probability and number theory: an overview of the erdos
TRANSCRIPT
![Page 1: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/1.jpg)
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
![Page 2: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/2.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 3: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/3.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 4: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/4.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 5: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/5.jpg)
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)
6 2 · 330
18722012
Alex Beckwith
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 6: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/6.jpg)
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)
6 2 · 3 230
18722012
Alex Beckwith
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 7: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/7.jpg)
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)
6 2 · 3 230 2 · 3 · 5 3
1872 24 · 32 · 13 32012 22 · 503 2
Alex Beckwith
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 8: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/8.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 9: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/9.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 10: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/10.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 11: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/11.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 12: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/12.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 13: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/13.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 14: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/14.jpg)
The Erdos-Kac Theorem
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 15: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/15.jpg)
The Erdos-Kac Theorem
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 16: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/16.jpg)
The Erdos-Kac Theorem
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 17: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/17.jpg)
The Erdos-Kac Theorem
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 18: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/18.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 19: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/19.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 20: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/20.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 21: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/21.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 22: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/22.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 23: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/23.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 24: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/24.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 25: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/25.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 26: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/26.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 27: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/27.jpg)
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
![Page 28: Probability and Number Theory: an Overview of the Erdos](https://reader030.vdocuments.mx/reader030/viewer/2022012801/61bd079761276e740b0ea243/html5/thumbnails/28.jpg)
A proof sketch
The atomic distribution function for N ∈ N is
FN(x) = νN
{n ≤ N :
ω(n)− log logN√log logN
≤ x
}=
1
N#
{n ≤ N :
ω(n)− log logN√log logN
≤ 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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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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)
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Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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A proof sketch continued
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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A proof sketch continued
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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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A proof sketch continued
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 :
ω(n)− log logN√log logN
≤ 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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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A proof sketch continued
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 :
ω(n)− log logN√log logN
≤ 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
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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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
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Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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References
Steuding, J. Probabilistic Number Theory 2002.
Gowers, T. The Importance of Mathematics.
Alex Beckwith
Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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Probability and Number Theory: an Overview of the Erdos-Kac Theorem
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Alex Beckwith
Probability and Number Theory: an Overview of the Erdos-Kac Theorem