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

Transcription

1 Probability and umber Theory: an Overview of the Erdős-Kac Theorem December 12, 2012 Probability and umber Theory: an Overview of the Erdős-Kac Theorem

2 Probabilistic number theory? Pick n with n 10, 000, 000 at random. How likely is it to be prime? How many prime divisors will it have? Probability and umber Theory: an Overview of the Erdős-Kac Theorem

3 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. 1 Probability and umber Theory: an Overview of the Erdős-Kac Theorem

4 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem

5 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem

6 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem

7 The prime divisor counting function ω Definition The function ω : defined by ω(n) := {p:p n} is called the prime divisor counting function; ω(n) yields the number of distinct prime divisors of n. n prime factorization ω(n) Probability and umber Theory: an Overview of the Erdős-Kac Theorem

8 An Illustration ω(n) n Probability and umber Theory: an Overview of the Erdős-Kac Theorem

9 An Illustration 1 2 3

10 An Illustration # {n : ω(n) } = Probability and umber Theory: an Overview of the Erdős-Kac Theorem

11 The Erdős-Kac Theorem Theorem Let. Then as, { ω(n) log log ν n : log log } = Φ(). That is, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

12 An Illustration Probability and umber Theory: an Overview of the Erdős-Kac Theorem

13 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } An Illustration = 10 = 20 = 30 = 40 = 50 = = 5000 = 1000 = 500 = 100 = = = = Probability and umber Theory: an Overview of the Erdős-Kac Theorem

14 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). Probability and umber Theory: an Overview of the Erdős-Kac Theorem

15 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

16 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. 3. The events p divides n, with p a small prime, are roughly independent (Brun sieve). Probability and umber Theory: an Overview of the Erdős-Kac Theorem

17 The Erdős-Kac Theorem Heuristically: 1. Most numbers near a fied have log log prime factors (Hardy and Ramanujan, Turán). 2. Most prime factors of most numbers near are small. 3. The events p divides n, with p a small prime, are roughly independent (Brun sieve). 4. If the events were eactly independent, a normal distribution would result. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

18 Erdős-Kac vs. Central Limit Theorem Theorem Let X 1, X 2,... be a sequence of independent and identically distributed random variables, each having mean µ and variance σ 2. Then the distribution of X X n nµ σ n tends to the standard normal as n. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

19 Uniform probability law By ν we denote the probability law of the uniform distribution with weight 1 on {1, 2,..., }. That is, for A, ν A = { 1 λ n with λ = n 0 n >. n A Probability and umber Theory: an Overview of the Erdős-Kac Theorem

20 Weak convergence We say that a sequence {F n } of distribution functions converges weakly to a function F if lim F n() = F () n for all points where F is continuous. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

21 Limiting distributions Let f be an arithmetic function. Let. Define F (z) := ν {n : f (n) z} = 1 # {n : f (n) z}. We say that f possess a limiting distribution function F if the sequence F converges weakly to a limit F that is a distribution function. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

22 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } Limiting distributions Let f be an arithmetic function. Let. Define F (z) := ν {n : f (n) z} = 1 # {n : f (n) z}. We say that f possess a limiting distribution function F if the sequence F converges weakly to a limit F that is a distribution function. = 10 = 50 = 500 = 5000 = Probability and umber Theory: an Overview of the Erdős-Kac Theorem

23 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

24 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Fact: A distribution function is completely characterized by its characteristic function. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

25 Characteristic functions Definition Let F be a distribution function. Then its characteristic function is given by ϕ F (τ) := ep(iτ z)df (z). The characteristic function is uniformly continuous on the real line. Fact: A distribution function is completely characterized by its characteristic function. Lemma The characteristic function of the standard normal distribution Φ is given by ϕ Φ (τ) = ep ( τ 2 ). 2 Probability and umber Theory: an Overview of the Erdős-Kac Theorem

26 Levy s continuity theorem Theorem Let {F n } be a sequence of distribution functions and {ϕ Fn } be the corresponding sequence of their characteristic functions. Then {F n } converges weakly to a distribution function F if and only if ϕ Fn converges pointwise on R to a function ϕ that is continuous at 0. Additionally, in this case, ϕ is the characteristic function of F. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

27 The Erdős-Kac Theorem Theorem Let. Then as, { ω(n) log log ν n : log log } = Φ(). That is, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

28 A proof sketch The atomic distribution function for is { ω(n) log log F () = ν n : log log = 1 { # ω(n) log log n : log log } }. We denote by ϕ F (τ) the characteristic function of F. We have ϕ F (τ) = e iτz df (z) Let P = { < 1 < 0 < 1 < < i } be a partition of R. Then we have ϕ F (τ) equal to Probability and umber Theory: an Overview of the Erdős-Kac Theorem

29 A proof sketch continued = = lim mesh(p)0 = lim mesh(p)0 [ = 1 = 1 = 1 e iτz df (z) e iτz (F ( k ) F ( k 1 )) k k lim mesh(p)0 ( 1 e iτz # {n : f (n) k } 1 ) # {n : f (n) k 1 } e iτz (# {n : f (n) k } # {n : f (n) k 1 } )] k ma{ω(n):n } n k=0 iτf (n) e iτf (n) e Probability and umber Theory: an Overview of the Erdős-Kac Theorem

30 A proof sketch continued et, we find some bounds for ϕ F (τ): ϕ F (τ) = ep ( τ 2 ) ( ( )) ( ) τ + τ O + O. 2 log log log (Informally, we write f () = O(g()) when there eists a positive function g such that f does not grow faster than g.) Probability and umber Theory: an Overview of the Erdős-Kac Theorem

31 A proof sketch continued et, we find some bounds for ϕ F (τ): ϕ F (τ) = ep ( τ 2 ) ( ( )) ( ) τ + τ O + O. 2 log log log (Informally, we write f () = O(g()) when there eists a positive function g such that f does not grow faster than g.) Take the limit as : ϕ F (τ) ep ( τ 2 ) = ϕ Φ (τ). 2 Probability and umber Theory: an Overview of the Erdős-Kac Theorem

32 A proof sketch continued In other words, the sequence of characteristic functions ϕ F converges pointwise to the characteristic function of the normal distribution. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

33 A proof sketch continued In other words, the sequence of characteristic functions ϕ F converges pointwise to the characteristic function of the normal distribution. Apply Levy s continuity theorem: { ω(n) log log ν n : log log } = Φ(). Thus, the limit distribution of the prime-divisor counting function ω(n) is the normal distribution with mean log log and variance log log. This completes the proof. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

34 # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } # {n : ω(n) } An Illustration = 10 = 20 = 30 = 40 = 50 = = 5000 = 1000 = 500 = 100 = = = = Probability and umber Theory: an Overview of the Erdős-Kac Theorem

35 References Steuding, J. Probabilistic umber Theory Gowers, T. The Importance of Mathematics. Probability and umber Theory: an Overview of the Erdős-Kac Theorem

36 Probability and umber Theory: an Overview of the Erdős-Kac Theorem

37 Probability and umber Theory: an Overview of the Erdős-Kac Theorem