probability and number theory: an overview of the erdos

Probability and Number Theory: an Overview ofthe Erdos-Kac Theorem

Alex Beckwith

December 12, 2012

Alex Beckwith

Probability and Number Theory: an Overview of the Erdos-Kac Theorem

Probabilistic number theory?

Pick n ∈ N with n ≤ 10, 000, 000 at random.

I How likely is it to be prime?

I How many prime divisors will it have?

Alex Beckwith


The prime divisor counting function ω

DefinitionThe function ω : N→ N defined by

ω(n) :=∑{p:p|n}

1

is called the prime divisor counting function; ω(n) yields thenumber of distinct prime divisors of n.

n prime factorization ω(n)

630

18722012

Alex Beckwith




ω(n) :=∑{p:p|n}

1



6 2 · 330

18722012

Alex Beckwith




ω(n) :=∑{p:p|n}

1



6 2 · 3 230

18722012

Alex Beckwith




ω(n) :=∑{p:p|n}

1



6 2 · 3 230 2 · 3 · 5 3

1872 24 · 32 · 13 32012 22 · 503 2

Alex Beckwith


An Illustration

n ∈ N

ω(n)

0

1

2

3

4

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 501 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Alex Beckwith


An Illustration

N ∈ N1 2 3

→

x

# {n ≤ N : ω(n) ≤ x}N

N = 50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

Alex Beckwith


The Erdos-Kac Theorem

TheoremLet N ∈ N. Then as N →∞,

νN

{n ≤ N :

ω(n)− log logN√log logN

≤ x

}= Φ(x).

That is, the limit distribution of the prime-divisor countingfunction ω(n) is the normal distribution with mean log logN andvariance log logN.

Alex Beckwith


An Illustration

x

# {n ≤ N : ω(n) ≤ x}N

N = 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 40

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

↓

x

# {n ≤ N : ω(n) ≤ x}N

N = 10000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 5000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 1000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 100

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

↓

x

# {n ≤ N : ω(n) ≤ x}N

N = 20000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 30000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 40000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

Alex Beckwith



Heuristically:

1. Most numbers near a fixed N ∈ N have log logN prime factors(Hardy and Ramanujan, Turan).

2. Most prime factors of most numbers near N are small.

3. The events “p divides n, with p a small prime, are roughlyindependent (Brun sieve).

4. If the events were exactly independent, a normal distributionwould result.

Alex Beckwith


Erdos-Kac vs. Central Limit Theorem

TheoremLet X1,X2, . . . be a sequence of independent and identicallydistributed random variables, each having mean µ and variance σ2.Then the distribution of

X1 + · · ·+ Xn − nµ

σ√n

tends to the standard normal as n→∞.

Alex Beckwith


Uniform probability law

By νN we denote the probability law of the uniform distributionwith weight 1

N on {1, 2, . . . ,N}. That is, for A ⊂ N,

νNA =∑n∈A

λn with λN =

{1N n ≤ N

0 n > N.

Alex Beckwith


Weak convergence

We say that a sequence {Fn} of distribution functions convergesweakly to a function F if

limn→∞

Fn(x) = F (x)

for all points where F is continuous.

Alex Beckwith


Limiting distributions

Let f be an arithmetic function. Let N ∈ N. Define

FN(z) := νN{n : f (n) ≤ z} =1

N#{n ≤ N : f (n) ≤ z}.

We say that f possess a limiting distribution function F if thesequence FN converges weakly to a limit F that is a distributionfunction.

x

# {n ≤ N : ω(n) ≤ x}N

N = 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 5000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

Alex Beckwith


Characteristic functions

DefinitionLet F be a distribution function. Then its characteristic function isgiven by

ϕF (τ) :=

∫ ∞−∞

exp(iτz)dF (z).

The characteristic function is uniformly continuous on the real line.

Fact: A distribution function is completely characterized by itscharacteristic function.

LemmaThe characteristic function of the standard normal distribution Φ isgiven by

ϕΦ(τ) = exp

(−τ

2

2

).

Alex Beckwith


Levy’s continuity theorem

TheoremLet {Fn} be a sequence of distribution functions and {ϕFn} be thecorresponding sequence of their characteristic functions. Then{Fn} converges weakly to a distribution function F if and only ifϕFn converges pointwise on R to a function ϕ that is continuous at0. Additionally, in this case, ϕ is the characteristic function of F .

Alex Beckwith



TheoremLet N ∈ N. Then as N →∞,

νN

{n ≤ N :


≤ x

}= Φ(x).

That is, the limit distribution of the prime-divisor countingfunction ω(n) is the normal distribution with mean log logN andvariance log logN.

Alex Beckwith


A proof sketch

The atomic distribution function for N ∈ N is

FN(x) = νN

{n ≤ N :


≤ x

}=

1

N#

{n ≤ N :


≤ x

}.

We denote by ϕFN(τ) the characteristic function of FN . We have

ϕFN(τ) =

∫ ∞−∞

e iτzdFN(z)

Let P = {· · · < x−1 < x0 < x1 < · · · < xi · · · } be a partition of R.Then we have ϕFN

(τ) equal to

Alex Beckwith


A proof sketch continued

=

∫ ∞−∞

e iτzdFN(z)

= limmesh(P)→0

∑k

e iτz (FN(xk)− FN(xk−1))

= limmesh(P)→0

∑k

e iτz(

1

N# {n ≤ N : f (n) ≤ xk} −

1

N# {n ≤ N : f (n) ≤ xk−1}

)

=1

N

[lim

mesh(P)→0

∑k

e iτz(

# {n ≤ N : f (n) ≤ xk} − # {n ≤ N : f (n) ≤ xk−1})]

=1

N

max{ω(n):n≤N}∑k=0

e iτ f (n)

=1

N

∑n≤N

e iτ f (n)

Alex Beckwith



Next, we find some bounds for ϕFN(τ):

ϕFN(τ) = exp

(−τ

2

2

)(1 + O

( |τ |+ |τ |3√log logN

))+ O

(1

logN

).

(Informally, we write f (x) = O(g(x)) when there exists a positivefunction g such that f does not grow faster than g .)

Take the limit as N →∞:

ϕFN(τ)→ exp

(−τ

2

2

)= ϕΦ(τ).

Alex Beckwith



Next, we find some bounds for ϕFN(τ):

ϕFN(τ) = exp

(−τ

2

2

)(1 + O

( |τ |+ |τ |3√log logN

))+ O

(1

logN

).

(Informally, we write f (x) = O(g(x)) when there exists a positivefunction g such that f does not grow faster than g .)Take the limit as N →∞:

ϕFN(τ)→ exp

(−τ

2

2

)= ϕΦ(τ).

Alex Beckwith



In other words, the sequence of characteristic functions ϕFN

converges pointwise to the characteristic function of the normaldistribution.

Apply Levy’s continuity theorem:

νN

{n ≤ N :


≤ x

}= Φ(x).

Thus, the limit distribution of the prime-divisor counting functionω(n) is the normal distribution with mean log logN and variancelog logN. This completes the proof.

Alex Beckwith



In other words, the sequence of characteristic functions ϕFN

converges pointwise to the characteristic function of the normaldistribution.Apply Levy’s continuity theorem:

νN

{n ≤ N :


≤ x

}= Φ(x).

Thus, the limit distribution of the prime-divisor counting functionω(n) is the normal distribution with mean log logN and variancelog logN. This completes the proof.

Alex Beckwith


An Illustration

x

# {n ≤ N : ω(n) ≤ x}N

N = 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 40

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

↓

x

# {n ≤ N : ω(n) ≤ x}N

N = 10000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 5000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 1000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

←x

# {n ≤ N : ω(n) ≤ x}N

N = 100

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

↓

x

# {n ≤ N : ω(n) ≤ x}N

N = 20000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 30000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 40000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

→x

# {n ≤ N : ω(n) ≤ x}N

N = 50000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

Alex Beckwith


References

Steuding, J. Probabilistic Number Theory 2002.

Gowers, T. The Importance of Mathematics.

Alex Beckwith


Alex Beckwith


probability and number theory: an overview of the erdos

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