemann zeta function are on the critical line

Levinson, Montgomery (1974): ζ(s) and ζ0(s) have essentially the same number of zeros in <(s) < 1/2.

Horizontal distribution of zeros of the derivative of the Riemann zeta function

Littlewood’s lemma (1924): For an analytic function F(s),

b 1/2 a

Z T +U

T

log |F(b + it)|dt − Z T +U

T

log |F(a + it)|dt + Z a

b

arg F(σ + i(T + U))dσ − Z a

b

arg F(σ + iT)dσ = 2π X dist > 2π(a − 1/2) NL

T T+U

Zeros of F(s)

(Eg. ζ0(s))

NL: number of zeros of ζ0(s) to the left of the 1/2-line

R T

0 log |ζ0(a + it)|dt

So number theorists are interested in:

=

d dv

R T

0 |ζ0(a + it)|vdt

• v=0

Random Unitary Matrices

A is an N × N unitary matrix: AA† = A†A = I eigenvalues of A: eiθn A is chosen randomly with respect to Haar measure on U(N)

Using the first 100000 Riemann zeros – Picture by Andrew Odlyzko Probability density for distances between consecutive eigenvalues/ zeros

Nearest Neighbor Spacing

zeros around the th zero

Picture by

• A. Odlyzko

billion zeros around the 1016 th zero

15

†

Moments of the Riemann Zeta Function

(Hardy and Littlewood, 1918) (Ingham, 1926)

ζ(s) =

∞

X

n=1

1 ns , Res > 1 = Y

p

(1 − 1/ps)−1

4

Moments of the Riemann Zeta Function: Conjecture

Z

U(N)

|ΛA(1)|2λdAHaar ∼ fλ,U(N)N λ2

Moments of characteristic polynomials: Theorem

N ∼ log T

Conjecture (Keating and Snaith, 2000): fλ = fλ,U(N)

Zeros of a characteristic polynomial (blue) and its derivative (green)

N=20 N=60

Pictures by Phan Toan

Horizontal (radial) distribution of zeros of the derivative of zeta (characteristic polynomial)

Mezzadri: J. Phys. A 36 (2003) Pictures from Dueñez, Farmer, Froelich, Hughes, Mezzadri, Phan: Nonlinearity 23 (2010) Distribution of distance from the unit circle of zeros of , N=100 Distribution of real part of the zeros of for 100000 zeros around T=1000000.

Λ0

A(s)

ζ0(s)

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

v = 1 Winn: Commun. Math. Phys. 315 (2012)

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

We can calculate:

R

U(N) |Λ0 A(eiθ)|2K2M|ΛA(eiθ)|2MdAHaar

K-M integer

Hughes (2001); Conrey, Rubinstein, Snaith (2006); Dehaye (2008); Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein, Snaith (preprint); Basor, Bleher, Buckingham, Grava, Its, Its, Keating (preprint)

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

We can calculate:

R

U(N)

• Λ0

A

ΛA (e−α)

• 2K

dAHaar, K integer

Conrey, Snaith: Commun. Number Th. and Physics. Vol. 2, Num.3 (2008) Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein, Snaith: preprint

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

We can calculate:

R

SO(2N) (Λ00(1))K dAHaar, K integer

Altuğ, Bettin, Petrow, Rishikesh, Whitehead (2014)

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

We can calculate:

R

SO(2N) Λ(1)rΛ0(eφ) Λ(eφ)

dAHaar, <(r) > 1/2

Ian Cooper (thesis)

R T

0 |ζ0(a + it)|vdt

R

U(N) |Λ0 A(s)|vdAHaar

Important to number theorists: We would like to calculate:

non-integer

v

We can calculate:

R

SO(2N)

⇣

Λ0

A

ΛA (e−α)

⌘K dAHaar, for integer K

Mason and Snaith (2018) Emilia Alvarez (thesis)

For large N Z

SO(2N)

✓Λ0

A

ΛA (ea/N ◆K dAHaar → I · · · I ∆(u2

1, u2 2, . . . , u2 K)∆(u1, u2, . . . , uK)f(u1) · · · f(uK)du1 · · · duK

Vandermonde: ∆(u1, u2, . . . , uK) = det

• 1

u1 u2

1

· · · uK1

1

1 u2 u2

2

· · · uK1

2

. . . . . . . . . ... . . . 1 uK u2

K

· · · uK1

K

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∆(u2

1, u2 2, . . . , u2 K)∆(u1, u2, . . . , uK)

= X

σ∈SK

sgn(σ)

K

Y

i=1

u2σ(i)−2

i

! X

τ∈SK

sgn(τ)

K

Y

k=1

uτ(k)−1

k

! → K! X

σ∈SK

sgn(σ)u2σ(1)−2

1

u2σ(2)−1

2

u2σ(3)

3

· · · u2σ(K)+K−3

K

! = K! det(u2j+i−3

i

)K

i,j=1

For large N Z

SO(2N)

✓Λ0

A

ΛA (ea/N ◆K dAHaar → I · · · I K! det(u2j+i3

i

)K

i,j=1f(u1) · · · f(uK)du1 · · · duK

→ K! det ✓I f(ui)u2j+i3

i

dui ◆K

i,j=1

Emilia Alvarez (thesis)