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Transcript of The Jensen-Pأ³lya Program for the Riemann Hypothesis and ...math.gmu.edu/~wmorris/Jens آ ...

  • 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

    Takeaway about Taylor Polynomials

    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

    Takeaway about Taylor Polynomials

    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

    Takeaway about Taylor Polynomials

    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