Keywords: prime counting function, prime probability function, Brocard's conjecture, Chebyshev's Theorem, complete system of incongruent residues, computational complexity, Dirichlect primes, Euler's constant γ, expected value, factorising 2010 Mathematics Subject Classification. 11A07, 11A41, 11A51, 11N36, 11Y05, 11Y11, 11Y16

is polynomial time; Hardy-Littlewood conjecture; integer factorising algorithm; Law of Large Numbers; Mertens' theorem; mutually independent prime divisors; polynomial time algorithm; prime counting function π(n); prime density; primes in an arithmetic progression; Prime Number Theorem; probability model; probabilistic number theory; twin primes.

Abstract. All the known approximations of π(n) for finite values of n are derived from real-valued functions that are asymptotic to π(x), such as x logex , Li(x) and Riemann's function R(x) = ∞ n=1 µ(n)

(n) li(x 1/n ). The degree of approximation for finite values of n is determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield a closer approximation than others to the actual values of π(n) for computable values of n. None of these can, however, claim to estimate π(n) uniquely for all values of n. We show that statistically the probability of n being a prime is

π( √ j) i=1 (1 -1 p i
), and that statistically the expected value of the number π(n) of primes less than or equal to n is given uniquely by

n j=1 π( √ j) i=1 (1 -1 p i
) for all values of n. We then demonstrate how this yields elementary probability-based proofs of the Prime Number Theorem, Dirichlect's Theorem, and the Twin-Prime Conjecture.

Introduction

"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood. Describing the distribution of primes is at the heart of much mathematics...". 1

In the first half of this investigation we address the thesis that what makes the distribution of primes 'mysterious', and difficult to engage with for emerging scholars, is a curious-apparently implicitreluctance to define prime probability statistically; an issue which may need to be addressed more extensively elsewhere.

In the second half, we explore the structure of divisibility (and, ipso facto, of primality) 2 , and statistically define the probability of a number being prime.

We then show how this yields elementary (and unexpectedly related) probability-based proofsderived from first principles-of fundamental prime properties such as the Prime Number Theorem, Dirichlect's Theorem, the Twin Prime Conjecture and the P vN P problem. 3 1. Introduction 2

1. Introduction

1.A. The functions π(x) and x logex : A historical perspective

To place this investigation in an appropriate historical perspective, we note that Adrien-Marie Legendre and Carl Friedrich Gauss are reported 4 to have independently conjectured in 1796 that, if π(x) denotes the number of primes less than x, then π(x) is asymptotically equivalent to x logex . Around 1848/1850, Pafnuty Lvovich Chebyshev proved that π(x)

x logex , and confirmed that if π(x)/ x logex has a limit, then it must be 1 5 .

Fig. 1: The asymptotic behaviour of the primes Fig. 1: Graph showing ratio of the prime-counting function π(x) to two of its approximations, x ln x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x ln x converges from above very slowly, while the ratio for Li(x) converges more quickly from below. 6 The question of whether π(x)/ x logex has a limit at all, or whether it oscillates, was answered-it has a limit-first by Jacques Hadamard and Charles Jean de la Vallée Poussin independently in 1896, using advanced argumentation involving functions of a complex variable 7 ; and again independently by Paul Erdös and Atle Selberg 8 in 1949/1950, using only elementary-but still abstruse-methods without involving functions of a complex variable.

1.B. A better approximation to π(x): The integral Li(x)

We also note that, reportedly 9 : "In a handwritten note on a reprint of his 1838 paper 'Sur l'usage des séries infinies dans la théorie des nombres', which he mailed to Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlect conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to π(x) is given by the offset logarithmic integral Li(x) defined by: based proofs are shown to be both simple and capable of being taught to, and reproduced by, any interested first-year undergraduate student of mathematics, or even a GCSE A level+ amateur enthusiast, with a spirit of enquiry. See §7., Appendix IV for the resources needed by a reader for following, and reproducing, the proofs of this paper.

Li(x) =

x 2 1 loget .dt = li(x) -li(2)." 10 We further note that in 1889 Jean de la Vallée Poussin proved 11 (cf. Fig. 1): ". . . that Li(x) represents π(x) more exactly than x logex and its remaining approximations We note that all the known approximations of π(n) for finite values of n are derived from realvalued functions that are asymptotic to π(x), such as x logex , Li(x) and Riemann's function

R(x) = ∞ n=1 µ(n) (n) li(x 1/n
). The degree of approximation for finite values of n is determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield the closest approximation upon comparison with the actual values of π(n) for computable values of n (eg. Fig. 2). (n) li(x 1/n ). 12 The question remains:

• Is there a function which best approximates π(n) for all values of n?

1.D. Is there a unique function which best approximates π(n) for all values of n?

In this investigation we shall answer the above question affirmatively by: 10 Where li(x) =

x 0 1 loget .dt. 11 [Di52], p.440.

• first, defining the statistical probability of an integer n being a prime; and

• second, showing that the statistically expected distribution of the primes-and hence the best approximation for π(n) for all finite n-is given by the unique statistical prime counting function (cf. Fig. 3):

π L (n) = n j=1 π( √ j) i=1 (1 -1 p i
).

Fig. 3: Statistically expected distribution of the primes ).

We shall then demonstrate how this yields elementary, probability-based, proofs of the Prime Number Theorem, Dirichlect's Theorem, and the Twin-Prime Conjecture.

2. The curious reluctance to define prime probability statistically 2.A. Prime probability: conventional wisdom Now, the explicit thesis of this investigation is that lack of recognition of π L (n) as the prime counting function for the number of primes ≤ n is, apparently, reflection of a curious-albeit implicitreluctance to accept a statistical definition of prime probability as legitimate.

For instance, conventional number-theory wisdom appears to be that the distribution of primes suggested by the Prime Number Theorem 14 , π(n) ∼ n logen , is such that the probability P(n ∈ {p}) of an integer n being a prime p can only be heuristically estimated as 1 logen 15 ; apparently reflecting an implicit faith in G. H. Hardy and J. E. Littlewood's 1922 dictum that 16 : "Probability is not a notion of pure mathematics, but of philosophy or physics".

It is a dictum that can reasonably be taken by the laity to suggest, with some authority, that the statistical probability P(n ∈ {p}) of an integer n being a prime p is also not capable of being welldefined statistically 17 independently of the Theorem.

2.B. Statistical probability that a prime p divides n

However, such a conclusion would be misleading, since any lay investigation of such a probability from first principles:

(1) would begin naturally by considering if, and only if, conditions for i to be a divisor of n;

(2) would move fairly straightforwardly to an elementary residue function such as r i (n) 18 , defined (Definition 1) for all n ≥ 2 and all i ≥ 2 by:

n + r i (n) ≡ 0 (mod i) where i > r i (n) ≥ 0 since r i (n) = 0 if, and only if, i is a divisor of n;
(3) would then (Theorem 3.3) note for any i ≥ 2 that:

M i = {(0, 1, 2, . . . , i -1), r i (n), 1 i }
is a probability model 19 for the values of r i (n) for n ≥ 2;

(4) which would further imply:

(i) first (Corollary 3.4) that, by the standard definition of the statistical probability P(e) of an event e 20 , the probability P(p|n) that r p (n) = 0-whence the prime p divides n-is:

P(p|n) = 1 p
and that the probability P(p | n) that r p (n) = 0-whence the prime p does not divide n-is:

P(p | n) = 1 -1 p
since the p numbers 0, 1, . . . , (p -1) are all incongruent and form a complete system of residues 21 ;

(ii) second (Lemma 3.5) that:

(a) the product of the individual probability that r p i (n) = 0-whence the prime p i divides the integer n-and the individual probability that r p j (n) = 0-whence the prime p j = p i divides n-is:

P(p i |n).P(p j |n) = 1 p i . 1 p j
17 See, for instance, [St02], Chapter 2, p.9, Theorem (sic) 2.1! 18 Depicted graphically in §5., Appendix II(A), Fig. 6. (b) the joint probability P(p i |n ∩ p j |n) that r p i (n) = 0 and r p j (n) = 0whence both the primes p i = p j divide the integer n-is:

P(p i |n ∩ p j |n) = 1 p i .p j
since the p i .p j numbers v.p i + u.p j , where p i > u ≥ 0 and p j > v ≥ 0, are also all incongruent and form a complete system of residues 22 ;

(iii) and third (Theorem 3.8) that the prime divisors of any integer n are thus mutually independent by the standard definition of the 'mutual independence' of two events e 1 and e 2 23 .

2.C. Statistical probability of n being a prime

Since it is easily shown that n is a prime if, and only if, it is not divisible by any prime p ≤ √ n, it would immediately then follow:

(i) first (Theorem 3.11) that the statistical probability of n being a prime p is given 24 by the prime probability function (cf. Fig. 4):

P(n ∈ {p}) = π( √ n) i=1 (1 -1 p i ) ∼ 2e -γ logen , 25
where 2.e -λ ≈ 1.12292 . . .; 26

(x) = (x -p 2 n ) n i=1 (1 -1 p i ) + n-1 j=1 (p 2 j+1 -p 2 j ) j i=1 (1 -1 p i
) + 2 (see Fig. 5).

22 Ibid., p.52, Theorem 59. 23 See §4., Appendix I; also [Ko56], Chapter VI, §1, Definition 1, p.57 and §2, p.58; see also [Ka59], p.54.

24 Compare [HL23], pp.36-37. 25 The asymptotic equivalence follows by Mertens's Theorem p≤x (1 (ii) and second that (Theorem 3.13), by the Law of Large Numbers 27 , the expected value 28 of the number π(n) of primes less than or equal to n is (Definition 4) the prime counting function π L (n) (cf. Fig. 5), such that:

-1 p ) ∼ e -λ logex , [
π(n) ∼ π L (n) = n j=1 π( √ j) i=1 (1 -1 p i ).
(x) = π L (x) = (x -p 2 n ) n i=1 (1 -1 p i ) + n-1 j=1 (p 2 j+1 -p 2 j ) j i=1 (1 -1 p i ) + 2 in the interval (p 2 n , p 2 n+1 ). Note that the gradient in the interval (p 2 n , p 2 n+1 ) is n i=1 (1 -1 p i
).

2.D. The anomaly in approximating π(n) heuristically: conventional wisdom

However conventional number theory wisdom-whilst reasonably conceding 29 that the heuristic probability of an integer n being prime could also be naïvely assumed as

√ n i=1 (1-1 p i
)-seems to unreasonably argue against such naïvety, by concluding that the number π(n) of primes less than or equal to n suggested by such probability would then be approximated by the heuristic prime counting function:

π H (n) = n j=1 π( √ n) i=1 (1 -1 p i ) = n. π( √ n) i=1 (1 -1 p i
) ∼ 2.e -γ n logen . For instance, Hardy and Littlewood note that: "In the first place we observe that any formula in the theory of primes, deduced from considerations of probability, is likely to be erroneous in just this way. Consider, for example, the problem 'what is the chance that a large number n should be prime?' We know that the answer is that the chance is approximately 1 log n .

Now the chance that n should not be divisible by any prime less than a fixed x is asymptotically equivalent to 27 See §4., Appendix I; also [Ko56], Chapter VI, §3, p.61. 28 See §4., Apendix II. Compare also [HL23], pp.36-37. See also §6., Appendix III for the expected values π L (n), and the actual values π(n), for 4 ≤ n ≤ 1500.

29 [Gr95], p.13.

<x

(1 -1 )

and it would be natural to infer 1 that the chance required is asymptotically equivalent to

< √ x (1 - 1 ) But < √ x (1 - 1 ) ∼ 2e -C log n
and our inference is incorrect, to the extent of a factor 2e -C .

1 One might well replace < √ x by < x, in which case we should obtain a probability half as large. This remark is in itself enough to show the unsatisfactory character of the argument." . . . pp.36-37, G.H Hardy and J.E. Littlewood, Some problems of 'partitio numerorum:' III: On the expression of a number as a sum of primes, Acta Mathematica, December 1923, Volume 44, pp.1-70. Now, even if we ignore the incongruity of treating x as 'fixed ', the 'character ' of the argument in Hardy and Littlewood's footnoted remark can be considered 'unsatisfactory' only if we conflate necessity with sufficiency!

Otherwise, what we ought to reasonably conclude from the argument is that: Lemma 2.1. Whilst the statistical probability that n should not be divisible by any prime less than

x is <x (1 -1 ) if x ≤ √ n, it is defined by < √ n (1 -1 )-and not by <x (1 -1 )-if x > √ n.
Proof : We shall show in §3.A. of this investigation that whilst-if x > √ n-the terms of the former product do, those of the latter product do not, define the statistical probabilities of the necessary and sufficient-mutually independent-conditions that jointly define the primality of n under the probability model (see §3.B.):

M i = {(0, 1, 2, . . . , i -1), r i (n), 1 i }.
Moreover, the argument that we may treat π H (n) as a heuristic approximation to π(n) is 'unreasonable' since an apparent anomaly does, then, surface when we express π(n) and the function π H (n) in terms of the number of primes determined by each function respectively in each interval (p 2 n , p 2 n+1 ) as follows:

π(p 2 n+1 ) = n j=1 (π(p 2 j+1 ) -π(p 2 j )) + π(p 2 1 ) π H (p 2 n+1 ) = p 2 n+1 . π( p 2 n+1 ) i=1 (1 -1 p i ) = ( n j=1 (p 2 j+1 -p 2 j ) + p 2 1 ). n i=1 (1 -1 p i ) = n j=1 (p 2 j+1 . n i=1 (1 -1 p i ) -p 2 j . n i=1 (1 -1 p i )) + p 2 1 . n i=1 (1 -1 p i )
Reason: By Corollary 3.13, π L (n) is the expected value of π(n), and, for any given k > 1:

π L (p 2 k+1 ) -π L (p 2 k ) > 0 as n → ∞;
whilst, for any given k > 1 30 :

p 2 k+1 . n i=1 (1 -1 p i ) -p 2 k . n i=1 (1 -1 p i ) → 0 as n → ∞.
More specifically, by Corollary 3.13 and Mertens' Theorem 31 , the expected value of the number of primes between the prime squares p 2 k and p 2 k+1 (see Fig. 4), for any k > 1, is given by:

π(p 2 k+1 ) -π(p 2 k ) ∼ π L (p 2 k+1 ) -π L (p 2 k ) as k → ∞ π L (p 2 k+1 ) -π L (p 2 k ) = (p 2 k+1 -p 2 k ). k i=1 (1 -1 p i ) ≥ ((p k + 2) 2 -p 2 k ). k i=1 (1 -1 p i ) ≥ 4(p k + 1). k i=1 (1 -1 p i ) ∈ O( p k logep k ) as k → ∞ → ∞ as k → ∞
So, if we were to contrarily accept both π L (n) and π H (n) as prime counting functions, then the anomaly noted by Hardy and Littlewood would, indeed, follow from the Prime Number Theorem π(n) ∼ n logen , since π H (n) ∼ 2.e -γ n logen ! Brocard's conjecture: We note without further comment that Brocard's conjecture:

π(p 2 k+1 ) -π(p 2 k ) ≥ 4
would follow if we could show that, for k > 1, the difference between π(n) and π L (n) is always less than 4(p k + 1). k i=1 (1 -

1 p i ) + 1. 32
2.E. The 'second' Hardy-Littlewood conjecture concerning prime density

We next note that the 'heuristic' definition of the probability of a number being prime, albeit discounted by Hardy and Littlewood as 'unsatisfactory', is not only justifiable statistically (as shown in §3.D.), but that Definition 4 immediately implies:

Theorem 2.2. π L (m + n) ≤ π L (m) + π L (n) for all integers m, n ≥ 2 Proof : The m terms of the summation π L (m) = m j=1 π( √ j) i=1 (1 -1 p i ) are identical to the first m terms of π L (m + n) = m+n j=1 π( √ j) i=1 (1 -1 p i ); whilst the k th term π( √ k) i=1 (1 -1 p i ) of π L (n) is greater than the corresponding (m + k) th term π( √ m+k) i=1 (1 -1 p i ) of π L (m + n) for m ≥ 1, k ≥ 1 33 .
We further have, by the Law of Large Numbers, that:

Corollary 2.3. π(m + n) ≤ π(m) + π(n) as m → ∞ 30
Compare with what appears to be a similar argument in [St02], Chapter 2, p.9, Theorem (sic) 2.1. 31 i.e., p≤x (1 -1 p ) ∼ e -λ logex , [HW60], Theorem 429, p.351. 32 cf. Wikipedia: Brocard's conjecture. 33 As is graphically obvious from Fig. 4.

The significance of Theorem 2.2 is seen if we compare: (i) Theorem 2.2 with the definition of the 'second' Hardy-Littlewood 1923 conjecture in Richards 34 concerning the estimated density of primes as: 'π(x + y) ≤ π(x) + π(y) for all integers x, y ≥ 2' where the author claims: "We show that this assertion is probably false";

(ii) and Corollary 2.3 with the original conjecture 35 , where Hardy and Littlewood define:

" (x) = lim n→∞ (π(n + x) -π(n))"
and remark that: "It is plain that the determination of a lower bound for (x) is a problem of exceptional depth. . . . The problem of an upper bound has greater possibilities. . . . An examination of the primes less than 200 suggests forcibly that: (x) ≤ π(x) (x ≥ 2)".

An elementary probability-based approach to estimating prime counting functions statistically

In the rest of this investigation we demonstrate the far-reaching significance of defining the statistical probability of n being a prime by giving elementary probability-based proofs that:

(i) The Prime Number Theorem: First, by the Law of Large Numbers, we have π(x) ∼ π L (x) since p 2 n+1 -p 2 n → ∞ (Corollary 3.13). Second, we note the function π L (x)/ x logex is differentiable in the interval (p 2 n , p 2 n+1 ) with derivative (π L (x)/ x logex ) ∈ o(1) (Lemma 3.15). We conclude that both π L (x)/ x logex and π(x)/ x logex do not oscillate as x → ∞.

Chebyshev's Theorem 36 , π(x)

x logex , then yields the Prime Number Theorem (Theorem 3.16): π(x) ∼ x logex .

(ii) Dirichlect's Theorem: By the Law of Large Numbers, the expected value of the number π (a,d) (n) of Dirichlect primes of the form a + m.d which are less than or equal to n, where a, d are co-prime and 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k (q i prime)
, is given by the Dirichlect prime counting function π D (n) (Definition 6), such that:

π (a,d) (n) ∼ π D (n) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .π L (n) → ∞.
(iii) Twin Prime Theorem: By the Law of Large Numbers, the expected value of the number π 2 (n) of twin primes ≤ n is given by the twin-prime counting function:

π T (n) = n j=1 P(j ∈ {p} ∩ j + 2 ∈ {p}).
We conclude that there are infinitely many twin primes since we show that (Corollary 3.34):

π 2 (n) ∼ π T (n) ∼ e -2γ . n log 2 e n
.

3.A. The residues r i (n).

We begin by formally defining the residues r i (n) for all n ≥ 2 and all i ≥ 2 as below 37 :

Definition 1. n + r i (n) ≡ 0 (mod i) where i > r i (n) ≥ 0.
Since each residue r i (n) cycles over the i values (i -1, i -2, . . . , 0), these values are all incongruent and form a complete system of residues 38 mod i.

It immediately follows that:

Lemma 3.1. r i (n) = 0 if, and only if, i is a divisor of n.

3.B. The probability model

M i = {(0, 1, 2, . . . , i -1), r i (n), 1 i }
By the standard definition of the probability P(e) of an event e 39 , we have by Lemma 3.1 that:

Lemma 3.2. For any n ≥ 2, i ≥ 2 and any given integer i > u ≥ 0:

• the probability P(r i (n) = u) that r i (n) = u is 1 i ; • u=i-1 u=0 P(r i (n) = u) = 1; • and the probability P(r i (n) = u) that r i (n) = u is 1 -1 i .
By the standard definition of a probability model 40 , we conclude that:

Theorem 3.3. For any i ≥ 2, M i = {(0, 1, 2, . . . , i -1), r i (n), 1 i } is a probability model for the values of r i (n).

Corollary 3.4. For any n ≥ 2 and any prime p ≥ 2, the probability P(r p (n) = 0) that r p (n) = 0, and that p divides n, is 1 p ; and the probability P(r p (n) = 0) that r p (n) = 0, and that p does not divide n, is 1 -1 p .

We also note the standard definition 41 : Definition 2. Two events e i and e j are mutually independent for i = j if, and only if, P(e i ∩ e j ) = P(e i ).P(e j ). 37 The residues ri(n) can also be graphically displayed variously as shown in the Appendix II in §5.. 38 [HW60], p.49. 39 See §4., Appendix I; also [Ko56], Chapter I, §1, Axiom III, pg.2. 40 See §4., Appendix I. 41 See §4., Appendix I; also [Ko56], Chapter VI, §1, Definition 1, pg.57 and §2, pg.58.

3.C. The prime divisors of any integer n are mutually independent

We then have that:

Lemma 3.5. If p i and p j are two primes where i = j then, for any n ≥ 2, we have:

P((r p i (n) = u) ∩ (r p j (n) = v)) = P(r p i (n) = u).P(r p j (n) = v)
where p i > u ≥ 0 and p j > v ≥ 0.

Proof : The p i .p j numbers v.p i + u.p j , where p i > u ≥ 0 and p j > v ≥ 0, are all incongruent and form a complete system of residues 42 mod (p i .p j ). Hence:

P((r p i (n) = u) ∩ (r p j (n) = v)) = 1 p i .p j
By Lemma 3.2:

P(r p i (n) = u).P(r p j (n) = v) = ( 1 p i )( 1 p j
).

The lemma follows.

If u = 0 and v = 0 in Lemma 3.5, so that both p i and p j are prime divisors of n, we immediately conclude by Definition 2 that:

Corollary 3.6. P((r p i (n) = 0) ∩ (r p j (n) = 0)) = P(r p i (n) = 0).P(r p j (n) = 0).

We can also express this as:

Corollary 3.7. P(p i |n ∩ p j |n) = P(p i |n).P(p j |n).

We thus conclude that:

Theorem 3.8. The prime divisors of any integer n are mutually independent.

3.C.a. Integer Factorising cannot be polynomial-time

We digress briefly from our investigation of prime counting functions to note that Theorem 3.8 immediately yields the actively pursued 43 (although prima facie unconnected) computational complexity consequence that no deterministic algorithm 44 can compute a factor of any randomly given integer n in polynomial time 45 ! We note the standard definition 46 : Definition 3. A deterministic algorithm computes a number-theoretical function f (n) in polynomialtime if there exists k such that, for all inputs n, the algorithm computes f (n) in ≤ (log e n) k + k steps.

It then follows from Theorem 3.8 that:

Corollary 3.9. Any deterministic algorithm that always computes a prime factor of n cannot be polynomial-time.

Proof : Any computational process that successfully identifies a prime divisor of n must necessarily appeal to at least one logical operation for identifying such a factor.

Since n is a prime if, and only if, it is not divisible by any prime p ≤ √ n, and n may be the square of a prime, it follows from Theorem 3.8 that we necessarily require at least one logical operation for each prime p ≤ √ n in order to logically determine whether p is a prime divisor of n.

Since the number of such primes is of the order O(n/log e n), the number of computations required by any deterministic algorithm that always computes a prime factor of n cannot be polynomial-time-i.e. of order O((log e n) c ) for any c-in the length of the input n. The corollary follows.

3.D. The statistical probability P(n ∈ {p}) that n is a prime

Since n is a prime if, and only if, it is not divisible by any prime p ≤ √ n, it follows immediately from Lemma 3.2 and Lemma 3.5 that: Lemma 3.10. For any n ≥ 2, the probability P(n ∈ {p}) of an integer n being a prime p is the probability that

r p i (n) = 0 for any 1 ≤ i ≤ k if p 2 k ≤ n < p 2 k+1 .
By Corollary 3.4 we can express this by the statistical prime probability function (graphically illustrated in §2.C., Fig. 4) 47 :

Theorem 3.11. P(n ∈ {p}) = π( √ n) i=1 (1 -1 p i ) ∼ 2e -γ logen .
It immediately follows that, for any m > π( √ n):

Corollary 3.12.

P(n ∈ {p}) > m i=1 (1 -1 p i ).

3.E. The statistical prime counting function π L (n)

It now follows from Theorem 3.11 that, since p 2 n+1 -p 2 n → ∞ as n → ∞, by the Law of Large Numbers 48 , the expected value 49 of the number π(n) of primes less than or equal to n is given by the prime counting function (graphically illustrated in §2.C., Fig. 5): Gr95], p.13). 48 See §4., Appendix I; also [Ko56], Chapter VI, §3, p.61; [El79b], pp.52-57. 49 See §4., Apendix III. Compare also [HL23], pp.36-37.

Definition 4. π L (n) = n j=1 π( √ j) i=1 (1 -1 p i ). Corollary 3.13. π(n) ∼ π L (n). 47 We note that Ltn→∞logen. π( √ n) i=1 (1 -1 p i ) = 2.e -λ ≈ 1.12292 . . . ([

3.F. The interval (p 2

n , p 2 n+1 )

It also follows immediately from the definition of π(x) as the number of primes less than or equal to x that:

Lemma 3.14.

π( √ x) i=1 (1 -1 p i ) = π( √ x+1) i=1 (1 -1 p i ) for p 2 n ≤ x < p 2 n+1 .
We can also generalise the number-theoretic function of Definition 4 as the real-valued function:

Definition 5. π L (x) = π L (p 2 n ) + (x -p 2 n ) n i=1 (1 -1 p i ) for p 2 n ≤ x < p 2 n+1 .
We note that the graph of

π L (x) in the interval (p 2 n , p 2 n+1 ) for n ≥ 1 is now a straight line with gradient n i=1 (1 -1 p i
), as illustrated in §2.C., Fig. 5 where we defined π L (x) equivalently by:

π L (x) = (x) = (x -p 2 n ) n i=1 (1 -1 p i ) + n-1 j=1 (p 2 j+1 -p 2 j ) j i=1 (1 -1 p i ) + 2 3.G. The function π L (x)/ x logex We consider next the function π L (x)/ x logex in the interval (p 2 n , p 2 n+1 ): π L (x)/ x logex = (π L (p 2 n ) + (x -p 2 n ) n i=1 (1 -1 p i ))/ x logex This now yields the derivative (π L (x). logex x ) in the interval (p 2 n , p 2 n+1 ) as: π L (x).( logex x ) + (π L (x)) . logex x (π L (p 2 n ) + (x -p 2 n ) n i=1 (1 -1 p i )).( logex x ) + (π L (p 2 n ) + (x -p 2 n ) n i=1 (1 -1 p i )) . logex x (π L (p 2 n ) + (x -p 2 n ) n i=1 (1 -1 p i )).( 1 x 2 -logex x 2 ) + ( n i=1 (1 -1 p i )). logex x
Since p 2 n ≤ x < p 2 n+1 and π L (x) ∼ π(x) by the Law of Large Numbers, by Mertens' 50 and Chebyshev's Theorems we can express the above as:

∼ (π L (p 2 n ) + e -γ (x-p 2 n ) logen ).( 1 x 2 -logex x 2 ) + e -γ .logex x.logen ∼ ( π L (p 2 n ) x + e -γ logen (1 - p 2 n x )). (1-logex) x + e -γ .logex x.logen ∼ ( π L (p 2 n ) p 2 n . p 2 n x + e -γ
logen (1 -

p 2 n x )). (1-2.logep n ) p 2 n + 2.e -γ .logep n p 2 n .logen
Since each term → 0 as n → ∞, we conclude that the function π L (x)/ x logex does not oscillate but tends to a limit as x → ∞ since:

Lemma 3.15. (π L (x)/ x logex ) ∈ o(1).

3.H. An elementary probability-based proof of the Prime Number Theorem

The above now yields an elementary probability-based proof that:

Theorem 3.16. π(x) ∼ x/log e x Proof : By Lemma 3.15 (π L (x)/ x logex ) ∈ o(1); whence the function π L (x)/ x logex does not oscillate but tends to a limit as x → ∞.

Since p 2

n+1 -p 2 n → ∞ as n → ∞, and π(x) ∼ π L (x) by the Law of Large Numbers, the theorem follows from Chebyshev's Theorem that π(x) x/log e x.

3.I. Primes in an arithmetic progression

We consider next Dirichlect's Theorem, which is the assertion that if a and d are co-prime and 1 ≤ a < d, then the arithmetic progression a+m.d, where m ≥ 1, contains an infinitude of (Dirichlect) primes.

We first note that Lemma 3.5 can be extended to prime powers in general 51 : Lemma 3.17. If p i and p j are two primes where i = j then, for any n ≥ 2, α, β ≥ 1, we have:

P((r p α i (n) = u) ∩ (r p β j (n) = v)) = P(r p α i (n) = u).P(r p β j (n) = v) where p α i > u ≥ 0 and p β j > v ≥ 0. Proof : The p α i .p β j numbers v.p α i + u.p β j
, where p α i > u ≥ 0 and p β j > v ≥ 0, are all incongruent and form a complete system of residues 52 mod (p α i .p β j ). Hence:

P((r p α i (n) = u) ∩ (r p β j (n) = v)) = 1 p α i .p β j
By Lemma 3.2:

P(r p α i (n) = u).P(r p β j (n) = v) = ( 1 p α i )( 1 p β j ).
The lemma follows.

If u = 0 and v = 0 in Lemma 3.17, so that both p i and p j are prime divisors of n, we immediately conclude by Definition 2 that:

Corollary 3.18. P((r p α i (n) = 0) ∩ (r p β j (n) = 0)) = P(r p α i (n) = 0).P(r p j β (n) = 0).
We can also express this as:

Corollary 3.19. P(p α i |n ∩ p β j |n) = P(p α i |n).P(p β j |n).
We thus conclude that:

Theorem 3.20. For any two primes p = q and natural numbers n, α, β ≥ 1, whether or not p α divides n is independent of whether or not q β divides n.

51 Hint: The following arguments may be easier to follow if we visualise the residues rpα i (n) and r p β i (n) as they would occur in §5., Fig. 6 and Fig. 7. 52 [HW60], p.52, Theorem 59.

3.I.a. The probability that n is a prime of the form a + m.d

We note next that:

Lemma 3.21. For any co-prime natural numbers 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k
where:

q 1 < q 2 < . . . < q k are primes and α 1 , α 2 . . . α k ≥ 1 are natural numbers;

the natural number n is of the form a + m.d for some natural number m ≥ 1 if, and only if:

a + r q α i i (n) ≡ 0 (mod q α i i ) for all 1 ≤ i ≤ k
where 0 ≤ r i (n) < i is defined for all i > 1 by:

n + r i (n) ≡ 0 (mod i) . Proof : First, if n is of the form a + m.d for some natural number m ≥ 1, where 1 ≤ a < d = q α 1 1 .q α 2 2 . . . q α k k , then: n ≡ a (mod d) and : n + r q α i i (n) ≡ 0 (mod q α i i ) f or all 1 ≤ i ≤ k whence : a + r q α i i (n) ≡ 0 (mod q α i i ) f or all 1 ≤ i ≤ k Second: If : a + r q α i i (n) ≡ 0 (mod q α i i ) f or all 1 ≤ i ≤ k and : n + r q α i i (n) ≡ 0 (mod q α i i ) f or all 1 ≤ i ≤ k then : n -a ≡ 0 (mod q α i i ) f or all 1 ≤ i ≤ k whence : n ≡ a (mod d)
The Lemma follows.

By Lemma 3.2, it follows that:

Corollary 3.22. The probability that a + r

q α i i (n) ≡ 0 (mod q α i i ) for any 1 ≤ i ≤ k is 1 q α i i
.

By Theorem 3.20, it further follows that:

Corollary 3.23. The joint probability that a + r

q α i i (n) ≡ 0 (mod q α i i ) for all 1 ≤ i ≤ k is k i=1 1 q α i i .
We conclude by Lemma 3.21 that:

Corollary 3.24. The probability that n is of the form a+m.d for some natural number m ≥ 1, where

1 ≤ a < d = q α 1 1 .q α 2 2 . . . q α k k is k i=1 1 q α i i .

It follows that:

Corollary 3.25. The probability P(n ∈ {p} ∩ n ∈ {a + m.d}) that n is a Dirichlect prime of the form a + m.d for some natural number m ≥ 1, where 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k is: k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .P(n ∈ {p}).
Proof : Since a, d are co-prime, we have by Lemma 3.21 that if n is of the form a + m.d for some natural number m ≥ 1, where 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k , we have that: n ≡ a (mod q i ) f or all 1 ≤ i ≤ k whilst : n + r i (n) ≡ 0 (mod i) f or all 1 ≤ i whence : a + r q i (n) ≡ 0 (mod q i ) f or all 1 ≤ i ≤ k r q i (n) = 0 f or all 1 ≤ i ≤ k and : q i | n f or all 1 ≤ i ≤ k
Hence, if n is of the form a + m.d for some natural number m ≥ 1, where 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k
and (a, d) = 1, the probability that q i |n for all 1 ≤ i ≤ k is 1.

By Lemma 3.10, Theorem 3.11 and Theorem 3.20, the probability that any n ≥ q 2 k is a Dirichlect prime of the form a + m.d is thus:

k i=1 1 q α i i . 2≤p≤ √ n p =q i f or1≤i≤k (1 -1 p ) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 . 2≤p≤ √ n (1 -1 p ) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 . π( √ n) j=1 (1 -1 p j ) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .P(n ∈ {p})
The Corollary follows.

3.I.b. An elementary probability-based proof of Dirichlect's Theorem

It further follows from Theorem 3.11 that, since p 2 n+1 -p 2 n → ∞ as n → ∞, by the Law of Large Numbers 53 the expected value of the number π (a,d) (n) of Dirichlect primes, of the form a + m.d for some natural number m ≥ 1 and 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k ,
that are less than or equal to any n ≥ q 2 k is given by the Dirichlect prime counting function:

Definition 6. π D (n) = n l=1 ( k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .P(l ∈ {p})).
We conclude that:

Lemma 3.26. π (a,d) (n) ∼ π D (n) → ∞ as n → ∞.
Proof : If a, d are co-prime and 1 ≤ a < d = q

α 1 1 .q α 2 2 . . . q α k k , we have for any n ≥ q 2 k :
53 See §4., Appendix I; also [Ko56], Chapter VI, §3, pg. 61.

π D (n) = n l=1 ( k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .P(l ∈ {p})) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .π L (n) = k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 . n l=1 π( √ l) j=1 (1 -1 p j ) ≥ k i=1 1 q α i i . k i=1 (1 -1 q i ) -1 .n. π( √ n) j=1 (1 -1 p j )
The lemma follows since, by Mertens' Theorem, p≤x (1 -1 p ) ∼ e -λ logex , we have that:

n. π( √ n) j=1 (1 -1 p j ) ∼ 2e -γ n loge(n) → ∞ as n → ∞.
We conclude by the Law of Large Numbers when applied to the interval,

p 2 n+1 -p 2 n → ∞ as n → ∞, that:
Theorem 3.27. There are an infinity of primes in any arithmetic progression a + m.d where (a, d) = 1 54 .

3.J. An elementary probability-based proof that there are infinitely many twinprimes

We next note that, by Theorem 3.11, we can define the twin-prime counting function π T (n), which gives the expected value of the number π 2 (n) of twin primes (p i , p i+1 = p i + 2) for 3 ≤ p i ≤ n as:

Definition 7. π T (n) = n j=1 P(j ∈ {p} ∩ j + 2 ∈ {p})
In order to estimate π T (n), we first define: Definition 8. An integer n is a TW integer if, and only if,

r p i (n) = 0 and r p i (n) = 2 for all 1 ≤ i ≤ π( √ n).
Since n is a prime if, and only if, it is not divisible by any prime p ≤ √ n, we then have that:

Lemma 3.28. If n is a TW integer, then n is a prime.

Proof : The lemma follows immediately from Definition 8, Definition 1 and Lemma 3.1.

Lemma 3.29. If n is a TW integer, then n + 2 is either a prime or p 2 π( √ n)+1
. Proof : By Definition 8 and Definition 1:

r p i (n) = 2 f or all 1 ≤ i ≤ π( √ n) n + 2 = λ.i f or all 2 ≤ i ≤ p π( √ n) , λ ≥ 1 Hence, if n + 2 is divisible by p π( √ n)+1 , then n + 2 = p 2 π( √ n)+1
; else it is a prime. Since each residue r i (n) cycles over the i values (i -1, i -2, . . . , 0), these values are all incongruent and form a complete system of residues mod i. It thus follows from Definition 8 and Section 3.B. that the probability of n ≥ 9 being a TW integer is:

Lemma 3.30. P(n ∈ {TW}) = π( √ n) i=2 (1 -2 p i
).

The number π TW (n) of TW integers ≥ 9 but ≤ n is thus:

Lemma 3.31. π TW (n) = n j=9 π( √ j) i=2 (1 -2 p i
).

Since the number of TW integers such that n

+ 2 = p 2 π( √ n)+1 is not more than π( √ n), it also follows that, for n ≥ 9: Lemma 3.32. π T (n) ≥ n j=9 π( √ j) i=2 (1 -2 p i ) -π( √ n).
We further note that:

Theorem 3.33. π T (n) → ∞ as n → ∞.
Proof : We have by Lemma 3.32 that, for n ≥ 9:

π T (n) ≥ (n -9). π( √ n) i=2 (1 -2 p i ) -π( √ n) ≥ (n -9). π( √ n) i=2 (1 -1 p i )(1 -1 (p i -1) ) -π( √ n) ≥ (n -9). π( √ n) i=2 (1 -1 p i )(1 -1 p i-1 ) -π( √ n) ≥ (n -9). π( √ n) i=2 (1 -1 p i-1 ) 2 -π( √ n) ≥ (n -9). n i=1 (1 -1 p i ) 2 -π( √ n)
Now, by Chebyshev's and Mertens' Theorems, we have that:

(n -9). n i=1 (1 -1 p i ) 2 -π( √ n) ∼ (n -9).( e -γ logen ) 2 -π( √ n) ∼ e -2γ . n log 2 e n -9e -2γ log 2 e n -O( √ n logen ) → ∞ as n → ∞ The theorem follows. Since p 2 n+1 -p 2 n → ∞ as n → ∞, it follows by the Law of Large Numbers that π 2 (n) ∼ π T (n) ∼ π T W (n).
We conclude that there are infinitely many twin primes, and that 55 :

Corollary 3.34. π 2 (n) ∼ e -2γ . n log 2 e n .
3.K. The Generalised Prime Counting Function:

n j=1 π( √ j) i=a (1 -b p i )
We note that the argument of Theorem 3.33 in §3.J. is a special case of the limiting behaviour of the Generalised Prime Counting Function

n j=1 π( √ j) i=a (1 -b p i
), which estimates the number of integers ≤ n such that there are b values that cannot occur amongst the residues r p i (n) for a ≤ i ≤ π( √ j) 56 :

55 Where e -2γ = 0.3152373316 . . .; compare [HW60], p.371, §22.20:

π 2 (n) ∼ 2C2. n
n j=1 π( √ j) i=a (1 -b p i ) → ∞ as n → ∞ if p a > b ≥ 1.
Proof : For p a > b ≥ 1, we have that:

n j=1 π( √ j) i=a (1 -b p i ) ≥ n j=p 2 a π( √ j) i=a (1 -b p i ) ≥ n j=p 2 a π( √ n) i=a (1 -b p i ) ≥ (n -p 2 a ). π( √ n) i=a (1 -b p i ) ≥ (n -p 2 a ). n i=a (1 -b p i )
The theorem follows if:

log e (n -p 2 a ) + n i=a log e (1 -b p i ) → ∞ (i)
We note first the standard result for |x| < 1 that:

log e (1 -x) = -∞ m=1 x m m
For any p i > b ≥ 1, we thus have:

log e (1 -b p i ) = -∞ m=1 (b/p i ) m m = -b p i -∞ m=2 (b/p i ) m m
Hence:

n i=a log e (1 -b p i ) = -n i=a ( b p i ) -n i=a ( ∞ m=2 (b/p i ) m m ) (ii) We note next that, for all i ≥ a: c < (1 -b p a ) → c < (1 -b p i )
It follows for any such c that:

∞ m=2 (b/p i ) m m ≤ ∞ m=2 ( b p i ) m = (b/p i ) 2 1-b/p i ≤ b 2 c.p 2 i Since: ∞ i=1 1 p 2 i = O(1)
it further follows that: Probability model58 : A probability model is a mathematical representation of a random phenomenon. It is defined by its sample space, events within the sample space, and probabilities associated with each event.

n i=a ( ∞ m=2 (b/p i ) m m ) ≤ n i=a ( b 2 c.p 2 i ) = O(1) (
• The sample space S for a probability model is the set of all possible outcomes.

• An event A is a subset of the sample space S.

• A probability is a numerical value assigned to a given event A.

Distribution Function59 : Let X be a random variable which denotes the value of the outcome of a certain experiment, and assume that this experiment has only finitely many possible outcomes. Let Ω be the sample space of the experiment (i.e., the set of all possible values of X, or equivalently, the set of all possible outcomes of the experiment). A distribution function for X is a real-valued function m whose domain is Ω and which satisfies:

1. m(ω) ≥ 0, for all ω ∈ n, and

2. ω∈Ω m(ω) = 1.
For any subset E of Ω, we define the probability of E to be the number P (E) given by

P (E) = ω∈E m(ω)
Some notations60 : Let A and B be two sets. Then the union of A and B is the set

A ∪ B = {x | x ∈ A or x ∈ B}
The intersection of A and B is the set

A ∩ B = {x | x ∈ A and x ∈ B}
The difference of A and B is the set

A -B = {x | x ∈ A and x / ∈ B}
The set A is a subset of B, written A ⊂ B, if every element of A is also an element of B. Finally, the complement of A is the set

A = {x | x ∈ Ω and x / ∈ A}.
Mutual Independence 61 : A set of events {A 1 , A 2 , . . . , A n } is said to be mutually independent if for any subset {A i , A j , . . . , A m } of these events we have

P (A i ∩ A j ∩ . . . ∩ A m ) = P (A i )P (A j ) . . . P (A m ),
or equivalently, if for any sequence A 1 , A 2 , . . . , A n with A j = A j or A j ,

P (A i ∩ A j ∩ . . . ∩ A m ) = P (A i )P (A j ) . . . P (A m ).
Expected Value 62 : Let X be a numerically-valued discrete random variable with sample space Ω and distribution function m(x). The expected value E(X) is defined by:

E(X) = x∈Ω xm(x),
provided this sum converges absolutely. We often refer to the expected value as the mean, and denote E(X) by µ for short. If the above sum does not converge absolutely, then we say that X does not have an expected value.

Law of Large Numbers 63 : Let X 1 , X 2 , . . . , X n be an independent trials process, with finite expected value µ = E(X j ) and finite variance σ 2 = V (X j ). Let S n = X 1 + X 2 + . . . + X n . Then for any > 0,

P (| Sn n -µ| ≥ ) → 0
as n → ∞. Equivalently,

P (| Sn n -µ| < ) → 1
as n → ∞.

Appendix II: The residue function r i (n)

We graphically illustrate how the residues r i (n) occur naturally as values of:

A A: The natural-number based residue functions R i (n)

The residues r i (n) can be defined for all n ≥ 1 as the values of the natural-number based residue functions R i (n), defined for all i ≥ 1 as below in Fig. 6. We note that each function R i (n) cycles through the values (i -1, i -2, . . . , 0) with period i. Function:

R 1 n R 2 n R 3 n R 4 n R 5 n R 6 n R 7 n R 8 n R 9 n R 10 n R 11 n . . . R n n
n = 1 0 1 2 3 4 5 6 7 8 9 10 . . . n-1 n = 2 0 0 1 2 3 4 5 6 7 8 9 . . . n-2 n = 3 0 1 0 1 2 3 4 5 6 7 8 . . . n-3 n = 4 0 0 2 0 1 2 3 4 5 6 7 . . . n-4 n = 5 0 1 1 3 0 1 2 3 4 5 6 . . . n-5 n = 6 0 0 0 2 4 0 1 2 3 4 5 . . . n-6 n = 7 0 1 2 1 3 5 0 1 2 3 4 . . . n-7 n = 8 0 0 1 0 2 4 6 0 1 2 3 . . . n-8 n = 9 0 1 0 3 1 3 5 7 0 1 2 . . . n-9 n = 10 0 0 2 2 0 2 4 6 8 0 1 . . . n-10 n = 11 0 1 1 1 4 1 3 5 7 9 0 . . . n-11 n r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n . . . 0 Fig. 6: The natural-number based residue functions Ri(n)

B: The natural-number based residue sequences E(n)

The above residues r i (n) can also be viewed alternatively as values of the associated residue sequences,

E(n) = {r i (n) : i ≥ 1}
, defined for all n ≥ 1, as illustrated below in Fig. 7.

We note that:

• The sequences highlighted in red identify a prime 64 p (since r i (p) = 0 for 1 < i < p);

• The 'boundary' residues r 1 (n) = 0 and r n (n) = 0 are identified in cyan.

64 Conventionally defined as integers that are not divisible by any smaller integer other than 1.

Fig. 7: The natural-number based residue sequences E(n)

Function: R 1 n R 2 n R 3 n R 4 n R 5 n R 6 n R 7 n R 8 n R 9 n R 10 n R 11 n . . . R n n
E(1): 0 1 2 3 4 5 6 7 8 9 10 . . . n-1 E(2): 0 0 1 2 3 4 5 6 7 8 9 . . . n-2 E(3): 0 1 0 1 2 3 4 5 6 7 8 . . . n-3 E(4): 0 0 2 0 1 2 3 4 5 6 7 . . . n-4 E(5):

0 1 1 3 0 1 2 3 4 5 6 . . . n-5 E(6): 0 0 0 2 4 0 1 2 3 4 5 . . . n-6 E(7): 0 1 2 1 3 5 0 1 2 3 4 . . . n-7 E(8): 0 0 1 0 2 4 6 0 1 2 3 . . . n-8 E(9): 0 1 0 3 1 3 5 7 0 1 2 . . . n-9 E(10): 0 0 2 2 0 2 4 6 8 0 1 . . . n-10 E(11): 0 1 1 1 4 1 3 5 7 9 0 . . . n-11 . . .

E(n):

r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n . . . 0 . . .

E N (1): 0 E N (2): 0 0 E N (3): 0 1 0 E N (4): 0 0 2 0 E N (5): 0 1 1 3 0 E N (6): 0 0 0 2 4 0 E N (7): 0 1 2 1 3 5 0 E N (
8): 0 0 1 0 2 4 6 0 E N (9): 0 1 0 3 1 3 5 7 0 E N (10): 0 0 2 2 0 2 4 6 8 0 E N (11): 0 1 1 1 4 1 3 5 7 9 0 . . . E N (n): r 1 n r 2 n r 3 n r 4 n r 5 n r 6 n r 7 n r 8 n r 9 n r 10 n r 11 n . . . 0 . . . We give below in Fig. 9 the output for 2 ≤ n ≤ 31 of a prime-number based algorithm E Q that computes the values q i (n) = r p i (n) of the sequence E P (n) for only each prime 2 ≤ p i ≤ n for any given n. E P (2): 0 E P (3): 1 0 E P (4): 0 2 E P (5): 1 1 0 E P (6): 0 0 4 E P (7): 1 2 3 0 E P (8): 0 1 2 6 E P (9): 1 0 1 5 E P (10): 0 2 0 4 E P (11): 1 1 4 3 0 E P (12): 0 0 3 2 10 E P (13): 1 2 2 1 9 0 E P (14): 0 1 1 0 8 12 E P (15): 1 0 0 6 7 11 E P (16): 0 2 4 5 6 10 E P (17): 1 1 3 4 5 9 0 E P (18): 0 0 2 3 4 8 16 E P (19): 1 2 1 2 3 7 15 0 E P (20): 0 1 0 1 2 6 14 18 E P (21): 1 0 4 0 1 5 13 17 E P (22): 0 2 3 6 0 4 12 16 E P (23): 1 1 2 5 10 3 11 15 0 E P (24): 0 0 1 4 9 2 10 14 22 E P (25): 1 2 0 3 8 1 9 13 21 E P (26): 0 1 4 2 7 0 8 12 20 E P (27): 1 0 3 1 6 12 7 11 19 E P (28): 0 2 2 0 5 11 6 10 18 E P (29): 1 1 1 6 4 10 5 9 17 0 E P (30): 0 0 0 5 3 9 4 8 16 28 E P (31): 1 2 4 4 2 8 3 7 15 27 0 . . . E P (n): q 1 n q 2 n q 3 n q 4 n q 5 n q 6 n q 7 n q 8 n q 9 n q 10 n q 11 n . . . 0 . . .

(n) = r p i (n) : 1 ≤ i ≤ π(n)} is shown only partially, partly in cyan) and E Q (whose output q i (n) = {r p i (n) : 1 ≤ i ≤ π( √ n)}
is highlighted in black and red, the latter indicating the generation of a prime sequence and, ipso facto, definition of the corresponding prime 65 .

Q 1 n Q 2 n Q 3 n Q 4 n Q 5 n Q 6 n Q 7 n Q 8 n Q 9 n Q 10 n Q 11 n . . . E Q (2): 0 (Prime by definition) E Q (3): 1 0 E Q (4): 0 2 E Q (5): 1 1 0 E Q (6): 0 0 4 E Q (7): 1 2 3 0 E Q (8): 0 1 2 6 E Q (9): 1 0 1 5 E Q (10): 0 2 0 4 E Q (11): 1 1 4 3 0 E Q (12): 0 0 3 2 10 E Q (13): 1 2 2 1 9 0 E Q (14): 0 1 1 0 8 12 E Q (15)
: 1 0 0 6 7 11 E Q (16): 0 2 4 5 6 10 E Q (17): 1 1 3 4 5 9 0 E Q (18): 0 0 2 3 4 8 16 E Q (19): 1 2 1 2 3 7 15 0 E Q (20): 0 1 0 1 2 6 14 18 E Q (21): 1 0 4 0 1 5 13 17 E Q (22): 0 2 3 6 0 4 12 16 E Q (23): 1 1 2 5 10 3 11 15 0 E Q (24): 0 0 1 4 9 2 10 14 22 E Q (25): 1 2 0 3 8 1 9 13 21 E Q (26): 0 1 4 2 7 0 8 12 20 E Q (27): 1 0 3 1 6 12 7 11 19 E Q (28): 0 2 2 0 5 11 6 10 18 E Q (29): 1 1 1 6 4 10 5 9 17 0 E Q (30): 0 0 0 5 3 9 4 8 16 28 E Q (31): 1 2 4 4 2 8 3 7 15 27 0 E Q (32): 0 1 3 3 1 7 2 6 14 26 30 E Q (33): 1 0 2 2 0 6 1 5 13 25 29 E Q (34): 0 2 1 1 10 5 0 4 12 24 28 E Q (35): 1 1 0 0 9 4 16 3 11 23 27 E Q (36): 0 0 4 6 8 3 15 2 10 22 26 E Q (37): 1 2 3 5 7 2 14 1 9 21 25 E Q (38): 0 1 2 4 6 1 13 0 8 20 24 E Q (39): 1 0 1 3 5 0 12 18 7 19 23 65 For informal reference and perspective, formal definitions of both the prime-number based algorithms E P and E Q are given in this work in progress Factorising all m ≤ n is of order Θ( n i=2 π(

√ i)).
E Q (40): 0 2 0 2 4 12 11 17 6 18 22 E Q (41): 1 1 4 1 3 11 10 16 5 17 21 E Q (42): 0 0 3 0 2 10 9 15 4 16 20 E Q (43): 1 2 2 6 1 9 8 14 3 15 19 E Q (44): 0 1 1 5 0 8 7 13 2 14 18 E Q (45): 1 0 0 4 10 7 6 12 1 13 17 E Q (46): 0 2 4 3 9 6 5 11 0 12 16 E Q (47): 1 1 3 2 8 5 4 10 22 11 15 E Q (48): 0 0 2 1 7 4 3 9 21 10 14 E Q (49): 1 2 1 0 6 3 2 8 20 9 13 E Q (50): 0 1 0 6 5 2 1 7 19 8 12 E Q (51): 1 0 4 5 4 1 0 6 18 7 11 E Q (52): 0 2 3 4 3 0 16 5 17 6 10 E Q (53): 1 1 2 3 2 12 15 4 16 5 9 E Q (54): 0 0 1 2 1 11 14 3 15 4 8 E Q (55): 1 2 0 1 0 10 13 2 14 3 7 E Q (56): 0 1 4 0 10 9 12 1 13 2 6 E Q (57): 1 0 3 6 9 8 11 0 12 1 5 E Q (58): 0 2 2 5 8 7 10 18 11 0 4 E Q (59): 1 1 1 4 7 6 9 17 10 28 3 E Q (60): 0 0 0 3 6 5 8 16 9 27 2 E Q (61): 1 2 4 2 5 4 7 15 8 26 1 E Q (62): 0 1 3 1 4 3 6 14 7 25 0 E Q (63): 1 0 2 0 3 2 5 13 6 24 30 E Q (64): 0 2 1 6 2 1 4 12 5 23 29 E Q (65): 1 1 0 5 1 0 3 11 4 22 28 E Q (66): 0 0 4 4 0 12 2 10 3 21 27 E Q (67): 1 2 3 3 10 11 1 9 2 20 26 E Q (68): 0 1 2 2 9 10 0 8 1 19 25 E Q (69): 1 0 1 1 8 9 16 7 0 18 24 E Q (70): 0 2 0 0 7 8 15 6 22 17 23 E Q (71): 1 1 4 6 6 7 14 5 21 16 22 E Q (72): 0 0 3 5 5 6 13 4 20 15 21 E Q (73): 1 2 2 4 4 5 12 3 19 14 20 E Q (74): 0 1 1 3 3 4 11 2 18 13 19 E Q (75): 1 0 0 2 2 3 10 1 17 12 18 E Q (76): 0 2 4 1 1 2 9 0 16 11 17 E Q (77): 1 1 3 0 0 1 8 18 15 10 16 E Q (78): 0 0 2 6 10 0 7 17 14 9 15 E Q (79): 1 2 1 5 9 12 6 16 13 8 14 E Q (80): 0 1 0 4 8 11 5 15 12 7 13 E Q (81): 1 0 4 3 7 10 4 14 11 6 12 E Q (82): 0 2 3 2 6 9 3 13 10 5 11 E Q (83): 1 1 2 1 5 8 2 12 9 4 10 E Q (84): 0 0 1 0 4 7 1 11 8 3 9 E Q (85): 1 2 0 6 3 6 0 10 7 2 8 E Q (86): 0 1 4 5 2 5 16 9 6 1 7 E Q (87): 1 0 3 4 1 4 15 8 5 0 6 E Q (88): 0 2 2 3 0 3 14 7 4 28 5 E Q (89): 1 1 1 2 10 2 13 6 3 27 4 E Q (90): 0 0 0 1 9 1 12 5 2 26 3 E Q (91): 1 2 4 0 8 0 11 4 1 25 2 E Q (92): 0 1 3 6 7 12 10 3 0 24 1 E Q (93): 1 0 2 5 6 11 9 2 22 23 0 E Q (94): 0 2 1 4 5 10 8 1 21 22 30 E Q (95): 1 1 0 3 4 9 7 0 20 21 29 E Q (96): 0 0 4 2 3 8 6 18 19 20 28 E Q (97): 1 2 3 1 2 7 5 17 18 19 27 E Q (98): 0 1 2 0 1 6 4 16 17 18 26 E Q (99): 1 0 1 6 0 5 3 15 16 17 25 E Q (100): 0 2 0 5 10 4 2 14 15 16 24 E Q (101): 1 1 4 4 9 3 1 13 14 15 23 E Q (102): 0 0 3 3 8 2 0 12 13 14 22 E Q (103): 1 2 2 2 7 1 16 11 12 13 21 E Q (104): 0 1 1 1 6 0 15 10 11 12 20 E Q (105): 1 0 0 0 5 12 14 9 10 11 19 E Q (106): 0 2 4 6 4 11 13 8 9 10 18 E Q (107): 1 1 3 5 3 10 12 7 8 9 17 E Q (108): 0 0 2 4 2 9 11 6 7 8 16 E Q (109): 1 2 1 3 1 8 10 5 6 7 15 E Q (110): 0 1 0 2 0 7 9 4 5 6 14 E Q (111): 1 0 4 1 10 6 8 3 4 5 13 E Q (112): 0 2 3 0 9 5 7 2 3 4 12 E Q (113): 1 1 2 6 8 4 6 1 2 3 11 E Q (114): 0 0 1 5 7 3 5 0 1 2 10 E Q (115): 1 2 0 4 6 2 4 18 0 1 9 E Q (116): 0 1 4 3 5 1 3 17 22 0 8 E Q (117): 1 0 3 2 4 0 2 16 21 28 7 E Q (118): 0 2 2 1 3 12 1 15 20 27 6 E Q (119): 1 1 1 0 2 11 0 14 19 26 5 E Q (120): 0 0 0 6 1 10 16 13 18 25 4 E Q (121): 1 2 4 5 0 9 15 12 17 24 3 . . . E Q (n): q 1 n q 2 n q 3 n q 4 n q 5 n q 6 n q 7 n q 8 n q 9 n q 10 n q 11 n . . . . . . Prime: p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 . . . p n . . . Divisor: 2 3 5 7 11 13 17 19 23 29 31 . . . p n . . . The following table gives the statistically expected values π L

(n) = n j=1 π( √ j)
i=1 (1 -1/p j ), and the actual values π(n), of the primes less than or equal to n for 4 ≤ n ≤ 1500.

n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(n) 1 2 0 1/2 1 0.0000 0 2 3 0 1/3 1 1/2 0.
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j)
n p n π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π(
π( √ n) n j=1 (1 -1/p j ) [( √ n)] π( √ n) j=1 (1 -1/p j ) π L (n) = n j=1 π( √ j) i=1 (1 -1/p j ) π (

  . Known approximations of π(n) for finite values of n

Fig. 2 :

 2 Fig.2: The distribution of the primes

Fig. 2 :

 2 Fig.2: The above graph compares the actual number π(x) (red) of primes ≤ x with the distribution of primes as estimated variously by the functions Li(x) (blue), R(x) (black), and x logex (green), where R(x) is Riemann's function ∞ n=1

Fig. 3 :

 3 Fig.3: The above graph compares the statistically expected values (red) vs actual values (blue) of π(n) for 4 ≤ n ≤ 1500 13 , where the statistically expected value π L (n) of π(n) is n j=1

Fig. 4 :

 4 Fig.4: The graph of y = π( √ x) i=1 (1 -1 p i )

Fig. 5 :Fig. 5 :

 55 Fig.5: The graph of y = (x) = π L (x)

  log 2 e n , where C2 = p≥3 p(p-2) (p-1) 2 ≈ 0.66016181584 . . .. 56 Thus b = 1 yields an estimate for the number of primes ≤ n, and b = 2 an estimate for the number of TW primes (Definition 8) ≤ n.Theorem 3.35.

  iii) From the standard result 57 : p≤x 1 p = log e log e x + O(1) + o(1) it then follows that: .(log e log e n + O(1) + o(1)) -O(1) The theorem follows since: log e (n -p 2 a ) -b.(log e log e n + O(1) + o(1)) -O(1) → ∞ and so: log e (n -p 2 a ) + n i=a log e (1 -b p i ) → ∞ 4. Appendix I: Definitions of some terms and concepts of Probability Theory

:

  The natural-number based residue functions R i (n); B: The natural-number based residue sequences E(n); and as the output of: C: The natural-number based algorithm E N ; D: The prime-number based algorithm E P ; E: The prime-number based algorithm E Q .

Fig. 6 :

 6 Fig.6: The natural-number based residue functions R i (n)

Fig. 7 :

 7 Fig.7: The natural-number based residue sequences E(n)

Fig. 8 :

 8 Fig.8: The output of the natural-number based algorithm E N

Fig. 8 :

 8 Fig.8: The output of the natural-number based algorithm E N

Fig. 9 :

 9 Fig.9: The output of the prime-number based algorithm E P

Fig. 9 :

 9 Fig.9: The output of the prime-number based algorithm E P E: The output of the prime-number based algorithms E P and E Q We give below in Fig.10 the output for 2 ≤ n ≤ 121 of the two prime-number based algorithms E P (whose output {q i (n) = r p i (n) : 1 ≤ i ≤ π(n)} is shown only partially, partly in cyan) and E Q (whose

Fig. 10 :

 10 Fig.10: The output of the prime-number based algorithms E P and E Q

Fig. 10 :

 10 Fig.10: The output of the prime-number based algorithms E P and E Q

cf. How Many Primes Are There? In The Prime Pages. Retrieved 10:29, September 27, 2015, from: https://primes.utm.edu/howmany.html.

The curious reluctance to define prime probability statistically

The curious reluctance to define prime probability statistically

B. S. Anand, The Curious Reluctance to Define Prime Probability Statistically

An elementary probability-based approach to estimating prime counting functions statistically

An elementary probability-based approach to estimating prime counting functions statistically

[START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF], p.351, Theorem 427.

cf. http://www.stat.yale.edu/Courses/1997-98/101/probint.htm.

Excerpted from [GS97], Chapter 1, §1.2, p.19.

Excerpted from[START_REF] Grinstead | Introduction to Probability[END_REF], Chapter 1, §1.2, p.21.

Appendix II: The residue function r i (n)

Appendix III: Distribution of primes ≤ 1500

Appendix III: Distribution of primes ≤ 1500

Appendix IV: Resources

The only resources needed by a reader for following, and reproducing, the proofs of this paper are knowledge of: Ideally, all the lemmas, theorems and corollaries of the paper could be given as exercises to students in any introductory course or textbook on the Theory of Numbers.

The thesis of the paper for a student of Number Theory is that all the known functions such as x/log e x, Li(x), and Riemann's R(x) are only proven to be asymptotically equivalent to π(x) (Fig. 2). There can be many such functions. However, none of them can claim to yield a mathematically unique estimate of π(n) for finite values of n.


Qingtian Zhang
email: zhangq@math.psu.edu

Global wellposedness of cubic Camassa-Holm equations

In this paper, we study the Cauchy problem of cubic Camassa-Holm equation or socalled Fokas-Olver-Rosenau-Qiao equation. This equation is derived from a model of shallow water dynamics. We prove the global existence of entropy weak solution to this problem in space H 1 ∩ Ẇ 2,1 . The stability and uniqueness of entropy weak solution are obtained in W 1,1 .

Introduction

We consider the cubic Camassa-Holm equation (cCH), or so-called Fokas-Olver-Rosenau-Qiao (FORQ) equation:

m t + [(u 2 -u 2 x )m] x = 0, m = u -u xx , (1.1)
or u t -u xxt + 3u 2 u x -u 3 x -4uu x u xx + 2u x u 2 xx + (u 2 x -u 2 )u xxx = 0. (1.2)
If written as a first-order equation of u, it is

u t + u 2 u x = 1 3 u 3 x -Φ(u), (1.3)
with

Φ(u) = (1 -∂ 2 x ) -1 [ 1 3 u 3 x ] + ∂ x (1 -∂ 2 x ) -1 [ 2 3 u 3 + uu 2 x ] . ( 1.4)
The initial data u| t=0 = u 0 (x). (1.5) This equation is a complete integrable water wave equation proposed by several different authors [START_REF] Fokas | On a class of physically important integrable equations[END_REF][START_REF] Olver | Tri-Hamiltonian duality between solitons and solitrary-wave solutions having compact support[END_REF][START_REF] Qiao | A new integrable equation with cuspons and W/M-shape-peaks solitons[END_REF]. It is a suitable approximation of incompressible Euler system without swirl. For smooth solutions to the equation, we have

(u 2 ) t + ( 1 2 u 4 ) x = 2 3 u 3 x u -2uΦ(u), (1.6) (u 2 x ) t + (u 2 u 2 x ) x = 2u 3 x u xx + 4 3 u 3 u x -2u x ∂ x (1 -∂ 2 x ) -1 ( 1 3 u 3 x ) -2u x (1 -∂ 2 x ) -1 [ 2 3 u 3 + uu 2 x ]. (1.7)
So we can check that the following two quantities are conserved:

H 1 (u) = ∫ IR (u 2 + u 2 x )dx, H 2 (u) = ∫ IR (u 4 + 2u 2 u 2 x - 1 3 u 4 x )dx. (1.8) 1
In fact, these two invariant quantities are the Hamiltonian functionals of (cCH), with the compatible hamiltonian operators

J 2 = -∂ x m∂ -1 x m∂ x , J 1 = - 1 4 (∂ x -∂ 3 x ).
(1.9)

With these notations, the equation can be written in the bi-Hamiltonian form

m t = J 2 δH 1 δm = J 1 δH 2 δm .
(1.10)

In recent decades, the partial differential equations from the fields of integrable systems have been studied adequately. Among them are KdV equation (KdV), modified KdV equations (mKdV), generalized KdV equations (gKdV), Camassa-Holm equations (CH), generalized Camassa-Holm equations (gCH), and so on. (KdV) and (mKdV) are completely integrable.

They have Lax pair formulation of the equations, which allows to apply inverse scattering techniques. However, few equations are completely integrable. From analytical point of view, the Hamiltonian quantities usually provide us the functional spaces for the solutions, which is not sufficient to obtain a solution. It usually requires further regularity to guarantee compactness. For example, (gKdV) is a semilinear dispersive equation. In [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF], the authors proved global well-posedness by using contract mapping theorem via dispersive estimates. Camassa-Holm equations are closely related with KdV equations in the sense that Camassa-Holm equations are the tri-Hamiltonian duality of KdV equations [START_REF] Olver | Tri-Hamiltonian duality between solitons and solitrary-wave solutions having compact support[END_REF]. However, unlike KdV equations, Camassa-Holm equations do not have that strong dispersive effect. In fact, they are more like quasilinear hyperbolic equations. They have the wave breaking phenomena which is not shared by the KdV equations. The problem of global well-posedness of Camassa-Holm equation is equivalent to the problem of how to continue the solution after the singularity appears. There are two methods to deal with it in general case. In [START_REF] Xin | On the weak solutions to a shallow water equation[END_REF][14], the author proved global existence of weak solution by compensated compactness method, and also proved a result of weak-strong uniqueness. The other way to obtain global well-posedness is by the method of characteristics. In [START_REF] Bressan | Global conservative solutions to the Camassa-Holm equation[END_REF] [3], conservative and dispersive weak solutions are constructed, and in [START_REF] Bressan | Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics[END_REF], the authors proved the uniqueness of conservative weak solutions. Different from the Camassa-Holm equation, the cubic Camassa-Holm equation has higher nonlinearity. This difficulty is manifested in the equation of characteristics:

dx dt = (u 2 -u 2 x )(t, x(t)). (1.11)
From the two conserved Hamiltonians, it is hard to expect the right-hand-side to be continuous with respect to x. So this characteristic ODE may not be wellposed when singularity occurs. The method of characteristics does not apply here.

Here I would like to mention another interesting equation -Novikov equation

u t -u xxt + 4u 2 u x -3uu x u xx -u 2 u xxx = 0. (1.12)
This equation shares the same conserved quantity with (cCH) but has lower order of nonlinearity. An advantage of this is reflected on the equation of characteristics:

dx dt = u 2 (t, x(t)). (1.13)
The right-hand-side is Hölder continuous with respect to x. So the method of characteristics works well here. For details, see [START_REF] Chen | Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation[END_REF].

For the cubic Camassa-Holm equation, local well-posedness of strong solutions has been proved in [START_REF] Fu | On the Cauchy problem for the integrable modified Ca-massaCHolm equation with cubic nonlinearity[END_REF] [7] and [START_REF] Himonas | The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation[END_REF]. In [START_REF] Gui | Wave-breaking and peakons for a modified Camassa-Holm equation[END_REF], the authors also revealed the blow-up mechanism of (cCH). They proved that inf x∈IR {(mu x )(t, x)} will approach -∞ in finite time. It means that the first-order and the second-order derivatives of the solution cannot both be bounded all the time. How to continue the solution after the singularity is an open question, which is our main concern in this paper.

Observe that in (cCH), there is a cubic term of u x . This implies that the (cCH) is more like a Hamilton-Jacobi equation, rather than a perturbed hyperbolic conservation law. Based on this observation, we take 1st order derivative to (cCH), and then view the obtained equation as a balance law for u x . This suggests us to use the technique in the theory of scalar conservation laws. Fortunately, the source term is under control and we can prove BV estimates of u x from the conservative form of equation of m, which leads us to use the Kružhikov's theory [START_REF] Kružhikov | First order quasi-linear equations in several independent variables[END_REF] to prove global wellposedness. Denote X to be the functional space

X := {f ∈ H 1 (IR) ∩ L 1 (IR)|f x ∈ BV }, (1.14) with the norm ∥f ∥ X = ∥f ∥ H 1 + ∥f ∥ L 1 + ∥f x ∥ BV .
Our main theorem is:

Theorem 1.1 (Well-posedness of entropy weak solution). Suppose the initial data u 0 ∈ X.

There exists an entropy weak solution to the Cauchy problem (1.3), (1.5), where the definition of the entropy weak solution is in Definition 3.2 in section 3.

The entropy weak solutions have following stability property. Fix any T > 0, and let u, v : IR + × IR → IR be two entropy weak solutions to (1.3), (1.5) with initial data u 0 , v 0 ∈ X, respectively. Then for any t ∈ (0, T )

∥u(t, •) -v(t, •)∥ W 1,1 ≤ e M T t ∥u 0 -v 0 ∥ W 1,1 , (1.15)
where M T is a constant related with T and H 1 ∩W 2,1 norms of initial data u 0 , v 0 . This implies uniqueness. We have the continuity in time

∥u(t 2 , •) -u(t 1 , •)∥ W 1,1 ≤ C T |t 2 -t 1 |, for all t 1 , t 2 ∈ [0, T ], (1.16)
for some constant C T . Consequently, the initial data can be interpreted in the following sense:

lim t→0+ ∥u(t, •) -u 0 (x)∥ W 1,1 = 0.
(1.17) Remark 1.1. This weak solution includes peakon and anti-peakon solutions with the expression u(t, x) = ae -|x-ct| , a, c ∈ IR.

Remark 1.2. This theorem partially answered the open question proposed in [START_REF] Gui | Wave-breaking and peakons for a modified Camassa-Holm equation[END_REF]. For an initial data outside Ẇ 2,1 , whether we can produce a global weak solution is still open.

The outline of this paper is following. In section 2, we prove the global well-posedness of approximate problem. In section 3, we prove the existence of entropy weak solutions by a compactness argument. The uniqueness and stability results are proved in section 4.

Global wellposedness of the approximate equation

We will prove the existence of solutions by using the viscous approximation. Consider the parabolic equation

m ε,t + [(u 2 ε -u 2 ε,x )m ε ] x = εm ε,xx . (2.18)
It can be written as

u ε,t -u ε,txx + [(u 2 ε -u 2 ε,x )(u ε -u ε,xx )] x = εu ε,xx -εu ε,xxxx , (2.19) or u ε,t + u 2 ε u ε,x = 1 3 u 3 ε,x -Φ(u ε ) + εu ε,xx . (2.20)
Rewrite the equation as

u ε,t -εu ε,xx = -u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ), (2.21)
and denote the right-hand-side as F (u ε , u ε,x ) for short. The initial data is the smooth approximation of the original initial data:

u ε (0, x) = u 0ε (x) = J ε * u 0 , (2.22)
where

J ε (x) = C ε exp(ε 2 /(x 2 + ε 2 )), C is a constant such that ∫ IR J ε (x)dx = 1. (2.23)
In this section, we assume ε > 0 fixed. We prove global wellposedness by two steps. First, we prove local wellposedness. Then we prove global wellposedness by a global a priori estimate.

Local well-posedness Proposition (Local wellposedness). There exists a time

T 0 > 0, such that the Cauchy problem (2.21), (2.22) is well-posed, u ∈ C([0, T 0 ]; H 2 (IR)).
The proof of local wellposedness is standard. We record it here for completeness.

Proof. We first consider the linearized equation

u ε,t -εu ε,xx = -ū 2 ε ūε,x + 1 3 ū3 ε,x -Φ(ū ε ), (2.24) or m ε,t -εm ε,xx = [(-ū 2 ε + ū2 ε,x ) mε ] x , (2.25)
where mε = ūε -ūε,xx .

By Duhamel principle, we can express the solution by the integral form,

u ε (t, x) = S(t)u 0ε + ∫ t 0 S(t -s)(-ū 2 ε ūε,x + 1 3 ū3 ε,x -Φ(ū ε ))(s, x)ds (2.26)
where

S(t)v = 1 √ 4πεt e -x 2 2πεt * v is the heat semigroup.
4

Define a map T as

T u ε = S(t)u 0ε + ∫ t 0 S(t -s)(-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ))(s, x)ds, (2.27) then (2.26) is equivalent to u ε = T ūε .
In the following, we will show that for T sufficiently small, T is a contraction mapping from L ∞ (0,

T 0 ; H 2 (IR)) to itself. (i) T is a map from L ∞ (0, T 0 ; H 2 (IR)) to itself. Suppose u ε ∈ L ∞ (0, T 0 ; H 2 (IR)). We prove T u ε ∈ L ∞ (0, T 0 ; H 2 (IR))
, where T 0 is a positive number to be determined later. In fact,

∥∂ 2 x T u ε (t, x)∥ L ∞ T 0 L 2 x ≤ ∂ 2 x S(t)u 0ε L ∞ T 0 L 2 x + ∫ t 0 ∫ IR 1 √ 4πε(t -s) ∂ x e - (x-y) 2 2πε(t-s) ∂ y (-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ))(s, y)dyds L ∞ T 0 L 2 x ≤ ∥∂ 2 x u 0ε ∥ L 2 x + ∫ t 0 1 √ 4πε(t -s) ∂ x e -x 2 2πε(t-s) L 1 x • (-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε )) x (s, •) L 2 x ds L ∞ T 0 ≤ ∥∂ 2 x u 0ε ∥ L 2 x + C √ ε √ T 0 (∥u ε ∥ 3 L ∞ T 0 H 2 x + ∥Φ∥ L ∞ T 0 L 2 x ).
(2.28) Here and in the following, we use C to express an absolute constant which may be different from line by line. By considering the convolution expression of Φ and Young's inequality, we have

∥Φ∥ L ∞ T 0 L 2 x ≤ 1 2 e -|x| L 1 x • 4 3 u 3 ε,x + 2u 2 ε u ε,x + 2u ε u ε,x u ε,xx L ∞ T 0 L 2 x ≤ C∥u ε ∥ 3 L ∞ T H 2 x (2.29) So ∂ 3 x T u ε (t, x) ∈ L ∞ (0, T ; L 2 (IR)). ∥T u ε (t, x)∥ L ∞ T 0 L 2 x ≤ ∥S(t)u 0ε ∥ L ∞ T 0 L 2 x + ∫ t 0 ∫ IR 1 √ 4πε(t -s) e - (x-y) 2 2πε(t-s) (-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ))(s, y)dyds L ∞ T 0 L 2 x ≤ ∥u 0ε ∥ L 2 x + ∫ t 0 1 √ 4πε(t -s) e -x 2 2πε(t-s) L 1 x • -u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε )(s, •) L 2 x ds L ∞ T 0 ≤ ∥u 0ε ∥ L 2 x + C √ ε T 0 (∥u ε ∥ 3 L ∞ T 0 H 2 x + ∥Φ∥ L ∞ T 0 L 2 x ).
(2.30) So we can conclude that T u ε ∈ L ∞ (0, T 0 ; H 2 (IR)) by interpolation.

(ii) When T 0 is sufficiently small, T is a contraction mapping.

∥T u -T v∥ L ∞ T 0 H 2 x ≤ C ∫ t 0 ∫ IR ∂ x e - (x-y) 2 2πε(t-s) √ 4πε(t -s) ∂ y [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, y)dyds L ∞ T 0 L 2 x +C ∫ t 0 ∫ IR e - (x-y) 2 2πε(t-s) √ 4πε(t -s) [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, y)dyds L ∞ T 0 L 2 x
(2.31) By Young's inequality, the first term

∫ t 0 ∫ IR ∂ x e - (x-y) 2 2πε(t-s) √ 4πε(t -s) ∂ y [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, y)dyds L ∞ T 0 L 2 x ≤ ∫ t 0 ∂ x e -x 2 2πε(t-s) √ 4πε(t -s) L 1 x ∂ x [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, x) L 2 x ds ≤ ∫ t 0 C √ εs ds(∥u∥ L ∞ T 0 H 2 x + ∥v∥ L ∞ T 0 H 2 x )∥u -v∥ L ∞ T 0 H 2 x ≤ C √ ε √ T 0 (∥u∥ L ∞ T 0 H 2 x + ∥v∥ L ∞ T 0 H 2 x )∥u -v∥ L ∞ T 0 H 2 x .
(2.32) The second term

∫ t 0 ∫ IR e - (x-y) 2 2πε(t-s) √ 4πε(t -s) [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, y)dyds L ∞ T 0 L 2 x ≤ ∫ t 0 e -x 2 2πε(t-s) √ 4πε(t -s) L 1 x [-u 2 ε u ε,x + 1 3 u 3 ε,x -Φ(u ε ) -(-v 2 ε v ε,x + 1 3 v 3 ε,x -Φ(v ε ))](s, x) L 2 x ds ≤ CT 0 (∥u∥ L ∞ T 0 H 2 x + ∥v∥ L ∞ T 0 H 2 x )∥u -v∥ L ∞ T 0 H 1 x (2.33) When T 0 is small enough, we have ∥T u -T v∥ L ∞ T 0 H 2 x ≤ C √ ε √ T 0 (∥u∥ L ∞ T 0 H 2 x + ∥v∥ L ∞ T 0 H 2 x )∥u -v∥ L ∞ T 0 H 2 x .
(2.34)

When T 0 is small enough, we can obtain that the map T is a contraction mapping. By contraction mapping theorem, there is a unique fixed point of the map T . So we proved the local wellposedness.

A priori estimates

1. L 1 estimate of m ε . Assume η(v) = |v|, η δ (v) = J δ (v) * η(v), so η δ (v) → η(v) as δ → 0.
Multiply equation (2.18) with η ′ δ (m ε ), take integral and let δ → 0,

∫ IR η(m ε (t, x))dx - ∫ IR η(m ε (0, x))dx ≤ ε ∫ t 0 ∫ IR -η ′′ (m ε )m 2 ε,x dx ≤ 0. (2.35) So we have ∥m ε (t, •)∥ L 1 ≤ ∥m ε,0 ∥ L 1 .
(2.36)

2. H 1 estimate. We have

(u 2 ε ) t + ( 1 2 u 4 ε ) x = 2 3 u 3 ε,x u ε -2u ε Φ(u ε ) + 2εu ε,xx u ε , (2.37) (u 2 ε,x ) t + (u 2 ε u 2 ε,x ) x = 2u 3 ε,x u ε,xx + 4 3 u 3 ε u ε,x -2u ε,x ∂ x (1 -∂ 2 x ) -1 ( 1 3 u 3 ε,x ) -2u ε,x (1 -∂ 2 x ) -1 [ 2 3 u 3 ε + uu 2 ε,x ] + 2εu ε,x u ε,xxx .
(2.38)

Take the summation and then integrate,

d dt ∫ IR u 2 ε + u 2 ε,x dx = -2ε ∫ IR u 2 ε,x + u 2 ε,xx dx (2.39)
So we have the energy estimate,

∫ IR u 2 ε (t, x) + u 2 ε,x (t, x)dx ≤ ∫ IR u 2 ε (0, x) + u 2 ε,x (0, x)dx.
(2.40)

3. L 1 estimate of u ε and u ε,xx . Multiply η ′ δ (u ε ) to the equation of u ε , take integration, and let δ → 0,

∫ IR η(u ε (t, x))dx - ∫ IR η(u ε (0, x))dx ≤ ∫ t 0 ∫ IR 1 3 |u ε,x | 3 + |Φ(u ε )|dx + ε ∫ IR η ′ (u ε )u ε,xx dx ≤ ∫ t 0 ∫ IR 1 3 |u ε,x | 3 dx + ∫ t 0 ∫ IR (e -|x| * 1 3 u 3 ε,x ) dx + ∫ t 0 ∫ IR e -|x| * ∂ x [ 2 3 u 3 ε + u ε u 2 ε,x ] dx -ε ∫ t 0 ∫ IR η ′′ (u ε )u 2 ε,x dx ≤ ∫ t 0 ( 2∥u ε ∥ 3 H 1 + ∥u ε ∥ 2 H 1 ∥u ε,xx ∥ L 1 + ∥u ε,x ∥ L ∞ ∥u ε,x ∥ 2 L 2 + 4∥u ε ∥ 3 H 1 ) dt (2.41) So ∥u ε (t, •)∥ L 1 -∥u ε (0, •)∥ L 1 ≤ ∫ t 0 6∥u ε ∥ 3 H 1 + ∥u ε ∥ 2 H 1 (∥u ε ∥ L 1 + ∥m ε ∥ L 1 )dt. (2.42) ∥u ε (t)∥ L 1 ≤ ∥u ε,0 ∥ L 1 e E 0 t + (6E 1/2 0 + ∥m ε,0 ∥ L 1 )(e E 0 t -1). (2.43)
Combining the estimate of m, we obtain the

L 1 estimate of u xx ∥u ε,xx (t)∥ L 1 ≤ ∥m ε,0 ∥ L 1 + ∥u ε,0 ∥ L 1 e E 0 t + (6E 1/2 0 + ∥m ε,0 ∥ L 1 )(e E 0 t -1).
(2.44)

Above three estimates are all uniform with respect to ε. Next, we derive the high order energy estimates, which are depend on ε.

4. H 2 estimate. u ε,xxt -εu ε,xxxx = ∂ 2 x F (u ε , u ε,x ), (2.45)
where the right-hand-side is

∂ 2 x F (u ε , u ε,x ) = ∂ x (-2u ε u 2 ε,x -u 2 ε u ε,xx + u 2 ε,x u ε,xx -∂ x Φ(u ε )) = -6u ε u ε,x u ε,xxx -2u 3 ε,x -u 2 ε u ε,xxx + 2u ε,x u 2 ε,xx + u 2 ε,x u ε,xxx -∂ 2 x Φ(u ε ).
(2.46) Multiply u ε,xx and then take the integral,

d dt ∫ IR 1 2 u 2 ε,xx dx + ε ∫ IR u 2 ε,xxx dx ≤ ∫ IR u ε,xxx (-2u ε u 2 ε,x -u 2 ε u ε,xx + u 2 ε,x u ε,xx -∂ x Φ(u ε )) dx ≤ ∫ IR ε 2 u 2 ε,xxx + 2 ε (-2u ε u 2 ε,x -u 2 ε u ε,xx + u 2 ε,x u ε,xx -∂ x Φ(u ε )) 2 dx (2.47) So we obtain d dt ∫ IR 1 2 u 2 ε,xx dx + ε 2 ∫ IR u 2 ε,xxx dx ≤ ∫ IR 8 ε [4u 2 ε u 4 ε,x + u 4 ε u 2 ε,xx + u 4 ε,x u 2 ε,xx + (∂ x Φ) 2 (u ε )]dx ≤ 32 ε ( ∥u ε ∥ 2 H 1 ∥u ε,xx ∥ 2 L 1 ∥u ε ∥ 2 H 1 + ∥u ε ∥ 4 H 1 ∥u ε ∥ 2 H 2 + ∥u ε,xx ∥ 4 L 1 ∥u ε ∥ 2 H 2 + ∥∂ x Φ(u ε )∥ 2 L 2
) .

(2.48) Considering (1.4), we have

∥∂ x Φ(u ε )∥ 2 L 2 ≤ 4 3 u 3 ε,x + 2u 2 ε u ε,x + 2u ε u ε,x u ε,xx L 2 x 1 2 e -|x| L 1 x ≤ 4 3 ∥u ε,xx ∥ 2 L 1 x ∥u ε ∥ H 1 x + 2∥u ε ∥ 3 H 1 x + 2∥u ε ∥ H 1 x ∥u ε,xx ∥ L 1 x ∥u ε ∥ H 2 x ≤ ( 4 3 ∥u ε,xx ∥ 2 L 1 x + 2∥u ε ∥ 2 H 1 x + 2∥u ε ∥ H 1 x ∥u ε,xx ∥ L 1 x )∥u ε ∥ H 2 x .
(2.49) Therefore, d dt

1 2 ∥u ε ∥ 2 H 2 x + ε 2 ∥u ε,xxx (t, •)∥ 2 L 2 x ≤ C ε [ ∥u ε ∥ 4 H 1 x ∥u ε,xx ∥ 2 L 1 x + ∥u ε,xx ∥ 4 L 1 x + ∥u ε ∥ 4 H 1 x + ∥u ε ∥ 2 H 1 x ∥u ε,xx ∥ 2 L 1 x +∥u ε ∥ 2 H 2 x (∥u ε ∥ 4 H 1 x + ∥u ε,xx ∥ 4 L 1 x + 1)
] .

(2.50)

By Gronwall inequality, we have for any fixed

T , ∥u ε ∥ L ∞ T H 2 x and ∥u ε,x ∥ L 2 T H 2
x are bounded by some constant C(ε, T ), which is related with the initial data, ε and T . 5. H ℓ estimate. We prove H ℓ (ℓ > 2) estimate by a induction argument. Assume that for any T, ε > 0, there is a constant

C(ε, T ), such that ∥u ε ∥ L ∞ T H ℓ-1 x + ∥u ε ∥ L 2 T H ℓ x < C(ε, T ), for every t ∈ [0, T ].
Then we derive H ℓ estimate. Take ℓth order derivative

(∂ ℓ x u ε ) t -ε∂ ℓ x u ε,xx = ∂ ℓ x F (u ε , u ε,x ).
(2.51)

d dt ∫ IR 1 2 |∂ ℓ x u ε | 2 dx + ε ∫ IR |∂ ℓ+1 x u ε | 2 dx = ∫ IR |∂ ℓ x F (u ε , u ε,x )∂ ℓ x u ε |dx ≤ 2 ε ∫ IR |∂ ℓ-1 x F (u ε , u ε,x )| 2 dx + ε 2 ∫ IR |∂ ℓ+1 x u ε | 2 dx (2.52) So d dt ∫ IR 1 2 |∂ ℓ x u ε | 2 dx + ε 2 ∫ IR |∂ ℓ+1 x u ε | 2 dx ≤ 2 ε ∥∂ ℓ-1 x F (u ε , u ε,x )∥ 2 L 2 x ≤ 2 ε [ ∥u 2 ε + u 2 ε,x ∥ L ∞ x ∥∂ ℓ x u ε ∥ 2 L 2 x + ∥Q(u ε , ..., ∂ ℓ-1 x u ε )∥ 2 L 2 x ]
(2.53) We only care about the term with ∂ ℓ-1

x u ε in the term Q, since the lower order terms like

∂ i x u ε ∂ j x u ε ∂ k x u ε with i, j, k < ℓ -1 can be estimated by ∥∂ i x u ε ∂ j x u ε ∂ k x u ε ∥ L 2 x ≤ ∥∂ i x u ε ∥ L ∞ x ∥∂ j x u ε ∥ L ∞ x ∥∂ k x u ε ∥ L 2 x ≤ ∥u ε ∥ 3 H ℓ-1 x < [C(ε, T )] 3 .
(2.54)

When ℓ = 3, there is term u ε,x u 2 ε,xx . To estimate this term, we need to use the boundedness

of ∥u ε,xxx ∥ L 2 T L 2 x . ∥u ε,x u 2 ε,xx ∥ L 2 T L 2 x ≤ ∥u ε,x ∥ L ∞ T L ∞ x ∥u ε,xx ∥ L 2 T L ∞ x ∥u ε,xx ∥ L ∞ T L 2 x ≤ ∥u ε,x ∥ L ∞ T L ∞ x ∥u ε,xx ∥ L 2 T H 1 x ∥u ε,xx ∥ L ∞ T L 2 x < ∞.
(2.55)

When ℓ ≥ 4, Q is linear with respect to ∂ ℓ-1 x u ε . ∥∂ i x u ε ∂ j x u ε ∂ ℓ-1 x u ε ∥ L ∞ t L 2 x ≤ ∥∂ i x u ε ∥ L ∞ t L ∞ x ∥∂ j x u ε ∥ L ∞ t L ∞ x ∥∂ ℓ-1 x u ε ∥ L ∞ t L 2 x ≤ ∥u ε ∥ 3 L ∞ t H ℓ-1 x < [C(ε, T )] 3 .
(2.56) So in every case, we can obtain

∥u ε ∥ L ∞ T H ℓ x + ∥u ε ∥ L 2 T H ℓ+1 x is bounded.

Global wellposedness and regularity

Then from the above a priori estimates, we have Proposition 2.2 (Global well-posedness and regularity). If ε is a fixed positive constant. For any

T > 0, ℓ ≥ 2. If the initial data u 0ε ∈ H ℓ (IR), then there is a unique solution u ε (t, x) ∈ L ∞ (0, T ; H ℓ (IR)) ∩ L 2 (0, T ; H ℓ+1 (IR)), u ε,t ∈ L ∞ (0, T ; H ℓ-2 (IR)) ∩ L 2 (0, T ; H ℓ-1 (IR)).

Existence of entropy weak solutions to (cCH) Definition 3.1 (Weak solution). We call a function u : IR

+ × IR → IR a weak solution of the Cauchy problem (1.3), (1.5) if (i) u ∈ L ∞ (IR + ; H 1 (IR) ∩ Ẇ 1,4 (IR)), and (ii) u satisfies (1.3), (1.5) in distributional sense, that is ∫ ∫ IR + ×IR uϕ t + 1 3 u 3 ϕ x + 1 3 u 3 x ϕ -Φ(u)ϕdxdt + ∫ IR u 0 (x)ϕ(0, x)dx = 0, (3.57)
where (ii) u ∈ L ∞ (0, T ; X) for any T > 0, and (iii) for any convex C 2 entropy η : IR → IR with corresponding entropy flux q : IR → IR defined by q ′ (u) = η ′ (u)u 2 there holds

Φ(u) = (1 -∂ 2 x ) -1 [ 4 3 u 3 x + 2u 2 u x + 2uu x u xx ] = 1 2 e -|x| * [ 4 3 u 3 x + 2u 2 u x + 2uu x u xx ] . ( 3
∂ t η(u x ) + ∂ x (u 2 η(u x ) -q(u x )) + η ′ (u x )∂ x Φ(u) + 2uu x (u x η ′ (u x ) -η(u x )) ≤ 0, that is, for any φ ∈ C ∞ c ([0, ∞) × IR), φ ≥ 0, ∫ ∫ IR + ×IR η(u x )φ t + (u 2 η(u x ) -q(u x ))φ x -η ′ (u x )∂ x Φ(u)φ -2uu x (u x η ′ (u x ) -η(u x ))φdxdt + ∫ IR η(u 0 (x))φ(0, x)dx ≥ 0.
(3.59)

Our main theorem in this section is Theorem 3.1 (Existence). Suppose the initial data u 0 ∈ X. There exists at least one entropy weak solution to the Cauchy problem (1.3), (1.5).

Proof. We take a sequence of positive numbers {ε k } ∞ k=1 tending to zero. Then we obtain a sequence of solutions {u ε k } ∞ k=1 . From section 2.2, we know this sequence satisfies uniform L 1 , H Further more, we can obtain that u ε k → u pointwise. So in order to prove (3.57), it suffices to prove the convergence of term Φ(u ε k ). In fact,

∥Φ(u ε k ) -Φ(u)∥ L p = 1 2 e -|x| * 1 3 (u 3 ε,x -u 3 x ) + 1 2 (sgn x)e -|x| * [( 2 3 u 3 ε k + u ε k u 2 ε k ,x ) -( 2 3 u 3 + uu 2 x )] L p ≤ 2(∥u 3 ε k ,x -u 3 x ∥ L p + ∥u 3 ε k -u 3 ∥ L p + ∥u ε k u 2 ε k ,x -uu 2 x ∥ L p ) → 0.
(3.63)

So the limit function u is a weak solution to (cCH). Then we shall prove the entropy inequality. Since for η ∈ C 2 with η ′′ > 0, the approximate solution u ε satisfies

∂ t η(u ε,x ) + (u 2 ε η(u ε,x ) -q(u ε,x )) x + 2u ε u ε,x (u ε,x η ′ (u ε,x ) -η(u ε,x )) +η ′ (u ε,x )∂ x Φ(u ε ) = εη ′ (u ε,x )(u ε,x ) xx . (3.64) Multiply a positive test function, - ∫ ∫ IR + ×IR η(u ε,x )φ t + [u 2 ε η(u ε ) -q(u ε,x )]φ x dxdt - ∫ IR η(u 0 (x))φ(0, x)dx + ∫ ∫ IR + ×IR 2u ε u ε,x (u ε,x η ′ (u ε,x ) -η(u ε,x ))φ + η ′ (u ε,x )∂ x Φ(u ε )dxdt = - ∫ ∫ IR + ×IR εη(u ε,x ) x φ x dxdt -ε ∫ ∫ IR + ×IR η ′′ u 2 ε,xx φdxdt ≤ - ∫ ∫ IR + ×IR εη ′ (u ε,x )u ε,xx φ x dxdt.
(3.65)

Since u ε,xx (t, •) ∈ L 1 (IR) and φ is compact support, the right-hand-side approaches 0 as ε → 0. So u is an entropy weak solution.

W 1,1 stability and uniqueness

Suppose u and v are two entropy weak solutions with initial data u 0 and v 0 respectively. We estimate W 1,1 norm of their difference.

u t + u 2 u x = 1 3 u 3 x -Φ(u), v t + v 2 v x = 1 3 v 3 x -Φ(v). (4.66) Take their difference (u -v) t + u 2 (u -v) x + (u + v)(u -v)v x = 1 3 (u x -v x )(u 2 x + u x v x + v 2 x ) -(Φ(u) -Φ(v)). (4.67)
Multiply sgn(u -v) and then take integral,

d dt ∫ IR |u-v|dx ≤ ∫ IR (|2uu x |+|v x ||u+v|)|u-v|dx+ 1 3 |u x -v x ||u 2 x +u x v x +v 2 x |+|Φ(u)-Φ(v)|dx. (4.68) By the expression of Φ, ∥Φ(u) -Φ(v)∥ L 1 x ≤ 1 3 1 2 e -|x| L 1 x ∥u x -v x ∥ L 1 x ∥u 2 x + u x v x + v 2 x ∥ L ∞ x + 1 2 e -|x| L 1 x ( 2 3 ∥u -v∥ L 1 x ∥u 2 + uv + v 2 ∥ L ∞ x + ∥u -v∥ L 1 x ∥u 2 x ∥ L ∞ x + ∥v(u x + v x )∥ L ∞ x ∥u x -v x ∥ L 1 x ) (4.69) Considering the boundedness of the Ẇ 2,1 -norm of u, v, we have d dt ∥(u -v)(t, •)∥ L 1 x ≤ K(T, ∥u 0 ∥ H 1 x , ∥v 0 ∥ H 1 x , ∥u 0 ∥ Ẇ 2,1 x , ∥v 0 ∥ Ẇ 2,1 x )∥(u -v)(t, •)∥ W 1,1 x (4.70)
where K is a constant related with T, Take φ(t, x, s, y) = ψ( t+s 2 , x-y 2 )δ h ( t-s 2 )δ h ( x-y 2 ), ∫ ∫ For the source term, we have the following estimate

∥u 0 ∥ H 1 x , ∥v 0 ∥ H 1 x , ∥u 0 ∥ Ẇ 2,1 x , ∥v 0 ∥ Ẇ 2,1 x . 2 (|u x -v x | L 1 estimate). Denote η k (u) = |u -k|. Consider the equation (u x ) t + (u 2 u x - 1 3 u 3 x ) x + Φ x = 0. (4.71) Multiply η ′ k (u x ), |u x -k| t + sgn(u x -k)(u 2 u x - 1 3 u 3 x ) x + sgn(u x -k)∂ x Φ = 0. (4.72) ∫ ∫ IR + ×IR |u x -k|φ t +sgn(u x -k)(u 2 u x - 1 3 u 3 x -u 2 k+ 1 3 k 3 )φ x -sgn(u x -k)(∂ x Φ(u)+2kuu x )φdxdt ≥ 0. (4.73) ∫ ∫ IR + ×IR |u x -v y |φ t +sgn(u x -v y )(u 2 u x - 1 3 u 3 x -u 2 v y + 1 3 v 3 y )φ x -sgn(u x -v y )(∂ x Φ(u)+2v y uu x )φdxdt ≥ 0. (4.74) Similarly, for v, ∫ ∫ IR + ×IR |v y -k|φ s +sgn(v y -k)(v 2 v y - 1 3 v 3 y -v 2 k+ 1 3 k 3 )φ y -sgn(v y -k)(∂ y Φ(v)+2kvv y )φdyds ≥ 0. (4.75) ∫ ∫ IR + ×IR |v y -u x |φ s +sgn(v y -u x )(v 2 v y - 1 3 v 3 y -v 2 u x + 1 3 u 3 x )φ y -sgn(v y -u x )(∂ y Φ(v)+2u x vv y )φdyds ≥ 0. ( 4
IR 2 + ×IR 2 |u x -v y |(φ t + φ s ) + sgn(u x -v y )[u 2 (u x -v y ) - 1 3 (u 3 x -v 3 y )]φ x -sgn(u x -v y )[-v 2 (u x -v y ) + 1 3 (u 3 x -v 3 y )]
IR + ×IR |u x -v x |ψ t (t, x) + sgn(u x -v x )[(u 2 + v 2 )(u x -v x ) - 2 3 (u 3 x -v 3 x )]ψ x (t, x) -sgn(u x -v x )(∂ x Φ(u) -∂ x Φ(v) + 2u x v x (u -v))ψ(t,

R+L(t-) -R-L(t-τ) τ

|∂ x Φ(u) -∂ x Φ(v) + 2u x v x (u -v)| = ∂ x (1 -∂ 2 x ) -1 [ 1 3 (u x -v x )(u 2 x + v 2 x + u x v x )] + (1 -∂ 2 x ) -1 [ 2 3 (u -v)(u 2 + uv + v 2 ) +u(u x -v x )(u x + v x ) + (u -v)v 2 x ] - 2 3 (u -v)(u 2 + uv + v 2 ) -(u -v)u 2 x -v(u x -v x )(u x + v x ) + 2u x v x (u -v) . (4.82) Take integral, ∫ IR |∂ x Φ(u) -∂ x Φ(v) + 2u x v x (u -v)|dx ≤ sup t∈[0,T ] ( 1 3 (u 2 x + v 2 x + u x v x ) L ∞ + 2 3 (u 2 + uv + v 2 ) L ∞ + ∥u 2 x ∥ L ∞ + ∥u(u x + v x )∥ L ∞ +2∥u x v x ∥ L ∞ ) ∥u -v∥ W 1,1 (4

.58) Definition 3 . 2 (

 32 Entropy weak solution). We call a function u : IR + × IR → IR an entropy weak solution of the Cauchy problem (1.3), (1.5) provided (i) u is a weak solution in the sense of Definition 3.1,

  φ y -sgn(u x -v y )(∂ x Φ(u) -∂ y Φ(v) + 2v y uu x -2u x vv y )φdxdydsdt ≥ 0.

Figure 1 :

 1 Figure 1: The domain in consideration. Consider a trapezoid Ω as in Figure 1. The straight line γ 1 and γ 2 have slopes as -L and L separately.Then we take ψ to be a smooth approximation to the characteristic function of the trapezoid

  1 , Ẇ 1,∞ , and Ẇ 2,1 estimates. So by Helly's theorem, as k → ∞,u ε k ,x → u x a.e. in IR + × IR,By the Ẇ 1,∞ estimate and the energy estimate, there holds u ε k ,x → u x in L p ((0, T ) × IR) for all T > 0, for all p ∈ [1, ∞).

	(3.62)
	(3.60)
	and hence
	u ε (3.61)

k ,x → u x in L p loc (IR + × IR) for all p ∈ [1, ∞).

  1 (|u -v| L 1 estimate). Since u, v ∈ Ẇ 2,1 , u, v are Lipschitz functions. The derivatives u x , v x exist almost everywhere. From the equation, we know u t , v t are also well-defined. The equations for u and v are

  x)dxdt ≥ 0.IR + ×IR |u x -v x |ψ t (t, x) + |u x -v x |[(u 2 + v 2 ) -

				(4.78)
	∫ ∫	2 3	(u 2
	2 3	(u 2 x + u x v x + v 2 x ) < L.	(4.80)
		t
	γ 1	Ω	γ 2
		τ
	-R		R	x

x + u x v x + v 2 x )]ψ x (t, x) -sgn(u x -v x )(∂ x Φ(u) -∂ x Φ(v) + 2u x v x (u -v))ψ(t, x)dxdt ≥ 0. (4.79)

By the Ẇ 2,1 estimate and the energy estimate, we can find a constant L such that (u 2 + v 2 ) -

  .83) We let R → +∞ in (4.81), and add (4.70)∥u(t, •) -v(t, •)∥ W 1,1 ≤ ∥u(0, •) -v(0, •)∥ W 1,1 + M T ∥u(s, •) -v(s, •)∥ W 1,1 ds, (4.84)where M T is a constant of order O(e CT ), and related with ∥u 0 ∥ H 1 ∩ Ẇ 2,1 and ∥v 0 ∥ H 1 ∩ Ẇ 2,1 . By Gronwall's inequality, we obtain∥u(t, •) -v(t, •)∥ W 1,1 ≤ e M T t ∥u(0, •) -v(0, •)∥ W 1,1 .(4.85)

	∫ t
	0

Acknowledgement. Q. Zhang is partially supported by NSF grant DMS-1411786.


Lpmm Fre

Stress analysis by Kossel microdiffraction on a nickel-based single crystal superalloy during an in situ tensile test -Comparison with classical X-Ray diffraction

Keywords: Kossel microdiffraction, in situ, scanning electron microscope, synchrotron radiation

A Kossel microdiffraction experimental set up is under development inside a Scanning Electron Microscope (SEM) in order to determine the crystallographic orientation as well as the inter-and intragranular strains and stresses. An area of about one cubic micrometer can be analysed using the microscope probe, which enables to study different kinds of elements such as a grain boundary, a crack, a microelectronic component, etc. The diffraction pattern is recorded by a high resolution Charge-Coupled Device (CCD) camera. The crystallographic orientation, the lattice parameters and the elastic strain tensor of the probed area are deduced from the pattern indexation using a homemade software. The purpose of this paper is to report some results achieved up to now to estimate the reliability of the Kossel microdiffraction technique.

Introduction

Due to the growing complexity of new materials and their applications, it is increasingly necessary to know the strain and the stress state at a lower scale, a micrometer one, particularly in the case of micro-and nanosystems. Kossel microdiffraction is a tool that has been adapted for use in the Scanning Electron Microscopes (SEM), which enables to determine not only the crystallographic orientation, but also the inter-and intragranular strain and stress state while observing the microstructure and its possible evolution during in situ mechanical tests. Different techniques are available to obtain residual stresses at the micrometer scale: for instance micro-Raman spectroscopy [1], X-Ray microdiffraction with a white synchrotron beam [2], convergent beam electron diffraction inside a transmission electron microscope [3]. Kossel microdiffraction is currently being developed because of some advantages comparing to these techniques [4]. Indeed, the use inside a SEM is easy, non destructive and can be applied to metallic components. Moreover, spatial and strain resolutions are competitive. In the 1970s, Kossel microdiffraction was used for the determination of crystallographic orientations and lattice parameters [5,6,7]. Patterns were recorded on photographic films. In the 1990s, the development of high resolution CCD cameras began allowing easier and faster data collection. Lattice strain measurements finally began to be feasible [8]. The aim of this study is to check the reliability of the Kossel technique. Kossel line patterns on single crystals under loading have been recorded and the goal is to compare the stresses given by the technique with the stresses applied to the samples. We also compare results with those obtained by classical X-Ray Diffraction (XRD).

Kossel technique

In a SEM, when an electron beam is focused on a material, the latter is excited and thus an X-rays spherical wave is emitted. A part of these rays is diffracted by the crystallographic planes forming the so-called Kossel cones, in accordance with the Bragg's law: λ = 2d sin θ (λ: wavelength of the Denis BOUSCAUD a , Raphaël PESCI b , Sophie BERVEILLER c , Etienne PATOOR d b excited element, d: interreticular length, θ: Bragg's angle). A CCD camera intercepts a part of these cones, which are all emitted at the same time (one diffraction cone for each (hkl) diffracting plane). Kossel conics (lines or complete circles) are calculated from the orientation of the cones and the distance between the sample and the screen (Fig. 1). It is possible to excite the material with different types of radiation. Conventionally, an electron beam is used to excite either the sample directly or a target placed close to the sample [9]. This latter technique is called pseudo-Kossel. Kossel microdiffraction can also be obtained by employing synchrotron radiation [10]. The obtained Kossel line pattern is then indexed using a semi-automatic program, KSLStrain, developed by Adam Morawiec [11]. Several parameters are fulfilled in the program: the radiation wavelength of the emitted X-rays that is known with relatively high accuracy, the unit cell of the material, the pixel size of the camera and approximate values of the camera geometry parameters (sample-to-detector distance and the location of the pattern centre). Then, conics on the pattern are marked manually selecting a number of points per each Kossel line (Fig. 3a). Starting with fixed reference values of lattice parameters, a procedure is used to find the geometry of the pattern. Once the crystal orientation is determined and the camera geometry parameters are tuned, the program can proceed with the refinement of lattice parameters. The program uses the so-called "K-line equation based scheme". Kossel conics manually marked by the operator are matched to corresponding conics in simulated patterns. At the end, the camera geometry parameters, the crystal orientation and the strain components are fitted together. Visual example of a result is shown in Fig. 3b. The strain resolution is estimated to be about 3.10 -4 [11].

Measurements on single crystals under loading

To check the consistency of the stress results given by the Kossel microdiffraction, we have performed measurements on a nickel-based single crystal superalloy during an in situ tensile test. The chemical composition (in weight %) of the material is the following: Base Ni -8% Cr -5% Co -8% W -2% Mo -5% Al -1.5% Ti -6% Ta. Its main feature is a high yield strength (about 900MPa). In fact, it is made of two coherent phases: a Ni matrix γ and ((Ni, Co, Cr) 3 (Al, Ti, Ta)) cubic precipitates γ' (Fig. 4a). We used a small tensile testing device (Fig. 4b) to apply loading to "dog bone" specimens up to about 700MPa. tensors were determined assuming a biaxial stress state, due to the low emission depth of X-rays (Fig. 5b). The same experiments were performed inside the SEM. Results are shown in Table 1 and an example of a Kossel line pattern is shown in Fig. 6. Single crystal XRD stress analyses, using the Ortner's method [12], were carried out within a fourcircle goniometer to compare with Kossel experiments (Table 2). The probed area was about a few mm².

Table 2: Stress tensors (MPa) obtained by single crystal XRD. σ 11 according to the tensile direction. Assumption: σ 33 =0.

With the assumption of a biaxial stress state for the stress calculation from Kossel patterns, in situ stress results given by the three techniques and the macroscopic stress applied by the machine were compared, showing a good agreement. XRD measurements in one grain take several hours, so a slight relaxation of the tensile device can explain why the σ 11 obtained is a little below the applied stress. The σ 22 compressive values, which are the same for the two loadings, are probably due to the surface preparation.

In synchrotron experiments, only one Kossel line pattern was obtained per loading. The synchrotron beam was focused at a random location on the single crystal surface. On the other hand, in the SEM, three patterns were recorded per loading and stress tensors shown in Table 1 correspond to an average of three measurements. Three electronic spot locations were chosen a few hundreds of micrometers apart. The results are consistent with the loading applied but we noticed high standard deviations. The σ 11 deviations are shown in Table 3. For the initial state, the non-uniform surface preparation could explain these deviations. Once the material is loaded, the standard deviations decrease.

Table 3: σ 11 standard deviations (∆σ 11 ) calculated from three different Kossel analyses.

These deviations are probably due to the stress distribution between the two phases, the lattice misfit or local deviations of residual stresses induced by the surface preparation. The last value (about 70MPa) can also give information about the stress resolution of the Kossel microdiffraction, when such a pattern as shown in Fig. 6 is obtained.

Conclusion

This paper reports on microscale stress measurements using a non-destructive method, Kossel microdiffraction. Kossel patterns were obtained during in situ tensile experiments on single crystals in the Scanning Electron Microscope and with a monochromatic synchrotron beam. The patterns, recorded by a high resolution CCD camera, contained enough lines to determine the crystallographic orientation of the crystal and the elastic deformation of the lattice. Nickel-based single crystal superalloys were analyzed: different loadings up to 700MPa were applied on these specimens with a tensile micromachine and compared with the values obtained using the Kossel technique. A good agreement was observed. The results show how Kossel microdiffraction can be used as a precise tool to obtain residual stresses in polycrystalline samples. This technique could be

Fig. 1 :

 1 Fig. 1: The principle of the Kossel microdiffraction inside a SEM

Fig. 2 :

 2 Fig. 2: a) SEM image of the microstructure showing the grain analysed by Kossel diffractionb) Experimental Kossel line pattern from an IF steel grain

Fig. 3 :

 3 Fig. 3: a) IF steel pattern with some conics markedb) Simulated pattern after the fittings superimposed with the experimental pattern

Fig. 4 :

 4 Fig. 4: a) SEM image of the nickel-based superalloy microstructureb) In situ uniaxial tensile/compressive testing device

Fig. 5 :

 5 Fig.5: a) Experimental set up used at ID13 (ESRF) to obtain Kossel line patterns from a monochromatic synchrotron beam -b) Stress tensors (MPa) obtained by Kossel diffraction with a synchrotron beam at different stress states. σ 11 according to the tensile direction. Assumption: biaxial stress sate.

Fig. 6 :

 6 Fig.6: Simulated pattern after the fittings superimposed with the experimental one (400MPa loading)

Table 1 :

 1 Stress tensors (in MPa) obtained by Kossel diffraction inside the SEM at different stress states. σ 11 according to the tensile direction. Assumption: biaxial stress sate.

  , for example, to analyse residual stresses in microelectronic components like Cu-MEMS (Micro Electro Mechanical Systems).

		Initial state	400MPa loading	680MPa loading
	∆σ 11 [MPa]	135	75

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