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

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).
ζ(s) :=
∞∑ n=1
1
ns .
2
) Γ(1− s) · ζ(1− s).
ζ(s) :=
∞∑ n=1
1
ns .
2
) Γ(1− s) · ζ(1− s).
Hilbert’s 8th Problem
Conjecture (Riemann Hypothesis)
Apart from negative evens, the zeros of ζ(s) satisfy Re(s) = 12 .
Hilbert’s 8th Problem
Conjecture (Riemann Hypothesis)
Apart from negative evens, the zeros of ζ(s) satisfy Re(s) = 12 .
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,... ).
Important Remarks
Important Remarks
Jensen-Pólya Program
J. W. L. Jensen George Pólya (1859−1925) (1887−1985)
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.
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.
Roots of Deg 100 Taylor Poly for Ξ (
1 2 + z
)
1 2 + z
)
1 2 + z
)
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.
) .
) .
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.
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.
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.
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.
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.
Jensen’s Crit

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

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