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### Transcript of The Jensen-Pólya Program for the Riemann Hypothesis and ...math.gmu.edu/~wmorris/Jens...

• Hyperbolicity of Jensen polynomials

The Jensen-Pólya Program for the Riemann Hypothesis and Related

Problems

Ken Ono (U of Virginia)

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Riemann’s zeta-function

Definition (Riemann)

For Re(s) > 1, define the zeta-function by

ζ(s) :=

∞∑ n=1

1

ns .

Theorem (Fundamental Theorem) 1 The function ζ(s) has an analytic continuation to C (apart

from a simple pole at s = 1 with residue 1).

2 We have the functional equation

ζ(s) = 2sπs−1 sin (πs

2

) Γ(1− s) · ζ(1− s).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Riemann’s zeta-function

Definition (Riemann)

For Re(s) > 1, define the zeta-function by

ζ(s) :=

∞∑ n=1

1

ns .

Theorem (Fundamental Theorem) 1 The function ζ(s) has an analytic continuation to C (apart

from a simple pole at s = 1 with residue 1).

2 We have the functional equation

ζ(s) = 2sπs−1 sin (πs

2

) Γ(1− s) · ζ(1− s).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Riemann’s zeta-function

Definition (Riemann)

For Re(s) > 1, define the zeta-function by

ζ(s) :=

∞∑ n=1

1

ns .

Theorem (Fundamental Theorem) 1 The function ζ(s) has an analytic continuation to C (apart

from a simple pole at s = 1 with residue 1).

2 We have the functional equation

ζ(s) = 2sπs−1 sin (πs

2

) Γ(1− s) · ζ(1− s).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Hilbert’s 8th Problem

Conjecture (Riemann Hypothesis)

Apart from negative evens, the zeros of ζ(s) satisfy Re(s) = 12 .

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Hilbert’s 8th Problem

Conjecture (Riemann Hypothesis)

Apart from negative evens, the zeros of ζ(s) satisfy Re(s) = 12 .

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Important Remarks

Fact (Riemann’s Motivation)

Proposed RH because of Gauss’ Conjecture that π(X) ∼ XlogX .

What is known? 1 The first “gazillion” zeros satisfy RH (van de Lune, Odlyzko).

2 > 41% of zeros satisfy RH (Selberg, Levinson, Conrey,... ).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Important Remarks

Fact (Riemann’s Motivation)

Proposed RH because of Gauss’ Conjecture that π(X) ∼ XlogX .

What is known? 1 The first “gazillion” zeros satisfy RH (van de Lune, Odlyzko).

2 > 41% of zeros satisfy RH (Selberg, Levinson, Conrey,... ).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Important Remarks

Fact (Riemann’s Motivation)

Proposed RH because of Gauss’ Conjecture that π(X) ∼ XlogX .

What is known? 1 The first “gazillion” zeros satisfy RH (van de Lune, Odlyzko).

2 > 41% of zeros satisfy RH (Selberg, Levinson, Conrey,... ).

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen-Pólya Program

J. W. L. Jensen George Pólya (1859−1925) (1887−1985)

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen-Pólya Program

Definition The Riemann Xi-function is the entire function

Ξ(z) := 1

2

( −z2 − 1

4

) π iz 2 − 1

4 Γ

( − iz

2 +

1

4

) ζ

( −iz + 1

2

) .

Remark RH is true ⇐⇒ all of the zeros of Ξ(z) are purely real.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen-Pólya Program

Definition The Riemann Xi-function is the entire function

Ξ(z) := 1

2

( −z2 − 1

4

) π iz 2 − 1

4 Γ

( − iz

2 +

1

4

) ζ

( −iz + 1

2

) .

Remark RH is true ⇐⇒ all of the zeros of Ξ(z) are purely real.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Roots of Deg 100 Taylor Poly for Ξ (

1 2 + z

)

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Roots of Deg 200 Taylor Poly for Ξ (

1 2 + z

)

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Roots of Deg 400 Taylor Poly for Ξ (

1 2 + z

)

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Red points are good approximations of zeros of Ξ ( 1 2 + z

) .

The “spurious” blue points are annoying.

As d→ +∞ the spurious points become more prevalent.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Red points are good approximations of zeros of Ξ ( 1 2 + z

) .

The “spurious” blue points are annoying.

As d→ +∞ the spurious points become more prevalent.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Red points are good approximations of zeros of Ξ ( 1 2 + z

) .

The “spurious” blue points are annoying.

As d→ +∞ the spurious points become more prevalent.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen Polynomials

Definition (Jensen)

The degree d and shift n Jensen polynomial for an arithmetic function a : N 7→ R is

Jd,na (X) :=

d∑ j=0

a(n+ j)

( d j

) Xj

= a(n+ d)Xd + a(n+ d− 1)dXd−1 + · · ·+ a(n).

Definition A polynomial f ∈ R[X] is hyperbolic if all of its roots are real.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen Polynomials

Definition (Jensen)

The degree d and shift n Jensen polynomial for an arithmetic function a : N 7→ R is

Jd,na (X) :=

d∑ j=0

a(n+ j)

( d j

) Xj

= a(n+ d)Xd + a(n+ d− 1)dXd−1 + · · ·+ a(n).

Definition A polynomial f ∈ R[X] is hyperbolic if all of its roots are real.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen’s Criterion

Theorem (Jensen-Pólya (1927))

If Λ(s) := π−s/2Γ(s/2)ζ(s) = Λ(1− s),

then define γ(n) by

( −1 + 4z2

) Λ

( 1

2 + z

) = ∞∑ n=0

γ(n)

n! · z2n.

RH is equivalent to the hyperbolicity of all of the Jd,nγ (X).

What was known? The hyperbolicity for all n is known for d ≤ 3 by work of Csordas, Norfolk and Varga, and Dimitrov and Lucas.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen’s Criterion

Theorem (Jensen-Pólya (1927))

If Λ(s) := π−s/2Γ(s/2)ζ(s) = Λ(1− s), then define γ(n) by

( −1 + 4z2

) Λ

( 1

2 + z

) =

∞∑ n=0

γ(n)

n! · z2n.

RH is equivalent to the hyperbolicity of all of the Jd,nγ (X).

What was known? The hyperbolicity for all n is known for d ≤ 3 by work of Csordas, Norfolk and Varga, and Dimitrov and Lucas.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen’s Criterion

Theorem (Jensen-Pólya (1927))

If Λ(s) := π−s/2Γ(s/2)ζ(s) = Λ(1− s), then define γ(n) by

( −1 + 4z2

) Λ

( 1

2 + z

) =

∞∑ n=0

γ(n)

n! · z2n.

RH is equivalent to the hyperbolicity of all of the Jd,nγ (X).

What was known? The hyperbolicity for all n is known for d ≤ 3 by work of Csordas, Norfolk and Varga, and Dimitrov and Lucas.

Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

• Hyperbolicity of Jensen polynomials Introduction

Jensen’s Crit