From Nuclear Physics to Number Theory How the Manhattan Project helped us understand primes

Steven J. Miller Brown University Providence, February 12^{th}, 2005

http://www.math.brown.edu/∼sjmiller

Acknowledgements

Random Matrices (with Peter Sarnak)

• Rebecca Lehman

• Yi-Kai Liu

• Chris Hammond

• Inna Zakharevich

Random Graphs (with Peter Sarnak, Yakov Sinai)

• Dustin Steinhauer

• Randy Qian

• Felice Kuan

• Leo Goldmakher

Fundamental Problem: Spacing Between Events

General Formulation: Studying system, observe values at t_{1}, t_{2}, t_{3}, . . ..

Question: what rules govern the spacings between the t_{i}?

Fundamental Problem: Spacing Between Events

General Formulation: Studying system, observe values at t_{1}, t_{2}, t_{3}, . . .. Question: what rules govern the spacings between the t_{i}? Examples:

• Spacings b/w Primes.

• Spacings b/w Energy Levels of Nuclei.

• Spacings b/w Eigenvalues of Matrices.

• Spacings b/w Zeros of Functions.

Goals of the Talk

• Determine correct scale to study spacings.

• See similar behavior in different systems.

• Discuss tools / techniques needed to prove the results.

PART I

NORMALIZED SPACINGS

Normalized Spacing

Example: Fractional Parts For α ∉ ℚ, set x_{n} = nα mod 1. Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1.

Normalized Spacing

Example: Fractional Parts For α ∉ ℚ, set x_{n} = nα mod 1. Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1. Expect spacings between adjacent y’s of size \frac{1}{N}.

Normalized Spacing

Example: Fractional Parts For α ∉ ℚ, set x_{n} = nα mod 1. Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1. Expect spacings between adjacent y’s of size \frac{1}{N}. Should study \frac{y_{n+1}−y_{n}}{1/N}.

Normalized Spacing

Example: Primes π(x) = #\{p : p prime, p ≤ x\} ≈ \frac{x}{log x} (Ave Spacing b/w Primes at mostx) = \frac{x}{π(x)} ≈ log x.

Normalized Spacing

Example: Primes π(x) = #\{p : p prime, p ≤ x\} ≈ \frac{x}{log x} (Ave Spacing b/w Primes at mostx) = \frac{x}{π(x)} ≈ log x. If p_{n}, p_{n+1} ≈ x, study \frac{p_{n+1}−p_{n}}{log x}.

Normalized Spacing

Example: Primes π(x) = #\{p : p prime, p ≤ x\} ≈ \frac{x}{log x} (Ave Spacing b/w Primes at mostx) = \frac{x}{π(x)} ≈ log x. If p_{n}, p_{n+1} ≈ x, study \frac{p_{n+1}−p_{n}}{log x}. One reason why twin primes are so hard: \frac{2}{log x} → 0.

Normalized Spacing

Example: Primes π(x) = #\{p : p prime, p ≤ x\} ≈ \frac{x}{log x} (Ave Spacing b/w Primes at mostx) = \frac{x}{π(x)} ≈ log x. If p_{n}, p_{n+1} ≈ x, study \frac{p_{n+1}−p_{n}}{log x}. One reason why twin primes are so hard: \frac{2}{log x} → 0.

Aside: Twin primes led to finding the pentium bug!

PART II

PROBABILITY AND LINEAR ALGEBRA REVIEW

Probability Review

Let p(x) be a probability density:

p(x) ≥ 0
∫_{−∞}^{∞} p(x)dx = 1.

Thus

Prob(x ∈ [a, b]) = ∫_{a}^{b} p(x)dx.

Probability Review

Let p(x) be a probability density:

p(x) ≥ 0
∫_{−∞}^{∞} p(x)dx = 1.

Thus

Prob(x ∈ [a, b]) = ∫_{a}^{b} p(x)dx.

Moments:

k^{th}-moment = ∫_{−∞}^{∞} x^{k}p(x)dx.

Good function uniquely determined by its Taylor Series, a good probability density is uniquely determined by its moments.

Probability Review (cont)

Important quantities:

1. Mean μ = ∫ xp(x)dx.

2. Variance σ^{2} = ∫ (x − μ)^{2}p(x)dx.

Probability Review (cont)

Important quantities:

1. Mean μ = ∫ xp(x)dx.

2. Variance σ^{2} = ∫ (x − μ)^{2}p(x)dx.

Uniform: μ = 0, σ^{2} = \frac{1}{12} Gaussian: μ = 0, σ^{2} = 1 Delta Spike: μ = 1, σ^{2} = 0

p(x) = \begin{cases}1 & if |x| ≤ \frac{1}{2} \\ 0 & otherwise \end{cases}  p(x) = \frac{1}{\sqrt{2π}}e^{−x^{2}/2}  δ(x − 1)

Linear Algebra Review

( \begin{matrix}a_{11} & ⋯ & a_{1N} \\ ⋮ & ... & ⋮ \\ a_{N1} & ⋯ & a_{NN} \end{matrix} ) ( v_{1} ; ⋮ ; v_{N} ) = ( w_{1} ; ⋮ ; w_{N} )

In general, in A\overrightarrow{v} = \overrightarrow{w}, \overrightarrow{w} will have different magnitude and direction than \overrightarrow{v}.

( \begin{matrix}a_{11} & ⋯ & a_{1N} \\ ⋮ & ... & ⋮ \\ a_{N1} & ⋯ & a_{NN} \end{matrix} ) ( v_{1} ; ⋮ ; v_{N} ) = ( w_{1} ; ⋮ ; w_{N} )

In general, in A\overrightarrow{v} = \overrightarrow{w}, \overrightarrow{w} will have different magnitude and direction than \overrightarrow{v}.

\overrightarrow{v} ≠ \overrightarrow{0} is an eigenvector with eigenvalue λ if

A\overrightarrow{v} = λ\overrightarrow{v}.

( \begin{matrix}a_{11} & ⋯ & a_{1N} \\ ⋮ & ... & ⋮ \\ a_{N1} & ⋯ & a_{NN} \end{matrix} ) ( v_{1} ; ⋮ ; v_{N} ) = ( w_{1} ; ⋮ ; w_{N} )

In general, in A\overrightarrow{v} = \overrightarrow{w}, \overrightarrow{w} will have different magnitude and direction than \overrightarrow{v}. \overrightarrow{v} ≠ \overrightarrow{0} is an eigenvector with eigenvalue λ if

A\overrightarrow{v} = λ\overrightarrow{v}.

Note

A^{2}\overrightarrow{v} = A(A\overrightarrow{v}) = A(λ\overrightarrow{v}) = λ^{2}\overrightarrow{v}.

Linear Algebra Review (cont)

Say \overrightarrow{v_{i}} eigenvectors with eigenvalues λ_{i}. Assume

\overrightarrow{v} = c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}}.

Linear Algebra Review (cont)

Say \overrightarrow{v_{i}} eigenvectors with eigenvalues λ_{i}. Assume

\overrightarrow{v} = c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}}.

Then

A^{m}\overrightarrow{v} = A^{m}(c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}})

Linear Algebra Review (cont)

Say \overrightarrow{v_{i}} eigenvectors with eigenvalues λ_{i}. Assume

\overrightarrow{v} = c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}}.

Then

A^{m}\overrightarrow{v} = A^{m}(c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}})
= A^{m}(c_{1}\overrightarrow{v_{1}}) + ⋯ + A^{m}(c_{k}\overrightarrow{v_{k}})

Linear Algebra Review (cont)

Say \overrightarrow{v_{i}} eigenvectors with eigenvalues λ_{i}. Assume

\overrightarrow{v} = c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}}.

Then

A^{m}\overrightarrow{v} = A^{m}(c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}})
= A^{m}(c_{1}\overrightarrow{v_{1}}) + ⋯ + A^{m}(c_{k}\overrightarrow{v_{k}})
= c_{1}A^{m}\overrightarrow{v_{1}} + ⋯ + c_{k}A^{m}\overrightarrow{v_{k}}

Linear Algebra Review (cont)

Say \overrightarrow{v_{i}} eigenvectors with eigenvalues λ_{i}. Assume

\overrightarrow{v} = c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}}.

Then

A^{m}\overrightarrow{v} = A^{m}(c_{1}\overrightarrow{v_{1}} + ⋯ + c_{k}\overrightarrow{v_{k}})
= A^{m}(c_{1}\overrightarrow{v_{1}}) + ⋯ + A^{m}(c_{k}\overrightarrow{v_{k}})
= c_{1}A^{m}\overrightarrow{v_{1}} + ⋯ + c_{k}A^{m}\overrightarrow{v_{k}}
= c_{1}λ_{1}^{m}\overrightarrow{v_{1}} + ⋯ + c_{k}λ_{k}^{m}\overrightarrow{v_{k}}.

PART III

RANDOM MATRIX THEORY

Origins of Random Matrix Theory

Classical Mechanics: 3 Body Problem Intractable.

Origins of Random Matrix Theory

Classical Mechanics: 3 Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out.

Origins of Random Matrix Theory

Classical Mechanics: 3 Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out.

Fundamental Equation:

Hψ_{n} = E_{n}ψ_{n}

H : matrix, entries depend on system E_{n} : are the energy levels ψ_{n} : are the energy eigenfunctions

Origins (cont)

Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations – most configurations close to system average.

Origins (cont)

Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations – most configurations close to system average. Nuclear physics: choose matrix at random, calculate eigenvalues, average over matrices. Look at: Real Symmetric (A = A^{T}), Complex Hermitian (\overline{A}^{T} = A).

Random Matrix Ensembles

Real Symmetric Matrices:

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{12} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{1N} & a_{2N} & a_{3N} & ⋯ & a_{NN} \end{matrix} ) = A^{T} , a_{ij} = a_{ji}

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{12} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{1N} & a_{2N} & a_{3N} & ⋯ & a_{NN} \end{matrix} ) = A^{T} , a_{ij} = a_{ji}

Fix p, define

Prob(A) = ∏_{1≤i≤j≤N} p(a_{ij}).

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{12} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{1N} & a_{2N} & a_{3N} & ⋯ & a_{NN} \end{matrix} ) = A^{T} , a_{ij} = a_{ji}

Fix p, define

Prob(A) = ∏_{1≤i≤j≤N} p(a_{ij}).

This means

Prob(A : a_{ij} ∈ [α_{ij}, β_{ij}]) = ∏_{1≤i≤j≤N} ∫_{x_{ij}=α_{ij}}^{β_{ij}} p(x_{ij})dx_{ij}.

Want to understand eigenvalues of A.

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}.

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}. To each A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}})

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}. To each A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}})

Equivalently

∫_{a}^{b} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [a, b]\}}{N}

Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x).

Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x).

Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then for almost all A, as N → ∞

μ_{A,N}(x) ⟶ \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

Random Matrix Theory: Semi-Circle Law

500 Matrices: Gaussian 400 × 400

p(x) = \frac{1}{\sqrt{2π}} e^{−x^{2}/2} Random Matrix Theory: Semi-Circle Law

Cauchy Distr: p(x) = \frac{1}{π(1+x^{2})}

GOE Conjecture

GOE Conjecture: As N → ∞, the probability density of the spacing b/w consecutive normalized eigenvalues approaches a limit independent of p.

GOE Conjecture

GOE Conjecture: As N → ∞, the probability density of the spacing b/w consecutive normalized eigenvalues approaches a limit independent of p. Only known if p is a Gaussian.

GOE Conjecture

GOE Conjecture: As N → ∞, the probability density of the spacing b/w consecutive normalized eigenvalues approaches a limit independent of p. Only known if p is a Gaussian. GOE(x) ≈ Axe^{−Bx^{2}}.

Uniform Distribution: p(x) = \frac{1}{2} for |x| ≤ 1

5000: 300 × 300 uniform on [−1, 1] Cauchy Distribution: p(x) = \frac{1}{π(1+x^{2})}

5000: 100 × 100 Cauchy

5000: 300 × 300 Cauchy

Poisson Distribution: p(n) = \frac{λ^{n}}{n!} e^{−λ}

5000: 300 × 300 Poisson, λ = 5

5000: 300 × 300 Poisson, λ = 20

Fat Thin Families

Need a family F A T enough to do averaging.

Fat Thin Families

Need a family F A T enough to do averaging. Need a family THIN enough so that everything isn’t averaged out.

Fat Thin Families

Need a family F A T enough to do averaging. Need a family THIN enough so that everything isn’t averaged out. Real Symmetric Matrices have \frac{N(N+1)}{2} independent entries.

Random Graphs

Degree of a vertex = number of edges leaving the vertex.

Random Graphs

Degree of a vertex = number of edges leaving the vertex. Adjacency matrix: a_{ij} = number edges b/w Vertex i and Vertex j.

A = ( \begin{matrix}0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 2 & 0 \end{matrix} )

These are Real Symmetric Matrices.

McKay’s Law

Density of Eigenvalues for d-regular graphs

f(x) = \begin{cases}\sqrt[\frac{d}{2π(d^{2}−x^{2})}]{4(d − 1) − x^{2}} & |x| ≤ 2\sqrt{d − 1} \\ 0 & otherwise. \end{cases}

d = 3.

McKay’s Law

d = 6.

Fat Thin: fat enough to average, thin enough to get something different than Semi-circle.

3-Regular, 2000 Vertices and GOE

55

PART IV NUMBER THEORY

Infinitude of Primes

Theorem (Euclid): There are infinitely many primes.

Proof: Assume not: p_{1}, …, p_{N}.

Proof: Assume not: p_{1}, …, p_{N}.

Look at p_{1} ⋯ p_{N} + 1.

Infinitude of Primes

Theorem (Euclid): There are infinitely many primes.

Proof: Assume not: p_{1}, …, p_{N}. Look at p_{1} ⋯ p_{N} + 1. Gives weak bound for number of primes at most x.

Riemann Zeta Function

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Geometric Series (and Extending Functions): If |u| < 1,

∑_{k=0}^{∞} u^{k} = 1 + u + u^{2} + u^{3} + ⋯ = \frac{1}{1 − u}

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Geometric Series (and Extending Functions): If |u| < 1,

∑_{k=0}^{∞} u^{k} = 1 + u + u^{2} + u^{3} + ⋯ = \frac{1}{1 − u}

Unique Factorization: n = p_{1}^{r_{1}} ⋯ p_{m}^{r_{m}}.

∏_{p} (1 − \frac{1}{p^{s}})^{−1} = [1 + \frac{1}{2^{s}} + (\frac{1}{2^{s}})^{2} + ⋯] [1 + \frac{1}{3^{s}} + (\frac{1}{3^{s}})^{2} + ⋯] ⋯

= ∑_{n} \frac{1}{n^{s}}.

Riemann Zeta Function (cont):

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Riemann Zeta Function (cont):

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes:

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes:

• lim_{s→1^{+}} ζ(s) = ∞, π(x) → ∞.

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes:

• lim_{s→1^{+}} ζ(s) = ∞, π(x) → ∞.

• ζ(2) = \frac{π^{2}}{6}, π(x) → ∞.

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = Γ(\frac{s}{2})π^{−\frac{s}{2}}ζ(s) = ξ(1 − s).

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = Γ(\frac{s}{2})π^{−\frac{s}{2}}ζ(s) = ξ(1 − s).

Riemann Hypothesis:

All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ.

Riemann Zeta Function (cont):

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = Γ(\frac{s}{2})π^{−\frac{s}{2}}ζ(s) = ξ(1 − s).

Riemann Hypothesis:

All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ.

Observation:

Spacings b/w zeros appear same as b/w eigenvalues of Complex Hermitian matrices(\overline{A}^{T} = A).

Zeros of ζ(s) vs GUE

70 million spacings b/w adjacent zeros of ζ(s), starting at the 10^{20}^{th} zero (from Odlyzko)

SKETCH OF PROOF: Eigenvalue Trace Lemma

Trace(A) = a_{11} + a_{22} + ⋯ + a_{NN}.

PART V SKETCH OF PROOFS THEOREM: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

THEOREM: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

k = 1 Case: Say \overrightarrow{v} ≠ \overrightarrow{0} is an eigenvector with eigenvalue λ:

A\overrightarrow{v} = λ\overrightarrow{v} ⇔ (λI − A)\overrightarrow{v} = \overrightarrow{0}.

THEOREM: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

k = 1 Case: Say \overrightarrow{v} ≠ \overrightarrow{0} is an eigenvector with eigenvalue λ: A\overrightarrow{v} = λ\overrightarrow{v} ⇔ (λI − A)\overrightarrow{v} = \overrightarrow{0}.

Means λI − A isn’t invertible, so p(λ) = det(λI − A) = 0.

THEOREM: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

k = 1 Case: Say \overrightarrow{v} ≠ \overrightarrow{0} is an eigenvector with eigenvalue λ:

A\overrightarrow{v} = λ\overrightarrow{v} ⇔ (λI − A)\overrightarrow{v} = \overrightarrow{0}.

Means λI − A isn’t invertible, so p(λ) = det(λI − A) = 0.

Write p(λ) = (λ−λ_{1}) ⋯ (λ−λ_{N}), compare coefficients with det(λI −A).

SKETCH OF PROOF: Probability Review

Probability density p of mean μ = 0, variance σ^{2} = 1:

μ = ∫_{−∞}^{∞} xp(x)dx
σ^{2} = ∫_{−∞}^{∞}(x − μ)^{2}p(x)dx.

SKETCH OF PROOF: Probability Review

Probability density p of mean μ = 0, variance σ^{2} = 1:

μ = ∫_{−∞}^{∞} xp(x)dx
σ^{2} = ∫_{−∞}^{∞}(x − μ)^{2}p(x)dx.

As σ^{2} = 1, expect x^{2} ≈ 1.

SKETCH OF PROOF: Correct Scale

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}

∑_{i=1}^{N} λ_{i}(A)^{2} ∼ N^{2}

SKETCH OF PROOF: Correct Scale

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}
∑_{i=1}^{N} λ_{i}(A)^{2} ∼ N^{2}

Gives NAve(λ_{i}(A)^{2}) ∼ N^{2} SKETCH OF PROOF: Correct Scale

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}
∑_{i=1}^{N} λ_{i}(A)^{2} ∼ N^{2}

Gives NAve(λ_{i}(A)^{2}) ∼ N^{2} or Ave(λ_{i}(A)) ∼ \sqrt{N}.

SKETCH OF PROOF: Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}.

SKETCH OF PROOF: Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}.

To each A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}})

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}})

Equivalently

∫_{a}^{b} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [a, b]\}}{N}

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}})

Obtain:

k^{th}-moment = ∫_{−∞}^{∞} x^{k}μ_{A,N}(x)dx = \frac{1}{N} ∑_{i=1}^{N} \frac{λ_{i}(A)^{k}}{(2\sqrt{N})^{k}} = \frac{Trace(A^{k})}{2^{k}N^{\frac{k}{2}+1}}

SKETCH OF PROOF: Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x).

Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then for almost all A, as N → ∞

μ_{A,N}(x) ⟶ \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

SKETCH OF PROOF: Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x).

Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then for almost all A, as N → ∞

μ_{A,N}(x) ⟶ \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

Trace formula converts sums over eigenvalues to sums over entries of A.

SKETCH OF PROOF: Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x). Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then for almost all A, as N → ∞

μ_{A,N}(x) ⟶ \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

Trace formula converts sums over eigenvalues to sums over entries of A. Expected value of k^{th}-moment of μ_{A,N}(x) is

∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} \frac{Trace(A^{k})}{2^{k}N^{\frac{k}{2}+1}} ∏_{i≤j} p(a_{ij})da_{ij}

SKETCH OF PROOF: 2^{nd}-Moment

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2}.

SKETCH OF PROOF: 2^{nd}-Moment

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2}.

Substituting into expansion gives

\frac{1}{2^{2}N^{2}} ∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ji}^{2} · p(a_{11})da_{11} ⋯ p(a_{NN})da_{NN}

SKETCH OF PROOF: 2^{nd}-Moment

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2}.

Substituting into expansion gives

\frac{1}{2^{2}N^{2}} ∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ji}^{2} · p(a_{11})da_{11} ⋯ p(a_{NN})da_{NN}

Integration factors as

∫_{a_{ij}=−∞}^{∞} a_{ij}^{2}p(a_{ij})da_{ij} · ∏_{{(k,l)≠(ij) \atop k<l}} ∫_{a_{kl}=−∞}^{∞} p(a_{kl})da_{kl} = 1.

Have N^{2} summands, answer is \frac{1}{4}.

SKETCH OF PROOF: Zero Knowledge (Heuristic)

P (x) polynomial, zeros r_{1}, …, r_{n}.

SKETCH OF PROOF: Zero Knowledge (Heuristic)

P (x) polynomial, zeros r_{1}, …, r_{n}. Then

P (x) = (x − r_{1})(x − r_{2}) ⋯ (x − r_{n})
= x^{n} + a_{n−1}(r_{1}, …, r_{n})x^{n−1} + ⋯ + a_{0}(r_{1}, …, r_{n})

P (x) = (x − r_{1})(x − r_{2}) ⋯ (x − r_{n})
= x^{n} + a_{n−1}(r_{1}, …, r_{n})x^{n−1} + ⋯ + a_{0}(r_{1}, …, r_{n})

where

a_{n−1}(r_{1}, …, r_{n}) = −(r_{1} + ⋯ + r_{n})
⋮
a_{0}(r_{1}, …, r_{n}) = r_{1}r_{2} ⋯ r_{n}.

Knowledge of zeros gives info on coefficients.

PART VI CONCLUSIONS

Correspondences Similarities b/w Nuclei and Primes:

Energy Levels ⟷ Zeros of ζ(s)

Correspondences Similarities b/w Nuclei and Primes:

Energy Levels ⟷ Zeros of ζ(s)
Neutron Energy ⟷ Summation Lemmas

Correspondences Similarities b/w Nuclei and Primes:

Energy Levels ⟷ Zeros of ζ(s)
Neutron Energy ⟷ Summation Lemmas
Different Elements: U, Pu, ... ⟷ Different L-Functions

Summary

• Similar behavior in different systems.

Summary

• Similar behavior in different systems.

• Find correct scale.

Summary

• Similar behavior in different systems.

• Find correct scale.

• Average over similar elements.

Summary

• Similar behavior in different systems.

• Find correct scale.

• Average over similar elements.

• Need a Trace Lemma.

Summary

• Similar behavior in different systems.

• Find correct scale.

• Average over similar elements.

• Need a Trace Lemma.

• Thin subsets can exhibit very different behavior.

Open Problems

Real Symmetric Band Matrices

( \begin{matrix}a_{1,1} & a_{1,2} & 0 & ⋯ & 0 & 0 & 0 \\ a_{1,2} & a_{2,2} & a_{2,3} & ⋯ & 0 & 0 & 0 \\ 0 & a_{2,3} & a_{3,3} & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & a_{N−2,N−2} & a_{N−2,N} & 0 \\ 0 & 0 & 0 & ⋯ & a_{N−2,N} & a_{N−1,N−1} & a_{N−1,N} \\ 0 & 0 & 0 & ⋯ & 0 & a_{N−1,N} & a_{N,N} \end{matrix} )

( \begin{matrix}a_{1,1} & a_{1,2} & 0 & ⋯ & 0 & 0 & 0 \\ a_{1,2} & a_{2,2} & a_{2,3} & ⋯ & 0 & 0 & 0 \\ 0 & a_{2,3} & a_{3,3} & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & a_{N−2,N−2} & a_{N−2,N} & 0 \\ 0 & 0 & 0 & ⋯ & a_{N−2,N} & a_{N−1,N−1} & a_{N−1,N} \\ 0 & 0 & 0 & ⋯ & 0 & a_{N−1,N} & a_{N,N} \end{matrix} )

Real Symmetric Toeplitz Matrices

( \begin{matrix}b_{0} & b_{1} & b_{2} & ⋯ & b_{N−1} \\ b_{1} & b_{0} & b_{1} & ⋯ & b_{N−2} \\ b_{2} & b_{1} & b_{0} & ⋯ & b_{N−3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ b_{N−1} & b_{N−2} & b_{N−3} & ⋯ & b_{0} \end{matrix} )

( \begin{matrix}a_{1,1} & a_{1,2} & 0 & ⋯ & 0 & 0 & 0 \\ a_{1,2} & a_{2,2} & a_{2,3} & ⋯ & 0 & 0 & 0 \\ 0 & a_{2,3} & a_{3,3} & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & a_{N−2,N−2} & a_{N−2,N} & 0 \\ 0 & 0 & 0 & ⋯ & a_{N−2,N} & a_{N−1,N−1} & a_{N−1,N} \\ 0 & 0 & 0 & ⋯ & 0 & a_{N−1,N} & a_{N,N} \end{matrix} )

Real Symmetric Toeplitz Matrices

( \begin{matrix}b_{0} & b_{1} & b_{2} & ⋯ & b_{N−1} \\ b_{1} & b_{0} & b_{1} & ⋯ & b_{N−2} \\ b_{2} & b_{1} & b_{0} & ⋯ & b_{N−3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ b_{N−1} & b_{N−2} & b_{N−3} & ⋯ & b_{0} \end{matrix} )

Rates of Convergence APPENDIX: Contour Integration and L-Fns

109

Zero Knowledge: (Contour Integration)

−\frac{ζ′(s)}{ζ(s)} = −\frac{d}{ds} log ζ(s)
= \frac{d}{ds} ∑_{p} log (1 − p^{−s})
= ∑_{p} \frac{log p · p^{−s}}{1 − p^{−s}}
= ∑_{p} \frac{log p}{p^{s}} + Good(s).

Contour Integration:

∫ −\frac{ζ′(s)}{ζ(s)}\frac{x^{s}}{s}ds vs ∑_{p} log p ∫ (\frac{x}{p})^{s} \frac{ds}{s}.

Knowledge of zeros gives info on coefficients.

Families of L-Functions

More generally, we may consider an L-function

L(s, f) = ∑_{n=1}^{∞} \frac{a_{n}(f)}{n^{s}} = ∏_{p} L_{p}(p^{−s}, f)^{−1}, Re(s) > s_{0}.

Examples:

• Dirichlet Characters: a_{n}(f) = χ_{f}(n).

• Elliptic Curves: y^{2} = x^{3} + A_{f}x + B_{f}, a_{p}(f) is related to number of solns mod p.

General Riemann Hypothesis: All L-functions (after normalization) have their zeros on the critical line.