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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 , 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 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 :

# 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 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 , 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

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

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

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 ⎠

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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].

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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 , defined by nonexponential functions f : . It improves the standard { p = f (n) : p is prime and n "1 } 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. 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(x) simplifies the evaluation of the finite sum: This technique effectively transforms the vonMangoldt function into an expression of the form

µ(d)logd # s . (21) d " x+t d

The associated Dirichlet series is an absolutely convergent complex valued function on the complex 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

1 " (x) = # p = f (n) # x $ w( f (n) + t)%( f (n)). (22) f { } w(x)log x & f (n) # x

Note 2. This primes counting technique does not work on subsets of primes defined by exponential functions. ! n For example, the set of Fermat numbers F 22 1: n 1 , and the set of Mersenne numbers { n = + ! }

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p , would have to be handled with different primes counting methods. { M p = 2 "1: p # 2 prime }

! 2.2 Some Lemmas Let n ∈ ℕ be an integer. The Mobius function is defined by

# (!1)v if n = p p !p , µ(n) = $ 1 2 v (23) % 0 if n " squarefree.

And the vonMangoldt function is defined by

% log p if n = pk, k #1, (24) "(n) = & k ' 0 if n $ p , k #1.

The primes counting function is defined by

! x 1/2 !(x) = #{ p ! x : p is prime } = +O xe"c(log x) , (25) log x ( )

where c > 0 is a nonnegative constant. And the prime number theorem on arithmetic progression is defined by

x 1/2 !(x, a, q) = #{ p ! amodq " x : p is prime } = +O xe#c(log x) , (26) !(q)log x ( )

B where gcd(a, q) = 1, and q = O((log x) ), 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 x ≥ 1 be a sufficiently large real number. Then

x2 $ n!(n) = a1 +O(x ), (27) n " x, n#amodq q

2 where a1 = !!"(2) /!(2) is a constant.

Proof: Put the finite sum in the equivalent form

%n"(n) = & %n %µ(d)logd n # x, n $a mod q n # x, n $a mod q d | n (28) = &%µ(d)logd %n . d # x n # x / d , n $a mod q

! Primes Counting Methods:Twin Primes − 7

Substituting Lemma 6 returns

& 1 x2 x ) n (n) (d)logd O( ) $ ! = %$ µ ( 2 + + n " x, n#amodq d " x ' 2q d d * (29) x2 µ(d)logd & µ(d)logd ) = % +O(x % + . 2q $ d 2 ( $ d + d " x ' d " x *

Proceed to use Lemma 3, (or use equations (36) and (38)), and simplify to reach the expression

2 x ' !&(2) %1 %c(log x)1/2 * %c(log x)1/2 $ n!(n) = )% +O x e log x ,+O x 1+O(e log x) 2q ( !(2) ( )+ ( ( ) ) n " x, n#amodq (30) x2 = a1 +O(x ), q

2 where a1 = !!"(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 x ≥ 1 be a sufficiently large real number. Then

µ(n)logn !#(s) 1 s c(log x)1/2 = +O x $ e$ log x , (31) " s 2 ( ) n ! x n !(s) where c > 0 is a constant.

Proof: Write the finite sum in the form

µ(n)logn µ(n)logn µ(n)logn # s = # s % # s . (32) n " x n n $1 n n > x n

The constant is expressed in terms of the logarithmic derivative of the zeta function id est,

! d 1 µ(n)logn "& ( s) = #% s = # 2 . (33) ds "(s) n $1 n "(s)

!s Now, since the zeta function !(s) = #n"1 n is a decreasing function on the real half line ℜe(s) > 1, the "s derivative !!(s) = "$n#1(logn!)n is negative on the real half line s > 1. The result follows from these data. ■

The zeta function, and its derivative have the power series expansions

Primes Counting Methods:Twin Primes − 8

1 1 2 "(s) = + $ 0 # $1(s #1) + $ 2 (s #1) #!, (34) s #1 2

1 1 #" ( s) = $ $ % + % (s $1) $ % (s $1)2 +! (s $1)2 1 2 2 3 !

where γk is the kth Stieltjes constant, see [DL, 25.2.4]. These series can be used to compute numerical 2 approximations of the constants " s = #%$ ( s)/%(s) for s ≥ 1. Exampli gratia, the logarithmic derivative at s = 1 is ! !!(1) (s #1)2!!(s) = lim = #1, (35) 2 s "1 2 2 !(1) (s #1) !(s)!

and

µ(n)logn !#(1) c(log x)1/2 c(log x)1/2 = +O e$ log x = $1+O e$ log x , (36) " 2 ( ) ( ) n ! x n !(1)

where c > 0 is a constant. Similarly, the logarithmic derivative at s = 2 is

!!(2) (" 2 / 6)(# + log2" "12logC) 6(# + log2" "12logC) = " = " , (37) !(2)2 " 4 / 36 " 2

where γ is Euler constant, and C is Glaisher constant, see [DL, 5.17.7]. And

µ(n)logn !#(2) c(log x)1/2 6(! + log2" $12logC) c(log x)1/2 = +O e$ log x = $ +O e$ log x . (38) " 2 2 ( ) 2 ( ) n ! x n !(2) !

Lemma 4. Let x ≥ 1 be a sufficiently large real number, and let µ(n) be the Mobius function. Then

1/2 " µ(n)logn = O(xe#c(log x) log x) , (39) n ! x

where c > 0 is a constant.

Proof: Evaluate the Stieltjes integral

x "µ(n)logn = # (logt)dM(t) , (40) n!x 1

#c(log x)1/2 where M(x) = "n!x µ(n) = O(xe ) . ■

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

"(n) = # µ(d)logd. (41) $d | n

Proof: The logarithm derivative of the zeta function satisfies

! $# ( s) %(n) " = ' s . (42) $(s) n &1 n

On the other hand, the convolution of the Dirichlet series of the functions 1/ζ(s), and −ζʹ(s) respectively satisfies ! !"(s) % logn ( % µ(n)( $ µ(d)logn / d ! = ' * ' * = d|n . (43) '$ s * '$ s * $ s !(s) & n #1 n ) & n #1 n ) n #1 n

s Lastly, matching the coefficients of the powers of n− in (41) and (42) confirms the claim. ■

Various other identities and approximations of the vonMangoldt function are used in the literature. These include short formulas such as

"R (n) = $µ(d)logR /d , (44) d | n, d # R see [GP], the Vaughan identity, see [HN, p. 25], and the harmonic expansion

! c (q) !(n) = " $ n , (45) q #1 q see [MV, p. 114]. This later identity leads to the conjectured formula

!(n)cn 2 (q) #n"x + , (46) # !(n)!(n + 2) = $ # = 2c2 x + o(x) n " x q %1 q where c2 is the twin , see (12), and large real numbers x ≥ 1.

Extensive details of other identities and approximations of the vonMangoldt are discussed in [FI], et alii.

Primes Counting Methods:Twin Primes − 10

Power Sum Over Arithmetic Progressions

Lemma 6. Let a ≥ 1, and q ≥ 1 be integers. Let x ≥ 1 be a sufficiently large real number. Then

x2 # n = +O(x) . (47) n ! x, n"amodq 2q

Proof: The integers in the arithmetic progression are of the form n = qm + a, with 0 ≤ m ≤ (x − a)/q. Inserting this into the finite sum produces

# n = q # m + a # 1 n ! x, n"amodq m ! (x$a)/q m ! (x$a)/q (48) q % x $ a (% x $ a ( % x $ a ( = ' *' +1* + a' *. 2 & q )& q ) & q )

This last expression simplifies into the expression given above. ■

In this approximation the value x2 / 2q +O(x) !1 is nonnegative for all q in the range 1 ≤ q ≤ x, and a ≥ 1.

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3. References [AR] R.F. Arenstorf, There are infinitely many prime twins. http://arxiv.org/abs/math/ 0405509v1. Posted on May 26, 2004; withdrawn on June 9, 2004. [BC] Balog, Antal; Cojocaru, Alina-Carmen; David, Chantal. Average twin prime conjecture for elliptic curves. Amer. J. Math. 133 (2011), no. 5, 1179–1229. [BW] Bressoud, David; Wagon, Stan. A course in computational . Key College Publishing, Emeryville, CA; in cooperation with Springer-Verlag, New York, 2000. [CM] Cojocaru, Alina Carmen; Murty, M. Ram. An introduction to sieve methods and their applications. London Mathematical Society, 66. Cambridge University Press, Cambridge, 2006. [EU] Euclid Elements, Book VII, Proposition 20. [FI] Friedlander, John; Iwaniec, Henryk. Opera de cribro. American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010. [DL] Digital Library Mathematical Functions, http://dlmf.nist.gov. [GS] D. A. Goldston, Are There Infinitely Many Primes?, arXiv:0710.2123. [GP] Goldston, D. A.; Graham, S. W.; Pintz, J.; Yildirim, C. Y. Small gaps between almost primes, the parity problem, and some of Erdős on consecutive integers, arXiv:math.NT/0506067. [GU] Guy, Richard K. Unsolved problems in number theory. Third edition. Problem Books in . Springer-Verlag, New York, 2004. [GV] Granville, Andrew. Unexpected irregularities in the distribution of prime numbers. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 388–399, Birkhäuser, Basel, 1995. [HN] Harman, Glyn. Prime-detecting sieves. London Mathematical Society Monographs Series, 33. Princeton University Press, Princeton, NJ, 2007. [HW] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Sixth edition. Revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew Wiles. Oxford University Press, Oxford, 2008. [KR] Korevaar, Jacob. Prime pairs and the zeta function. J. Approx. Theory 158 (2009), no. 1, 69–96. [KT] Korevaar, Jaap; te Riele, Herman. Average prime-pair counting formula. Math. Comp. 79 (2010), no. 270, 1209–1229. [KV] Klyve, Dominic; Explicit bounds on twin primes and Brun's Constant. Thesis (Ph.D.)–Dartmouth College, Mass, 2007. [LA] Landau, Edmund Elementary number theory. Translated by J. E. Goodman. Chelsea Publishing Co., New York, N.Y., 1958. [MT] Miller, Steven J.; Takloo-Bighash, Ramin An invitation to modern number theory. With a foreword by Peter Sarnak. Princeton University Press, Princeton, NJ, 2006. [MV] Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical theory. Cambridge University Press, Cambridge, 2007. [NT] Nathanson, Melvyn B. Additive number theory. The classical bases. Graduate Texts in Mathematics, 164. Springer-Verlag, New York, 1996. [NW] Narkiewicz, W?adys?aw The development of prime number theory. From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. [PK] Pollack, Paul Not always buried deep. A second course in elementary number theory. American Mathematical Society, Providence, RI, 2009. [PP] Prime Pages, www.prime.utm.edu. [RN] Ribenboim, Paulo, The new book of prime number records, Berlin, New York: Springer-Verlag, 1996. [SD] Soundararajan, K. Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [SP] Shapiro, Harold N. Introduction to the theory of numbers. Pure and Applied Mathematics. A Wiley- Interscience Publication. New York, 1983. [ST] Stopple, Jeffrey. A primer of . From Pythagoras to Riemann. Cambridge University Press, Cambridge, 2003. [TM] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced

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Mathematics 46, (Cambridge University Press, Cambridge, 1995. [WA] Marek Wolf, The Skewes number for twin primes: counting sign changes of $p_2(x)-C_2 \Li_2(x)$, arXiv:1107.2809. [WB] M. Wolf, Generalized Brun's Constants , preprint IFTUWr 910//97, March 1997. [WK] Wikipedia.org. [WS] Weisstein, Eric W. "Twin Primes", http://mathworld.wolfram.com/TwinPrimes.html