A Dream About the Riemann Hypothesis
Julio Andrade
University of Bristol
Bristol, 6
th
Julio Andrade
October 2011
Number Theory, Physics and the Hilbert’s Dream
Introduction
We start with the Riemann zeta function:
ζ(s) =
∞
Y
X
1
=
(1 − p −s )−1
s
n
p
(R(s) > 1)
n=1
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Introduction
We start with the Riemann zeta function:
ζ(s) =
∞
Y
X
1
=
(1 − p −s )−1
s
n
p
(R(s) > 1)
n=1
Zeros of the Riemann zeta function
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Introduction
We start with the Riemann zeta function:
ζ(s) =
∞
Y
X
1
=
(1 − p −s )−1
s
n
p
(R(s) > 1)
n=1
Zeros of the Riemann zeta function
If s is a negative even integer so we have ζ(s) = 0 (trivial
zeros)
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Introduction
We start with the Riemann zeta function:
ζ(s) =
∞
Y
X
1
=
(1 − p −s )−1
s
n
p
(R(s) > 1)
n=1
Zeros of the Riemann zeta function
If s is a negative even integer so we have ζ(s) = 0 (trivial
zeros)
ζ(s) has infinitely many nontrivial zeros ρ = β + iγ in the
“critical strip” 0 ≤ σ ≤ 1.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Introduction
We start with the Riemann zeta function:
ζ(s) =
∞
Y
X
1
=
(1 − p −s )−1
s
n
p
(R(s) > 1)
n=1
Zeros of the Riemann zeta function
If s is a negative even integer so we have ζ(s) = 0 (trivial
zeros)
ζ(s) has infinitely many nontrivial zeros ρ = β + iγ in the
“critical strip” 0 ≤ σ ≤ 1.
Conjecture (The Riemann Hypothesis)
The nontrivial zeros of ζ(s) have real part equal to 21 .
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Hilbert’s Dream
Hilbert–Pólya Conjecture
Is the case that the imaginary parts t of the zeros
1
+ it
2
of the Riemann zeta function correspond to eigenvalues of an
unbounded self–adjoint operator (Hermitian Operator).
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Hilbert’s Dream
Hilbert–Pólya Conjecture
Is the case that the imaginary parts t of the zeros
1
+ it
2
of the Riemann zeta function correspond to eigenvalues of an
unbounded self–adjoint operator (Hermitian Operator).
Hilbert–Pólya operator: The operator is of the form 21 + iH,
where H is the Hamiltonian of a particle of mass m that is
moving under the influence of a potential V (x).
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Hilbert’s Dream
Hilbert–Pólya Conjecture
Is the case that the imaginary parts t of the zeros
1
+ it
2
of the Riemann zeta function correspond to eigenvalues of an
unbounded self–adjoint operator (Hermitian Operator).
Hilbert–Pólya operator: The operator is of the form 21 + iH,
where H is the Hamiltonian of a particle of mass m that is
moving under the influence of a potential V (x).
Riemann Hypothesis is equivalent to:
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Hilbert’s Dream
Hilbert–Pólya Conjecture
Is the case that the imaginary parts t of the zeros
1
+ it
2
of the Riemann zeta function correspond to eigenvalues of an
unbounded self–adjoint operator (Hermitian Operator).
Hilbert–Pólya operator: The operator is of the form 21 + iH,
where H is the Hamiltonian of a particle of mass m that is
moving under the influence of a potential V (x).
Riemann Hypothesis is equivalent to:
The Hamiltonian H is Hermitian.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Hilbert’s Dream
Hilbert–Pólya Conjecture
Is the case that the imaginary parts t of the zeros
1
+ it
2
of the Riemann zeta function correspond to eigenvalues of an
unbounded self–adjoint operator (Hermitian Operator).
Hilbert–Pólya operator: The operator is of the form 21 + iH,
where H is the Hamiltonian of a particle of mass m that is
moving under the influence of a potential V (x).
Riemann Hypothesis is equivalent to:
The Hamiltonian H is Hermitian.
V is real.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Quantum Gravity
Quantum gravity is the field of theoretical physics which attempts
to develop scientific models that unify:
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Quantum Gravity
Quantum gravity is the field of theoretical physics which attempts
to develop scientific models that unify:
QUANTUM MECHANICS ⇔ GENERAL RELATIVITY.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Quantum Gravity
Quantum gravity is the field of theoretical physics which attempts
to develop scientific models that unify:
QUANTUM MECHANICS ⇔ GENERAL RELATIVITY.
Candidate Theories for Quantum Gravity:
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Quantum Gravity
Quantum gravity is the field of theoretical physics which attempts
to develop scientific models that unify:
QUANTUM MECHANICS ⇔ GENERAL RELATIVITY.
Candidate Theories for Quantum Gravity:
String Theory.
Loop Quantum Gravity.
Supergravity.
Twistor Models.
Noncommutative Geometry.
...
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Quantum Gravity −→ Does this theory really exist?
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Quantum Gravity −→ Does this theory really exist?
If Yes −→ We can compute the Hamiltonian H of this Q.G.
Theory.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Quantum Gravity −→ Does this theory really exist?
If Yes −→ We can compute the Hamiltonian H of this Q.G.
Theory.
H will be a very fundamental Operator in this case −→ And
due to H be so fundamental WE EXPECT that will be a
Self-Adjoint operator.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Quantum Gravity −→ Does this theory really exist?
If Yes −→ We can compute the Hamiltonian H of this Q.G.
Theory.
H will be a very fundamental Operator in this case −→ And
due to H be so fundamental WE EXPECT that will be a
Self-Adjoint operator.
So MAYBE H is related with Prime Numbers and/or
Number Theory.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Julio’s Dream
Quantum Gravity −→ Does this theory really exist?
If Yes −→ We can compute the Hamiltonian H of this Q.G.
Theory.
H will be a very fundamental Operator in this case −→ And
due to H be so fundamental WE EXPECT that will be a
Self-Adjoint operator.
So MAYBE H is related with Prime Numbers and/or
Number Theory.
And we EXPECT in this case that the eigenvalues of H
correspond to zeros of ζ(s)
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Homework:
1
Construct a Quantum Gravity Theory mathematically
rigorous. (This will make you one of the most important
physicist).
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Homework:
1
Construct a Quantum Gravity Theory mathematically
rigorous. (This will make you one of the most important
physicist).
2
Compute the Hamiltonian of this New Theory. (Or if you
prefer the Hamiltonian of the Universe)
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Homework:
1
Construct a Quantum Gravity Theory mathematically
rigorous. (This will make you one of the most important
physicist).
2
Compute the Hamiltonian of this New Theory. (Or if you
prefer the Hamiltonian of the Universe)
3
Check if the eigenvalues of this Hamiltonian correspond to
the zeros of ζ(s).
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
Homework:
1
Construct a Quantum Gravity Theory mathematically
rigorous. (This will make you one of the most important
physicist).
2
Compute the Hamiltonian of this New Theory. (Or if you
prefer the Hamiltonian of the Universe)
3
Check if the eigenvalues of this Hamiltonian correspond to
the zeros of ζ(s).
If the answer was yes, you should receive a phone call for the
next Fields Medal and a prize of one million dollars. And I
personally guarantee that you will get a job position
wherever you want.
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
To Remember
We can ask if there is some relation between:
RIEMANN HYPOTHESIS ←→ QUANTUM GRAVITY
Zeros of Riemann Zeta Function are Eigenvalues of some
Gravity Hamiltonian?
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream
To Remember
We can ask if there is some relation between:
RIEMANN HYPOTHESIS ←→ QUANTUM GRAVITY
Zeros of Riemann Zeta Function are Eigenvalues of some
Gravity Hamiltonian?
I’ll stop here. Thank you for your attention
Julio Andrade
Number Theory, Physics and the Hilbert’s Dream