Brownian Motion and Random Prime Factorization

Transcription

1 Brownian Motion and Random Prime Factorization Kendrick Tang June 4, 202 Contents Introduction 2 2 Brownian Motion 2 2. Develoing Brownian Motion Measure Saces and Borel Sigma-Algebras A Collection of Continuous Paths A Short Introduction to Wiener Measure Three Defining Proerties of Browning Motion A Maing From C 0 [0, T ] to C 0 [0, ] Alying the Wiener Measure Distribution of the Greatest Positive Excursion Paul Levy s Arcsine Law The Gambler s Fortune A Different Prime Counting Function 8 3. Defining the Function Primes and Probability Brownian Motion and Primes 9 4. A Very Useful Aroximation Constructing and Correcting the Path Constructing the ath Correcting the Drift Correcting the Variance Correcting the Path Polygonality Proerties of Integers using Prime Factorization Conclusion 5 Bibliograhy 6

2 2 of 6 Introduction This aer will reflect the material covered in Patrick Billingsley s Prime Numbers and Brownian Motion aer from 973 []. First I will develo some concets of Brownian motion and introduce the Wiener measure for subsets of the set of continuous functions which ma the unit interval to the real line. Afterwards I will introduce a rime-factorization counting function and some roerties. In the final section of my aer, the two seemingly searate ideas will be tied together: I will construct a random ath from the integers using the rime-factorization counting function and use tools exlained in the revious sections to modify the ath to model Brownian motion. Then using a robability limit, I will conclude several interesting roerties of the rime factorization of numbers. 2 Brownian Motion 2. Develoing Brownian Motion Brownian motion is a mathematical model of a article that dislays inherently irregular and random movements. In this aer I will only be focusing on one dimensional Brownian motion. It is in this section where I introduce and define some necessary conditions for Brownian motion and notation, as well as the Wiener measure. However, I must first begin by defining measure saces and Borel sigma algebras. 2.. Measure Saces and Borel Sigma-Algebras The following material does not need to be fully understood for the reader to understand the rest of the aer. It is, however, an introduction to measure theory and borel sigma algebras which are both deely involved in the constraints of the big theorems of this aer. This section can be skied by the reader, and used as a reference when measure theory and borel sets are mentioned in the later arts of this aer. Definition 2.. A measurable sace defined by a large samle sace Ω, a subset of Ω, B, and is couled with a measure, µ to form a measure sace. A measure sace is notated by Ω, B, µ). In this aer we will be focusing on the set of continuous functions that ma the unit interval to the real line C 0, and sigma-algebras of C 0, secifically the Borel sigma-algebra. Definition 2.2. A sigma-algebra σ-algebra) is a collection of subsets of the samle sace that contains the emty set, is closed under countable unions, and is closed under comlements [2]. The largest sigma-algebra of Ω is the owerset of Ω, and the smallest sigmaalgebra is {, Ω} [2]. For this aer we will only consider the Borel sigmaalgebra:

3 3 of 6 Definition 2.3. The Borel sigma-algebra on Ω is the smallest sigma-algebra containing all oen sets of Ω, and a Borel set is an element of the Borel sigmaalgebra [2]. The measure in a measure sace can be any kind of measure. In this aer we will only consider the Wiener measure described in 2..3), which is a robalility measure, so we refer to our measure sace as a robability sace. Thus, we can write the measure sace this aer focuses on as C 0 [0, ], A, P ) where C 0 [0, ] is the set of continuous functions on the unit interval, A is a subset of C 0 [0, ], and P the Wiener measure of Borel sets on C 0 [0, ]. However, to form a sigma algebra on C 0 [0, ] we must be able to define an oen set. Thus, we define the metric for aths in C 0 [0, ] to be df, g) = max fx) gx), f, g C 0 [0, ] the maximum vertical dislacement of two functions in C 0 [0, ]. A set S is oen if for every x in S, there is a neighborhood around x that is comletely contained in S. In other words, for each x in S, there is an ɛ greater than 0 so that all y for which dx, y) < ɛ are contained in S. Again, these Borel sets are not of great imortance to understand the material in this aer, but are imortant to kee in mind because of some constraints on the Wiener measure which will be described later in the aer A Collection of Continuous Paths Consider the function xt) : [0, T ] R that reresents the vertical osition of a article that begins at the origin: x0) = 0. At each infinitesimal time interval, the article will either take an infinitesimal ste u or down. Thus, xt) C 0 [0, T ] where C 0 [0, T ] is the set of all continuous functions that ma [0, T ] to the real line and satisfy x0) = 0. For the majority of this aer, I will only either be working with the set of functions C 0 [0, ] continuous functions maing the unit interval to the real line), or a secial maing of functions in C 0 [0, T ] to C 0 [0, ] defined later in A Short Introduction to Wiener Measure The measureable sace C 0 [0, ] is of little interest in this aer. Instead we are interested in subsets A and the robability mesaure to derive interesting roerties of A. This measure, the Wiener measure P T A), determines the robability that a function x C 0 [0, T ] is in A. For examle, if A = {xt) : xt) < 0, t [0, T ]}, is the set of functions which always have negative dislacement, then P T A) would be the robability that a function randomly icked from C 0 0, T ) will also be in A. At this oint, the reader should note that it is not always ossible to define P T A) for all subsets of C[0, T ]. However, originally roven by Norbert Wiener in 923 [], the existance of P T A) does exist for all Borel sets

4 4 of 6 see 2.3). Also note that deriving the Wiener measure of a subset A is beyond the scoe of this aer; we will only be examining the Wiener measure of three different subsets Three Defining Proerties of Browning Motion Using the Wiener measure for a secific subset of C 0, I will define two of the criteria for Brownian motion. Consider the event A = [x : α xt) β], ) the set of functions where the article travels through the vertical gate [α, β] at time t. Note that the only restriction on α, β is that α < β. The Wiener measure of this event a relationshi taken for granted in this aer) is defined in [] to be P T [x : α xt) β] = 2πt β α e u2 /2t du. 2) Note that 2) is the integral of a gaussian with mean 0 and variance t. These will be the first two defining roerties of Brownian motion. These two roerties reflect non-drift since there is an equal chance of moving u or down) and the article s tendency to wander away from the origin. We will exect anything that exhibits Brownian motion to satisfy this roerty. The third roerty of Brownian motion is indeendence. Particles in Brownian motion move randomly, which means the direction a article moves in one time interval is comletely indeendent of how the article moves in another time interval. Consider t < t < s < s. If A is the subset of functions with roerty M defined on [t, t ] and B the subset of functions with another roerty N on [s, s ] then P T A B) = P T A)P T B). 3) The robability that x is in the intersection of A and B is the roduct of the robability of x A and the robability of x B because the robability of each event haening is indeendent of each other []. This can be generalized to multile events: Lemma 2.. If A,...A n are sets of functions that satisfy roerties for t τ [0, T ] where τ i τ j = for all i, j then P T A A 2... A n ) = P T A )P T A 2 )...P T A n ). 4) Proof. Since A A 2 A 3 = A A 2 A 3 ), and by 3), Continue by induction to show 4) P T A A 2 A 3 )) = P T A )P T A 2 A 3 ) P T A A 2 A 3 ) = P T A )P T A 2 )P T A 3 ).

5 5 of 6 In this aer, a article is in Brownian motion if it satisfies the three roerties restated below for the reader s convenience: For event ), the Wiener measure must follow the distribution of a gaussian with mean zero and variance t. Events over disjoint time intervals must be indeendent of each other A Maing From C 0 [0, T ] to C 0 [0, ] In this section I define a maing C 0 [0, T ] to C 0 [0, ] which also reserves Bronian motion. Given a article s ath xt) C 0 [0, T ], seed u the argument by T and shrink the amlitude by / T : yt) = T xtt ), t [0, ]. 5) Theorem 2.2. If yt) is defined by 5) then yt) C 0 [0, ] and yt) will exhibit Brownian motion as described in 2..4) []. Proof. Since xt) describes Brownian motion, the Wiener measure of xt) through the gate [α, β] at time t will be defined by a gaussian with mean 0 and variance t. By changing the time scale, xtt ) will have a similar Wiener measure, with mean 0, but variance now tt : P T [xtt ) : α xtt ) β] = 2πt β α e u2 /2tT du. By rescaling the random variable xt) the gaussian will change as follows: the mean will change by the scaling factor, and variance by the square of the scaling factor. Thus the mean remains zero, but the variance will be reduced to t ) 2 since tt T = t. Furthermore, indeedent events will remain indeendent through the uniform time and dislacement scaling. Therefore, the maing mas C 0 [0, T ] to C 0 [0, ] and reserves Brownian motion. This maing has an interesting consequence: Theorem 2.3. For any ɛ > 0, K > 0, define A to be functions with at least one chord with sloe exceeding K. Then P A) > ɛ []. Proof. Let ɛ > 0 b First begin with xt) C 0 [0, T ]. Choose T large enough so with robability larger than ɛ, xt) will have sloe larger than. This event isn t too strange and unfathomable that given enough time, it is guaranteed that xt) will roduce this given enough time. Also make sure T is larger than K 2. Now xt) is a function with sloe at least and belongs in C 0 [0, T ], T > K 2. Now ma xt) to yt) with 5). By shrinking the time scale, the sloe is increased from to T. By shrinking the vertical scale by / T the sloe is then reduced to T/ T > K 2 /K = K.

6 6 of 6 I will now resent, but not rove, some corollaries of 2.3, which are mentioned in []: Corollary 2.4. Since ɛ and K were chosen arbitrarily, with robability a randomly icked function from C 0 [0, ] will have at least one oint of unbounded ositive or negative variation. Corollary 2.5. It follows that the chords with unbounded sloes are infinitely dense in functions from C 0 [0, ]. This means with robability, a function from C 0 [0, ] will be nowhere differentiable. 2.2 Alying the Wiener Measure In this section I will rovide some well known Wiener measures and aly them to roduce some interesting robabilities. I will come back to these measures when examining rime number factorization in the last section of this aer Distribution of the Greatest Positive Excursion Consider the set of functions A so that x A if xt), t [0, ] exceeds some α. Since x) α and x) < α are mutually exclusive events, one or the other is guaranteed to haen which means P [x : max xt) α] =P [x : max xt) α and x) α] + P [x : max xt) α and x) < α]. Since the article has no drift, after arriving at height α the article has an equal robability of continuing to increase, or to decrease. Thus, the two events are equivalent and we can combine the robabilities into one: P [x : max xt) α] = 2P [x : max xt) α and x) α]. At this oint, the second restriction on x is redundant: if x) exceeds α then the original restriction is automatically satisfied. So finally, P [x : max xt) α] = 2P [x : x) α] which is ) where t = and β =. Thus, using 2) P [x : max xt) α] = 2 e u2 /2 du 6) 2π which is the Wiener measure of all functions which exceed α at some t [0, ] Paul Levy s Arcsine Law Let xt) C 0 [0, ] and define the set S to be the set of oints t so that xt) > 0. Then the Lebesgue measure of S is written as µ L {t : xt) > 0}) the total amount of time x is ositive. Now consider the set of functions α xt) : α µ L {t : xt) > 0}) β. 7)

7 7 of 6 The Wiener measure of this subset of C 0 [0, ] is given in [] as P A) = π β α du u u). This is a u-shaed distribution, which means for a fixed β α gate, the robability is lowest for α, β near 0.5 and highest for α, β near 0 and. Also, the grah is symmetric so the robability of assing the gate [α, β] is the same as assing the gate [ β, α] The Gambler s Fortune In this section I will construct continuous Brownian motion from a discreet random walk and aly the revious Wiener measures to rovide some interesting robabilities. Consider discreet random walks in one dimension. A article begins at the origin, and at every integer time value, it takes either one ste forward or one ste backwards. If we connect the total dislacement at each integer time value with straight lines to roduce a continuous function, then this article s ath belongs in C 0 [0, T ]. This motion, satisfies the two roerties of Brownian motion described in 2..4: at each ste the mean is and the variance )0.5) + )0.5) = 0 ) 2 0.5) + ) 2 0.5) = and the dislacement at each ste is surely indeendent. Let yt) be the olygonal ath defined by the random walk. The linearity of the aths between each integer time interval is not something we want in Brownian motion so to remove the olygonal blemish in yt) and reserve the roerties we want, take T to infinty and aly 5). Since we are contracting the time intervals by /T, as T the time intevals become infinitesimal, and we aroach continuous Brownian motion. Now consider a Gambler who s net fortune is maed in this random walk. At each hand or game he can win or lose one dollar. By alying 5) to a random walk and take T to infinity, the robability measure aroaches the Wiener measure []: Corollary 2.6. If yt) is defined by random walk, xt) the result of alying 5) to yt), and A any subset of C 0 [0, ] then P robabilityxt) A) P A) as T. Consider A = [x : α x) β]. If x) = h then yt ) = h/ T by 5). So by corollary 2.6 and 2) [ Probability α yt ) ] β β e u2 /2 as T. T 2π The following three examles are alications of the various Wiener measures alied to a Gambler who lays 00 hands: α

8 8 of 6 Consider ) and let α = β = 0.9, the Wiener measure evaluates to about 0.6. If the Gambler lays 00 hands T = 00) then the Gambler has a 60% chance of ending within = 9 dollars of what the Gambler started with. Now consider 6). The history of the Gambler s fortune belongs in A if at any oint during the night the Gambler s net winnings was larger than α T. If α =.7, the Wiener measure is about 0. and again if we set T = 00 then the Gambler will have a 0% chance that at some oint during the Gambler s 00 hands the Gambler will have 7 dollars more than what the Gambler started with. Now consider 7). We can estimate the robability that the Gambler s net fortune will be ositive for a certain ercentage of time. If α = 0.45, β = 0.55 then the Wiener measure is about 6% the Gambler has a 6% chance of being ositive for around half the night). Again, noting the interesting feature of Paul Levy s Arcsine Law, if α = 0.9, β = then the Wiener measure is near 20%. By the symmetry of the arcsine curve, the same is true for α = 0, β = 0.. Thus, the Gambler has a 20% chance of being ositive for 90% of the night, but also only for 0% of the night meaning he has a 20% chance of being negative for 90% of the very same night). While for the three A s above, the robability limit converges to the Wiener measure, this is not always the case. The roof is beyond the scoe of this aer, but it is imortant to note that the convergence is only the case for every Borel set see 2.3) A where for the boundary of A, denoted by A, satisfies P A) = 0 []. 3 A Different Prime Counting Function 3. Defining the Function Definition 3.. I will say that n divides m if m/n is zero and denote this with n m. Otherwise, n does not divide m, which we will be denote with n m. Let and define δ n) = fn) = { n 0 n rime δ n). In other words, f counts the number of distinct rimes in the factorization of n. This function increases very very slowly. In fact, f equals 2 first at n = 6, 3 at n = 30 and 4 at n = 20. Desite its growth rate, there are infinitely many rimes so f must eventually aroach infinity.

9 9 of Primes and Probability We can define P N S) = N ν{n : n [, N] and n S} where ν measures the number of integers in a set. This means P N S) is the ratio of integers in [, N] that are also in S. For examle, if S is the set of even numbers, P 0 S) = 5/0 = /2. Now consider P N n : α fn) β). This is the robability that a andomly icked integer between and N will have between α and β rimes in its factorization. The number of multiles of u to N is N/. For examle, if N = 50 and = 7 then there are exactly 50/7 = 7 numbers between and 50 that have 7 in its factorization. Thus, the robability that n, or that δ n) =, is P N [n : n] = N for large N[]. 8) N This aroximation is good for very large N since N/ N/ < and as N the difference between the floor of the quotient and regular division gets roortionally small. Now consider two relatively rime numbers a and b they share no rime factors). Then a n and b n if and only if ab n. Since two rime numbers are automatically relatively rime, we can deduce with 8) that P N n : n and q n) = P N n : q n) = N N q q Since P Nn : n) and q P Nn : q n) it is clear that rime factorization between two rimes are aroximately indeendent: P N n : n and q n) P N n : n)p N n : q n) 9) This theorem generalizes as follows: Theorem 3.. If,... k are k distinct rimes, then P N n : n, 2 n,... and k n) = P N n : n)...p N n : k n). Proof. If,... k are k distinct rimes then is relatively rime to Π k j=2 j which, by induction roduces the theorem. 4 Brownian Motion and Primes How can we relate the rime counting function described in the revious section and Brownian motion? First I will introduce a useful aroximation and then outline how the random dislacement will take be defined. The outut of this will satisfy indeendence due to Theorem 3. but due to the rareness of rimes and the slow growth rate of fn) there will be drift and variance roblems involved.

10 0 of 6 4. A Very Useful Aroximation Theorem 4.. The aroximation: If q rimes then for large enough, q log log. 0) q Proof. This roof is outline in [3], and is reroduced below: We first need four inequalities. Lemma 4.2. n k= k 2 < 5/3 Proof. The sum converges to π 2 /6 which is less than 5/3. Lemma x < e x for all x > 0. Proof. When x = both sides are equal to, and e x grows much faster than x. Lemma 4.4. logn + ) < Proof. The log term can be written as n+ dx x which can then be slit into a sum of integrals n i+ dx i= x. Since /x is a decreasing function, /i is the maximum value for each interval which means the sum of integrals is less than n i= i. Lemma 4.5. n i= n i= i. i + ) n k 2 n k= Proof. Each integer can be written as a roduct of a square free integer and an integer, that is i = k 2 or i = k 2, where i, k, R and i. Also + because each term on the right side is larger than. So logn + ) n + ) n k= k 2 n ex )) 5 3 = ex 5 3 n which means that log[logn + )3/5] n,

11 of 6 or that log logn + ) log5/3) This can be rearranged to be log logn + ) n+ n+ n +. log 5 3 n + and as n goes to infinity, the difference is bounded and retty small, so for large enough n this is a good aroximation. The time assed at rime is given by q /q and current height given by q δ qn) /q). By the aroximation 0) we can say that the time assed is about log log and the height about q δ qn) log log ). 4.2 Constructing and Correcting the Path 4.2. Constructing the ath Let N be given. Randomly choose n from [, N] and begin the ath at 0. At each rime starting at = 2), if n then move u one ste, else move one ste down. Then roceed to the next rime and reeat. While it seems like this ath will be redetermined once n has been icked, there is an inherent randomness in choosing n. It is from this randomness the Brownian motion will be derived. Imagine laying this game with a mysterious heler who icks the n and does not immediately reveal the number. It is only at each rime ste whether or not n and which way to move is revealed. With the hel of the mysterious layer, it is clear that the that will truly be random. A roblem arises from the fact that most rimes will not divide the number. Thus, there will be an intense drift in the negative direction since there will be many negative stes between each ositive ste. In fact, by 8) as goes to infinity, / goes to 0 which means the chances of dividing n goes to Correcting the Drift To correct the drift we change the magnitude of the stes. Everytime we move in the ositive direction, move forward by, and everytime we move in the negative direction, move backwards by. These changes in the stes taken will reduce the mean to be closer to 0 and damen the drift. Proof. ) P N n : n) + ) P N n : n) = ) ) ) = 0. + ) ) ) ) )

12 2 of Correcting the Variance With the mean fixed, there is now a roblem with the variance: ) 2 P N n : n) + ) 2 P N n : n) = = = ) 2 ) + ) 2 + ) 2 ) ) ) 2 ) + 2 ) 3 2 ) 3 As gets large, the variance goes to / which gets very very small. This is a roblem because this imlies that the Brownian motion will have very little vertical dislacement over each ste after a large enough amount of stes. To resolve this roblem, we need to change the amount of time sent at each ste. The correct thing to do is to send only / time units at each rime. Proof. Since the vertical variance at each rime is about /, and we are sending / time at each ste, the total vertical variance is aroximately q,q rime /q, which is the same as the variance of time. Thus, horizontal and vertical variances are roortional and we can say that the variance is aroximately t Correcting the Path Polygonality If we are checking numbers from [, N], at the last rime N the total amount of time sent will be q N /q log log N. Let T = log log N. At this oint we are faced with the same roblem we had with the random walk: the stes are too olygonal! To correct this, we do the same stes as before: send T to infinity and aly 5). After rescaling, the ath is now art of C 0 [0, ], has mean 0, variance t and event indeendence. Sending T to infinity will correct the olygonal shae of the ath, allowing the ath to aroach continuous Brownian motion. After transforming the ath, time t will transform to time log log / log log N and the vertical scale will be contracted to q δ qn) log log ). log log N 4.3 Proerties of Integers using Prime Factorization In this section I will make use of a useful aroximation and the aths constructed in the revious section to conclude some interesting roerties of n. These aths, denoted by ath N n) since they are deendent on both N and

13 3 of 6 n, are in C 0 [0, ]. The robability that ath N n) A where A is a subset of C 0 [0, ] is P N [n : ath N n) A] and, similarly to corollary 2.6: Theorem 4.6. [] If A is a Borel subset of C 0 [0, ] satisfying P A) = 0 then P N [n : ath N n) A] P A) as N ) The roof of this theorem goes beyond my ability... But, like the aroximations with the Gambler s fortune, we can determine some robabilitistic roerties of n. I will now use this limit to consider the following three sets of x C 0 so that a. the final value of x lies between α and β, so that P = 2π β b. the max value of x is greater than α, so that α e u2 /2 du; P = 2 e u2 /2 du; 2π c. and where the time with which x is ositive is between α and β, so that P = π β α α du u u). Examle 4.. Consider a). If α = β = 0.9 and N = 0 70 so log log N 5, then the robability that the ath is in the set aroaches the Wiener measure, which evaluates to 0.6. Thus, about 60% of n 0 70 have 3 to 7 rime factors. Examle 4.2. Definition 4.. The aarent comositeness of n u to a rime is δ q n) log log. q The larger this is, the more comosite n seems to be at a rime and the smaller this is, the more rime-like n seems to be. So the max aarent comositeness of n is max N δ q n) log log = q [ ] log max ath Nn) log N. N If we consider b) and let α =.7, N = 0 70 then the robability that P ath N n) has max value larger than.7 aroaches the Wiener measure in b), which evaluates to 0.. This means about 0% of n less than 0 70 has a max aarent comositeness of about 3.8.

14 4 of 6 Examle 4.3. Definition 4.2. A number n is excessive at a rime if δ q n) > log log. q That is, the number n is more comosite than the average number u to. It follows from the definition that a number n is excessive at if ath N n) > 0 at a. The total amount of time with which ath N n) is ositive is log log N : N and δ q n) > log log. q The first quotient is the factor time is scaled by when correcting the ath. Since we send / of unscaled time at each rime, and are considering only the where n is excessive at, the second art of the roduct is the sum of / where n is excessive at. If we comare this to c), again with N = 0 70, we can observe that 20% of n < N are excessive more than 90% of the time [], by the same symmetry discussed in 2.2.3) 20% of n < N are excessive less than only 0% of the time [], and only about 6% of n < N are excessive from 45% to 55% []. These three roerties indicate that very few integers are undecidedly comosite, that is, if a number is highly comosite at some or very rime-like), it is very likely that the number will be highly comosite or very rime-like) for most other. If we consider ϕ N n) = # : N and δ q n) > log log, q so that ϕ N n) is the number of rimes u to N where n is excessive, and the ratio of ϕ. The ratio of ϕ N n) and the regular rime counting function πn) is defined in [] as: Definition 4.3. R N n) = ϕ N n) πn) is the ratio of rimes where n is excessive over the total number of rimes. This is the number of where n is excessive, normalized by the number of rimes) Theorem 4.7. [] Let P R > α) be the robability that the ratio R is larger than α, α >. The given an ɛ > 0, there exists a K so when N is greater than K, P R < ɛ) /2 ɛ and P R > ɛ) /2 < ɛ.

15 5 of 6 This theorem follows from the fact that most integers are either excessive at most rimes, or not excessive at most rimes. Taking N large enough, and normalizing by the rime counting function roduces the theorem. In other words for large enough N, for about 50% of the time the ratio is at 0, and the rest of the time the ratio will be at. This means, almost exlicitly, that all integers are either excessive at all rimes or at none []. 5 Conclusion While this aer doesn t go into the details of find Wiener measures or the rigorous mathematics behind Brownian motion, I am still able to comute some interesting quantities, using only simle algebra, number theory and robability. These quantities are deely related to roerties of the integers and their relationshi with the rimes.

16 6 of 6 References [] Patrick Billingsley. Prime numbers and brownian motion. The American Mathematical Monthly, 800 Dec.)):099 5, 973. [2] Maya R. Guta. A measure theory tutorial measure theory for dummies). Technical Reort UWEETR , May [3] Wikiedia. Divergence of the sum of the recirocals of the rimes Wikiedia, the free encycloedia, 202. [Online; accessed -June-202].