{ } Primes Counting Methods: Twin Primes N. A. Carella, July 2012.

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1 Primes Counting Methods: Twin Primes N. A. Carella, July 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 " f ( = # p = f (n # : p is prime and n $1 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 "( = #{ p # : p is prime } is still incomplete. Indeed, the current asymptotic formula (25 of the 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 fied 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

2 Primes Counting Methods:Twin Primes 2 { p and p + 2 are primes } is infinite. Theorem 1. For any fied 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(dlog d d n+2 = $ # µ(dlogd # n!(n, d " +2, n%$2mod d (1 where!(n = "# µ(dlog d. This follows from Lemma 5 in Section 2.2, and inverting the summation. d n Applying Lemma 2, yields % # n!(n!(n + 2 = $ # 2 µ(dlog d ' c 1 & d +O( ( * d " +2, d odd = $ a 1 2 µ(dlogd % # + O $ µ(dlog d d " +2, d odd d ' # & d " +2, d odd ( *, (2 where a 1 =!! "(2 /! (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 # n!(n!(n + 2 = a O e $c(log 1/2 1/2 ( ( log + O O e$c(log ( ( log (3 = a O( 2 e $c(log 1/2 log, where a 2 = a 1 /2, and c > 0 is a nonnegative constant. Now, assume that there are finitely many twin primes p, and p + 2 < 0, where 0 > 0 is a large constant. For eample, Λ(nΛ(n + 2 = 0 for all n > 0. Then # n!(n!(n + 2 = c O 2 e $c(log 1/2 ( log " 3 0 (4 for any real number 1. But this is a contradiction for all sufficiently large real number > 0 2. 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 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].

3 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 comple valued function n!(n!(n + 2k D 2k (s = # = n"1 n s $ dr( %, (6 1 s where R( = # n!(n!(n + 2, k 1 is a fied integer, and Re(s 2. This series leads to another approach n" 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! 2 ( = # nlog(nlog(n + 2 $ # n=p k or n+2=q m ", k, m%2 dr(t = & +O( 1/2 log 2, t log(tlog(t n!(n!(n + 2 n log(nlog(n + 2 (7 where R( = # n!(n!(n + 2. This realizes a lower estimate of the twin primes counting function n"! 2 ( = #{ p! : p and p + 2 are primes } " c 1 log 2 +O(. (8 log 3 This lower estimate has the correct order of magnitude as determined by sieve method, that is,! 2 ( = #{ p! : p and p + 2 are primes }! c 3 log 2, (9 where c 3 > 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 a 1 log 2!! 2(! b 1 log 2, (10 where a 1, b 1 > 0 are constants. However, to establish the correct constant as conjectured where dt! 2 ( = 2c 2! = 2c log 2 2 t log 2 + O( log 3, (11 1+" ( c 2 = " 1! (p!1!2 = (12 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 epression

4 " 1 % "! 2 ( = 2( $ 1! ' 1! 1 % ( $ ' p>2# (p!1 2 & 2<p # p c & 2 Primes Counting Methods:Twin Primes 4 = 2c 2 log 2 +O( log 3, (13 where c = e!! = e! = 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 %! 2k ( = 2c 2 ( $ ' +O( 1/2+" (14 # p! 2 & 2<p k where k 1, and ε > 0 is an arbitrary small number. The error term is conditional, see [KR, p. 7]. The maimal order of magnitude is unconditionally bounded by! 2k (! c 2k "( 1+1/ p, (15 log 2 see [RN, p. 265]. The error term in this asymptotic formula is studied in [KT], and the numerical data for the change of signs " 2 ( # 2c 2 /log 2 is complied in [WF]. Note 1. This is reminiscent of the development of the prime number theorem: First, Chebyshev proved the inequality a 0 p k log!! (! b 0 log, (16 where a 0, b 0 > 0 are constants, see [LA, p. 88] for a proof. And about fifty years later, delavallee Poussin and Hadamard proved the asymptotic formula "! ( = 1! 1 % $ ' = # p & log +O( log 2, (17 2<p( c where c = e!! = e! = , and the number 1 1/p is the probability that the prime p does not divide the prime q. The product representation of π( 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! = , consult [BW], [HW] and [GV] for various levels of discussions. Brun Constant The Brun constant is defined by the twin primes series B = + = = , (18 p p p

5 Primes Counting Methods:Twin Primes 5 this is a numerical approimation, more accurately, < B < 2.347, consult [KV, p. 12] for the advanced theory and computational techniques. The other related constants for even n > 2 are considered in [WF]. 1 B n = p + 1, (19 p + n p 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 noneponential functions f : N N. It improves the standard prime detection method #!( f (n. The improved primes counting method is realized by augmenting or f (n" supplementing the weight function as in $ w( f (n + t"( f (n, (20 f (n # where w( > 0 is a suitable weight function, and t is a fied parameter. The new prime counting function detects eactly the same primes as the standard prime detection method, yet it has simpler and better analytical properties. Some possible choices of weight functions are w(n = n, and w(n = ϕ(n, et cetera. Under these conditions, the primes counting problem on a given sequence of primes is easier because the properly selected weight function w( simplifies the evaluation of the finite sum: This technique effectively transforms the vonmangoldt function into an epression of the form # µ(dlog d. (21 d " +t d s The associated Dirichlet series is an absolutely convergent comple valued function on the comple half plane Re(s > 1. Refer to [CM], [FI, p. 21], and similar literature for introductions to sieve methods. The corresponding primes counting function has a lower estimate of the form " f ( = #{ p = f (n # } $ 1 w(log & w( f (n + t%( f (n. (22 f (n # Note 2. This primes counting technique does not work on subsets of primes defined by eponential functions. For eample, the set of Fermat numbers { F n = 2 2n +1: n!1 }, and the set of Mersenne numbers

6 Primes Counting Methods:Twin Primes 6 { M p = 2 p "1: p # 2 prime }, would have to be handled with different primes counting methods. 2.2 Some Lemmas Let n N be an integer. The Mobius function is defined by # µ(n = (!1v if n = p 1 p 2! p v, $ % 0 if n " squarefree. (23 And the vonmangoldt function is defined by % "(n = log p if n = pk, k #1, & ' 0 if n $ p k, k #1. (24 The primes counting function is defined by! ( = #{ p! : p is prime } = (, (25 log +O 1/2 e"c(log where c > 0 is a nonnegative constant. And the prime number theorem on arithmetic progression is defined by! (, a, q = #{ p! amodq " : p is prime } =!(qlog +O e#c(log 1/2 (, (26 where gcd(a, q = 1, and q = O((log B, B > 0 constant, see [MV, p. 382], [NW], and [TM, p. 255]. Weighted Finite Sum Over Arithmetic Progressions This result is an important part of Theorem 1. The proof uses elementary methods. Other analytical methods are possible and provide alternative proofs of the twin primes conjecture. Lemma 2. Let a 1, and q 1 be integers. Let 1 be a sufficiently large real number. Then 2 $ n!(n = a 1, n#amodq q +O (, (27 where a 1 =!! "(2 /! (2 2 is a constant. Proof: Put the finite sum in the equivalent form % n"(n = & % n % µ(dlog d n #, n $a mod q n #, n $a mod q d n = &% µ(dlogd % n. d # n # / d, n $a mod q (28

7 Primes Counting Methods:Twin Primes 7 Substituting Lemma 6 returns $ n!(n = % $ µ(dlog d, n#amodq d " & 1 2 ( 2q d +O( ' 2 d + * = % 2 $ µ(dlog d & 2q d +O 2 ( % d " ' $ d " µ(dlog d d +. * (29 Proceed to use Lemma 3, (or use equations (36 and (38, and simplify to reach the epression ' $ n!(n = 2 2q %!&(2! (2 +O %1 e %c(log (, n#amodq 2 = a 1 q +O (, 1/2 ( log %c(log 1/2 ( ( log *,+O 1+O(e + (30 where a 1 =!! "(2 /! (2 2 > 0 is a constant. Twisted Finite Sums Lemma 3. Let s 1 be an integer, let µ(n be the Mobius function, and let 1 be a sufficiently large real number. Then where c > 0 is a constant. " µ(nlog n n =!#(s s! (s +O 2 1$s e $c(log n! Proof: Write the finite sum in the form 1/2 ( log, (31 # µ(nlog n = # µ(nlog n % # µ(nlog n. (32 n s The constant is epressed in terms of the logarithmic derivative of the zeta function id est, n $1 n s n > n s d 1 ds "(s = #% µ(nlog n n = # " & (s s "(s. (33 2 n $1 Now, since the zeta function! (s = # n!s is a decreasing function on the real half line Re(s > 1, the n"1 derivative! (s = " $ (lognn "s is negative on the real half line s > 1. The result follows from these data. n#1 The zeta function, and its derivative have the power series epansions

8 Primes Counting Methods:Twin Primes 8 "(s = 1 s #1 + $ 0 # $ 1 (s # $ 2 (s #12 #!, (34 1 #"(s = $ (s $1 $ % + % (s $1 $ % (s 3 $12 +! where γ k is the kth Stieltjes constant, see [DL, ]. These series can be used to compute numerical approimations of the constants " s = #% $ (s /%(s 2 for s 1. Eampli gratia, the logarithmic derivative at s = 1 is! (1! (1 = lim (s #1 2! (s = #1, (35 2 s "1 (s #1 2 2! (s and " µ(nlog n n =!#(1! (1 +O 2 e$c(log n! 1/2 ( log (, (36 1/2 = $1+O e$c(log log where c > 0 is a constant. Similarly, the logarithmic derivative at s = 2 is! (2! (2 = " (" 2 / 6(# + log2" "12logC 6(# + log2" "12logC = ", (37 2 " 4 / 36 " 2 where γ is Euler constant, and C is Glaisher constant, see [DL, ]. And " µ(nlog n n =!#(2 2! (2 +O 2 e$c(log n! 1/2 ( log 6(! + log2" $12 logc = $ +O e $c(log 1/2 ( log!. (38 2 Lemma 4. Let 1 be a sufficiently large real number, and let µ(n be the Mobius function. Then " µ(nlog n = O e n! #c(log 1/2 ( log, (39 where c > 0 is a constant. Proof: Evaluate the Stieltjes integral " µ(nlog n = # (logtdm(t, (40 n! 1 where M( = " µ(n = O(e #c(log 1/2. n!

9 Primes Counting Methods:Twin Primes 9 Formulae for the vonmangoldt Function A few of the various representations of the vonmangoldt function (24 are shown here. There is a simple proof of the identity (41 via the Mobius inversion formula, but an analytical proof is given to show its connection to the zeta function and beyond. Lemma 5. Let n 1 be an integer, and let Λ(n be the vonmangoldt function. Then Proof: The logarithm derivative of the zeta function satisfies "(n = # $ µ(dlog d. (41 d n $ " # (s $(s = ' %(n. (42 On the other hand, the convolution of the Dirichlet series of the functions 1/ζ(s, and ζʹ (s respectively satisfies!!"(s %! (s = logn ( % ' $ & n s * ' n #1 & $ n #1 µ(n n s n &1 n s $ d n ( * = $ µ(dlogn / d n. (43 s n #1 Lastly, matching the coefficients of the powers of n s in (41 and (42 confirms the claim. Various other identities and approimations of the vonmangoldt function are used in the literature. These include short formulas such as " R (n = $ µ(dlog R /d, (44 d n, d # R see [GP], the Vaughan identity, see [HN, p. 25], and the harmonic epansion see [MV, p. 114]. This later identity leads to the conjectured twin prime formula c!(n = " $ n (q q, (45 q #1!(nc n+2 (q #!(n!(n + 2 = $ # q = 2c + o(, (46 2 q %1 # n" where c 2 is the twin prime constant, see (12, and large real numbers 1. Etensive details of other identities and approimations of the vonmangoldt are discussed in [FI], et alii.

10 Primes Counting Methods:Twin Primes 10 Power Sum Over Arithmetic Progressions Lemma 6. Let a 1, and q 1 be integers. Let 1 be a sufficiently large real number. Then # n = 2 +O(. (47 2q n!, n"amodq Proof: The integers in the arithmetic progression are of the form n = qm + a, with 0 m ( a/q. Inserting this into the finite sum produces # n = q # m + a # 1 n!, n"amodq = q % ' 2 & m! ($a/q $ a q This last epression simplifies into the epression given above. m! ($a/q (% *' $ a & q +1 ( % * + a $ a ( ' *. & q (48 In this approimation the value 2 / 2q +O(!1 is nonnegative for all q in the range 1 q, and a 1.

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