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Primes Counting Methods: Twin Primes 1. Introduction

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Primes Counting Methods: Twin Primes 1. Introduction Primes Counting Methods: Twin Primes N. A. Carella, July 2012. Abstract: The cardinality of the subset of twin primes { p and p + 2 are primes } is widely believed to be an infinite subset of prime numbers. This work proposes a proof of this primes counting problem based on an elementary weighted sieve. Mathematics Subject Classifications: 11A41, 11N05, 11N25, 11G05. Keywords: Primes, Distribution of Primes, Twin Primes Conjecture, dePolignac Conjecture, Elliptic Twin Primes Conjecture, Prime Diophantine Equations. 1. Introduction The problem of determining the cardinality of the subset of primes { p = f (n) : p is prime and n !1 }, defined by a function f : ℕ → ℕ, as either finite or infinite is vastly simpler than the problem of determining the primes counting function of the subset of primes (x) # p f (n) x : p is prime and n 1 " f = { = # $ } The cardinality of the simplest case of the set of all primes P = { 2, 3, 5, … } was settled by Euclid over two millennia ago, see [EU]. After that event,! many other proofs have been discovered by other authors, see [RN, Chapter 1], and the literature. In contrast, the determination of the primes counting function of the set of all primes is still incomplete. Indeed, the current asymptotic formula (25) of the " (x) = #{ p # x : p is prime } primes counting function, also known as the Prime Number Theorem, is essentially the same as determined by delaVallee Poussin, and Hadamard. ! The twin primes conjecture claims that the Diophantine equation q = p + 2 has infinitely many prime pairs solutions. More generally, the dePolignac conjecture claims that for any fixed k ≥ 1, the Diophantine equation q = p + 2k has infinitely many prime pairs solutions, confer [RN, p. 265], [GS], [KV], [FI, p. 315], [AR], [KR], [NW, p. 337], [WK], [WS], [PP], and related topics in [BC], [GP], [GU, p. 31], [SD], et alii. This work proposes a proof of this primes counting problem. In particular, the cardinality of the subset of twin primes Primes Counting Methods:Twin Primes − 2 { p and p + 2 are primes } is infinite. Theorem 1. For any fixed k ≥ 1, the equation q = p + 2k has infinitely many prime pairs solutions. Proof: Without loss in generality, let k = 1, and consider the weighted finite sum over the integers: # n!(n)!(n + 2) = $# n!(n) # µ(d)logd n " x n " x d | n+2 (1) = $ # µ(d)logd # n!(n), d " x+2 n " x, n%$2modd where !(n) = "#d | n µ(d)logd . This follows from Lemma 5 in Section 2.2, and inverting the summation. Applying Lemma 2, yields % x2 ( # n!(n)!(n + 2) = $ # µ(d)logd 'c1 +O(x)* n " x d " x+2, d odd & d ) (2) % ( 2 µ(d)logd = $ a x + O'x $ µ(d)logd *, 1 # d ' # * d " x+2, d odd & d " x+2, d odd ) 2 where a1 = !!"(2) /!(2) is a constant. The restriction to odd d ≥ 1 stems from the relation Λ(n) = 0 for even d ≥ 2. Applying Lemmas 3 and 4, the previous equation becomes 1/2 1/2 n!(n)!(n + 2) = a x2 1+O e$c(log x) log x + O xO xe$c(log x) log x # 2 ( ( )) ( ( ) ) n " x (3) 1/2 = a x 2 + O x2e$c(log x) log x , 2 ( ) where a2 = a1/2, and c > 0 is a nonnegative constant. Now, assume that there are finitely many twin primes p, and p + 2 < x0, where x0 > 0 is a large constant. For example, Λ(n)Λ(n + 2) = 0 for all n > x0. Then 1/2 n!(n)!(n + 2) = c x 2 + O x2e$c(log x) log x " x3 (4) # 1 ( ) 0 n " x 2 for any real number x ≥ 1. But this is a contradiction for all sufficiently large real number x > x0 . Nil volentibus arduum. ■ It should be noted that other (twin primes sieve) weighted finite sums over the integers such as #! (n)!(n)!(n + 2) , #!(n)!(n)!(n + 2) , #! (n)!(n)!(n + 2), (5) n " x n " x n " x yield precisely the same result. The analysis of these later finite sums are quite similar to the Titchmarsh divisor problem over arithmetic progressions, see [CM, p. 172]. Primes Counting Methods:Twin Primes − 3 Counting Functions Previous works on the twin primes conjecture have studied certain Dirichlet series, see [AR], [KR], et cetera. The primes counting method introduced here suggests that a more suitable Dirichlet series is the complex valued function n!(n)!(n + 2k) $ dR(x) D (s) , (6) 2k = # s = % s n"1 n 1 x where R(x) = #n"x n!(n)!(n + 2) , k ≥ 1 is a fixed integer, and ℜe(s) ≥ 2. This series leads to another approach to study of the twin primes conjecture, and more generally, the dePolignac conjecture via the Wiener-Ikehara theorem introduced in [AR]. An application of partial summation to the finite sum yields the counting function n!(n)!(n + 2) n!(n)!(n + 2) ! 2 (x) = # $ # nlog(n)log(n + 2) k m nlog(n)log(n + 2) n " x n=p or n+2=q " x, k, m%2 (7) x dR(t) = & +O(x1/2 log2 x , 2 t log(t)log(t + 2) where R(x) = #n"x n!(n)!(n + 2) . This realizes a lower estimate of the twin primes counting function x x ! 2 (x) = #{ p ! x : p and p + 2 are primes } " c1 +O( ). (8) log2 x log3 x This lower estimate has the correct order of magnitude as determined by sieve method, that is, x ! 2 (x) = #{ p ! x : p and p + 2 are primes } ! c3 , (9) log2 x where c3 > 0 is a constant, see [RN, p. 265], [KV, p. 45], [MV, p. 91], [NT, p. 190]. Combining (8) and (9) puts the twin primes counting function in the form x x a ! ! (x) ! b , (10) 1 log2 x 2 1 log2 x where a1, b1 > 0 are constants. However, to establish the correct constant as conjectured x dt x x (x) 2c 2c O( ) , (11) ! 2 = 2 ! 2 = 2 2 + 3 1+" log t log x log x where !2 (12) c2 = "(1! (p !1) ) =.6601618158... p>2 is the twin primes constant, much more efforts will be required. Perhaps, the probabilistic way of determining the constant is correct. The probabilistic argument produces asymptotic expression Primes Counting Methods:Twin Primes − 4 2 " 1 % " 1 % x x , (13) ! 2 (x) = 2x($1! 2 ' ( $1! ' = 2c2 2 +O( 3 ) p>2 # (p !1) & 2<p)xc # p& log x log x where c = e!! = e!.5772156649... = 0.564307113... is a constant, and the number 1 – 1/p is the probability that the prime p does not divides the primes q nor q + 2. See [HW, p. 312], [KV, p. 38], and [ST, p. 180] for related details. The general case of the Hardy-Littlewood conjecture has the prime pairs counting function " p !1 % 1/2 +" (14) ! 2k (x) = 2c2 x ( $ '+O(x ) 2<p | k # p ! 2 & where k ≥ 1, and ε > 0 is an arbitrary small number. The error term is conditional, see [KR, p. 7]. The maximal order of magnitude is unconditionally bounded by x (x) c 1 1/ p , (15) ! 2k ! 2k 2 "( + ) log x p | k see [RN, p. 265]. The error term in this asymptotic formula is studied in [KT], and the numerical data for the 2 change of signs "2 (x) # 2c2 x /log x is complied in [WF]. Note 1. This is reminiscent of the development of the prime number theorem: First, Chebyshev proved the inequality ! x x a ! !(x) ! b , (16) 0 log x 0 log x where a0, b0 > 0 are constants, see [LA, p. 88] for a proof. And about fifty years later, delaVallee Poussin and Hadamard proved the asymptotic formula " 1 % x x (x) x 1 O( ) , (17) ! = ) $ ! ' = + 2 2<p(xc # p& log x log x where c = e!! = e!.5772156649... = 0.564307113... , and the number 1 – 1/p is the probability that the prime p does not divide the prime q. The product representation of π(x) was included to demonstrate that the probabilistic argument produces the correct asymptotic formula as proved by analytical method. Perhaps, the only puzzling part is the constant e!! = e!.5772156649... = 0.564307113... , consult [BW], [HW] and [GV] for various levels of discussions. Brun Constant The Brun constant is defined by the twin primes series 1 1 1 1 1 1 1 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ , (18) B = ∑⎜ + ⎟ = ⎜ + ⎟ + ⎜ + ⎟ + ⎜ + ⎟ + =1.902160... p ⎝ p p + 2 ⎠ ⎝ 3 5 ⎠ ⎝ 5 7 ⎠ ⎝11 13 ⎠ Primes Counting Methods:Twin Primes − 5 this is a numerical approximation, more accurately, 1.830 < B < 2.347, consult [KV, p. 12] for the advanced theory and computational techniques. The other related constants ⎛ 1 1 ⎞ , (19) Bn = ∑⎜ + ⎟ p ⎝ p p + n⎠ for even n > 2 are considered in [WF]. € 2. Elementary Foundation The elementary underpinning of Theorem 1 is assembled here. The basic primes detection principles, the definitions of several number theoretical functions, and a handful of Lemmas are recorded here. 2.1 Primes Counting Principles The basic primes detection method is simple, and works on many subsets of primes p = f (n) : p is prime and n "1 , defined by nonexponential functions f : → . It improves the standard { } ℕ ℕ prime detection method # f (n)"x !( f (n)). The improved primes counting method is realized by augmenting or supplementing the weight function as in ! $w( f (n) + t)"( f (n)), (20) f (n) # x where w(x) > 0 is a suitable weight function, and t is a fixed parameter. The new prime counting function detects exactly the same primes as the standard prime detection method, yet it has simpler and better analytical properties.
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