Twin Primes and the Zeros of the Riemann Zeta Function

Transcription

1 Iteratioal Joural of Mathematics Research. ISSN Volume 5, Number 2 203, pp Iteratioal Research Publicatio House wi Primes ad the Zeros of the Riema Zeta Fuctio H. J. Weber Departmet of Physics, Uiversity of Virgiia, Charlottesville, VA 22904, USA Abstract he Legedre-type relatio for the coutig fuctio of ordiary twi primes is reworked i terms of the iverse of the Riema zeta fuctio. Its aalysis sheds light o the distributio of the zeros of the Riema zeta fuctio i the critical strip ad their lik to the twi prime problem. AMS subject classificatio: A4, N05, M06. Keywords: zeta zeros, twi primes, o-raks, Perro itegrals.. Itroductio he pair sieve for ordiary twi primes [] leads to a formula for the twi prime coutig fuctio π 2 x that is aalogous to Legedre s formula [2] for the prime umber coutig fuctio πx. Before ad after separatig it ito mai ad error terms [], it is rewritte here usig relevat Dirichlet series. Sice the Riema zeta fuctio eds up i the deomiator of the cotour itegrals, this feature liks the zeta zeros to twi primes, much like πx or related coutig fuctios are expressed as Perro itegrals over ζ /ζ i aalytic umber theory [3], [4]. Our aalysis sheds light o the role of twi primes i the distributio of the otrivial zeros of the Riema zeta fuctio, which are those i the critical strip, as usual. I Sect. 2 the mai cocepts, such as twi raks, o-raks ad remats of the twi-prime pair sieve are recalled alog with its mai result, the Legedre type formula for π 2. I Sect. 3 it is rewritte as a Perro itegral ad aalyzed. I Sect. 4 the fidigs are summarized ad discussed.

2 2 H. J. Weber 2. Review of the Pair Sieve ad Notatios he prime umbers 2, 3 do ot play a active role here because they are ot of the stadard form 6m ±. his also applies to the first twi prime pair 3, 5. From ow o p deotes a prime umber or variable ad p j the jth prime with p = 2, p 2 = 3, p 3 = 5,....I our twi prime sieve p j plays the role of the variable x i Eratosthees sieve. Defiitio 2.. If 6m ± is a ordiary twi prime pair for some positive iteger m, the m is its twi rak. A positive iteger is a o-rak if 6 ± are ot both prime. he arithmetical fuctio Nx, x = + is eeded for o-raks. 2 Defiitio 2.2. If x is real the Nx is the iteger earest to x. he ambiguity for x = + with itegral will ot arise. 2 I Ref. [] we the prove Lemma 2.3. If p 5 is prime the the positive itegers k, p ± = p ± N p > 0, = 0,, 2,... 6 are o-raks. If a iteger k>0 is a o-rak, there is a prime p 5 so that Eq. holds with either + or sig. his meas that the pairs 6k + ± ad 6k ± each cotai at least oe composite umber. herefore, the primes p 5 orgaize all o-rak umbers i pairs of arithmetic progressios. hese pairs are twi prime aalogs of multiples p, >, of primes struck from the itegers i Eratosthees sieve. Give a prime p 5, whe all o-raks to primes 5 p <pare subtracted from the o-raks to p, the the o-raks to paret prime p are left formig the set A p. his process [] aturally itroduces the primorial Lp = p as the period of its 5 p p arithmetic progressios. Lp j is the twi prime sieve s aalog of the variable x i Eratosthees sieve. Defiitio 2.4. Let p p 5 be prime. he supergroup S p = A p cotais the sets of arithmetic o-rak progressios of all A p,5 p p. p p he umber Sp couts the o-raks of S p over oe period Lp. Defiitio 2.5. Sice [] Lp >Sp, there is a set R p of remats r i its first period such that r S p ; they are twi-raks or o-raks to primes p j <p, where p j 5is the jth prime. Let Mj + = 6 p2 j+. Whe all o-raks to primes p p j are removed from the first period [, Lp j ], all r Mj + are twi raks. hese frot

3 wi Primes ad the Zeros of the Riema Zeta Fuctio 3 twi raks play the role of the primes p x i Eratosthees sieve that are left over whe multiples of primes are removed. he prime p j is the twi sieve aalog of x there; p j ad Lp j correspod to the variable z ad P z = p z p, respectively, i more sophisticated sieves. 3. Reworkig the wi Prime Formula If p j is the jth prime, the we eed Lp j = p, x = Lp j Mj + 2 for the mai result of Ref. [], which is a Legedre-type formula for the umber of twi raks i the first period of legth Lp j of the supergroup S pj, where π 2 couts twi pairs below 6x + : π 2 6x + = R 0 + μ2 ν[ x ] + O 3 x, L j x where [x/] is the greatest iteger fuctio, L j x = R 0 = Lp j 2 p p j <p x p, ad Cx log log x 2, C>0[], p j log x 4 couts the umber of remats i S pj, that is, twi raks prime pairs at distace 2 ad o-raks to primes p j <p x. herefore, the i the of Eq. 3 ru over these primes [ oly ad their products, ad the upper limit is x because the greatest iteger x fuctio = 0 for >x.he twi pair coutig fuctio π 2 Mj + is the umber ] of frot twi-raks ad the aalog of π x i Legedre s formula for the prime coutig fuctio πx see, e.g., pp. 2-3, Ch. of Ref. [5]; they are icluded i R 0. he error term O i Eq. 3 accouts for the less tha perfect cacellatio at low values of x of R 0 ad the sum i Eq. 3, but Eq. 3 is oly relevat at large x i the followig. Let us briefly sketch the cacellatio of the too large R 0 agaist the sum i Eq. 3 at large log log x, upo decomposig [x/] = x/ {x/} as usual. Expadig the R 0 product ito a sum ad combiig it with the correspodig sum of the ratios x/ shifts the upper limit of the R 0 sum from p j to x, whe rewritte i its product form. his trasforms its etire asymptotics from log log x to log x. For more details we refer to heors. 5.7, 5.8 of Ref. []

4 4 H. J. Weber he asymptotic relatios 4 derive from log Lp j = log p = p j + Rp j = log x + O log 2 x, 5 x where the error term comes from Mj +, ad Rp j is the remaider of the prime umber theorem. he Dirichlet series characteristic of twi primes ad associated with R 0 are P j s = 2p s = 2 p s μ2 ν s. 6 p>p j p p j hey coverge absolutely for σ>, as is evidet from the majorat [6] ζ 2 s ζ 2s = = 2 ν = s, σ>. 7 Note that 2 ν log /ζ 2 i the iterval [, x] o average, which is show i of Ref. [3]. he correspodig Dirichlet series for primes is where ζ s = x p x p = s there. P 0 s = p 2 p s = ζ s, 8 is the Riema zeta fuctio, ad the aalog of R 0 here is We ow use the Perro formula i essetially the form proved i of Ref. [3]. Lemma 3.. Let the Dirichlet series As = Rs >. he a = σ +i As xs 2πi σ i s ds + O x = =, =x where the lhs x meas that for = x, a is reduced by /2. a be absolutely coverget for σ = s x σ a mi, log x, 9

5 wi Primes ad the Zeros of the Riema Zeta Fuctio 5 Corollary 3.2. For σ> [ x ] a x +O =, =x = σ +i Asζ s xs 2πi σ i x σ d a d mi s ds, log x. 0 Proof. his follows from Lemma 3. ad the proof of i Ref. [3] usig a = [ x ] a, Asζ s = N x s a. N x N N= N Lemma 3.3. P sζ 2 s = 2 s 2 p>2 + p s p s = 2 s 2 2 Ds 2 Ds = + p>2 ν=0 2 ν p ν+2s = + coverges absolutely for σ>/2. Here N=4 2 2r en+2r o N 2 r e N 2 r o N N s 3 r e N = m ν i, r o N = i= μ i + 3, r e N =, r o N = 4 ν i >0 μ i >0 i= are additive fuctios for i Eq. 3. Proof. Substitutig i the expasios N = p 2ν + e p 2ν m+ e m p 2μ +3 o p 2μ +3 o 5 p>2 2 p s p s 2 = p>2 + p s p s = + 2 p s , 7 p2s p s

6 6 H. J. Weber + p 2s 2 p s = + yields Eq. 3 with N of the form i Eq. 5. ν=0 2 ν, 8 p ν+2s hus for σ> P j s = ζ 2 s 2 s 2 2 p s + p s p s 2 2<p p j p>2 P s = 2<p p j 2 9 p s with P s from Eq. 2. We ow apply Cor. 3.2 to P j s. his yields the Legedre-type formula before it is split ito its mai ad error terms accordig to Ref. [] so that the leadig asymptotic term is R 0. heorem 3.4. For σ>, R 0 = Lp j 2 ad x>0from Eq. 2, p π 2 6x + = R 0 + 2πi ζ 3 σ x σ +O + O σ +i σ i x log 3 x 2 s 2 x s ds sζs 2<p p j 2 p s Ds with Ds from Eq. 3 ad >0 at least of order x c,0<c<. + O, 20 Proof. We replace i Eq. 3 the sum by the Perro itegral of Cor. 3.2 with As = P j s usig Lemma 3.3 for P s i cojuctio with Eq. 9. Cacelig the factor ζ s, this yields the Perro itegral i Eq. 20. he Euler product of Ds i Eq. 2 guaratees o zeros for σ>/2. Note that μf 2 νf = 2 νf = d 3 ñ d 3, 2 f f ñ where ñ is the product of differet prime divisors of ad, for ay f ñ, df = δ f = 2 νf 22 is the divisor fuctio. hus, we ca use the majorat d 3 i the error term i Cor. 3.2, where ζ 3 s = d 3 s. We split the sum ito three pieces as usual see, e.g., =

7 wi Primes ad the Zeros of the Riema Zeta Fuctio 7 heor of Ref. [3] with S =, S 2 = S 3 = <x/e x/e<<ex,. For S, S 3 we have >ex log x/. he total cotributio due to S ad S 3 is at most ζ 3 σ x σ /, which is the first error term i Eq. 20 with the costat implied by the O. For S 2, we divide the sum ito itervals of the type I k = [x ± 2 k x/, x ± 2 k+ x/ ] with 2 k+ / < ex, ad a shorter iterval at the ed if eeded. he umber of such itervals is O log. he cotributio of the sum over such a iterval to the remaider of Perro s formula is of order at most I k d 3 2 k = I k d 3 2 k. 23 he legth of I k is of order 2 k x/, which is larger tha x c, sice is at least of order x c for some 0 <c<. Now recall the estimate see Ref. [6], Ch. 2, formula 2..4: d 3 = yp 2 log y + Oy 2/3 log y, 24 <y P 2 beig a certai polyomial of degree 2. It follows that 2 k x log 2 x d 3 = O + Ox 2/3 log x. 25 I k If we sum over k the cotributio of S 2 is at most of order O x log3 x, which gives the secod error term i Eq. 20. he iterval x/e<<x ca be subdivided ad treated similarly leadig to the same boud. his completes the proof. Corollary 3.5. he Riema hypothesis RH is icompatible with the twi prime formula 20 of heor Proof. Assumig RH, we shift the lie of itegratio i Eq. 20 from σ>toσ = 2 +ε for ay ε>0usig Cauchy s theorem. Sice RH implies the Lidelöf hypothesis [6] Chapt. 3, we kow that ζ s = O t δ, s = σ + it, σ 2 + ε 26 for some small δ>0 that may deped o ε. We ote that the zeros s p = log 2/ log p of p p j 2/p s cacel the correspodig poles of Ds. Sice oly s > 0.5,

8 8 H. J. Weber we estimate for σ + ε 23 s Ds = 3 s s [ + p s p s 2 ] = O. 27 p 5 As 5 p p j log x i the product p 2/p s, the latter will be at most of order 2 p s 2 p = O log x 28 for σ /2 + ε as p j log x. Hece, o σ = 2 Perro itegral i heor. 3.4 obeys + ε the vertical part of the 2 +ε+i 2 s 2 x s ds 2 +ε i sζs 2<p p j 2 p s Ds = O δ x 2 +ε log log x, 29 with the log factor from the itegratio. O the horizotal lie segmets from + ε ± i to σ ± i the Perro itegral is 2 bouded by O δ x σ. he factor / log x from the itegratio cacels log x. he error terms of heor. 3.4 are slightly smaller tha these, respectively, ad ca be combied with them. akig σ = + ε, = x α ad equatig the expoets of x i both error terms determies α =. herefore, the Perro itegral plus error terms 2 i Eq. 20 are of order Ox ε++δ/2 ad caot reduce R 0 Cx/ log log x 2 to the kow boud [5] Ox/ log x 2 for π 2 6x +, q.e.a. We ext address the remaider of the twi prime formula 3 after extractig its asymptotic law [] usig the followig Perro itegral. Corollary 3.6. Let As be absolutely coverget for σ>, the for σ> { x } a = σ +i [ dsx s As 2πi <x σ i s ζ s ] s +O x σ a d mi, log x. 30 =, =x d Proof. Usig { x = } x [ x ] 3

9 wi Primes ad the Zeros of the Riema Zeta Fuctio 9 ad applyig Lemma 3. to xas + for the ratio x/, itegrated alog the lie σ>0, ad Cor. 3.2 we obtai the Perro itegral i Eq. 30 upo shiftig s s i the first term. Usig a a d, the error term of Lemma 3. combies with that of Cor. 3.2 d givig that of Eq. 30. We ow apply Cor. 3.6 to P j s which yields the Perro itegral for the error term R E of Ref. [] after separatig formula 3 ito its mai ad error terms so that the mai term obeys the proper asymptotic law expected for twi primes []. he error term is the same as i heor For the cacellatio ivolved i gettig the proper asymptotics we refer to the discussio below Eq. 4. Clearly, the sum i Eq. 3, represeted by the Perro itegral i heor. 3.4, is R 0 plus a asymptotic term cx/ log x 2, with c>0 calculated i Ref. []. hus it is large, ad a applicatio of the cotour deformatio to the Perro itegral i heor. 3.4 ito the kow zero-free regio of the Riema zeta fuctio fails to give a small value ucoditioally because the optimal a = 0 caot be reached at ay fiite. heorem 3.7. here are costats a>0, 0 <b<c,<α<2so that the twi prime remaider takes o the form R E = μ2 ν{ x } x + log a log 3 + O = O p j <<x, L j x x log 3 x +O + O x log b log 3 log x α + O x + a log 2 log x α log = O x exp c log x log x 3, = exp c log x. 32 Proof. We start from Cor. 3.6 for P j s i cojuctio with the error terms of heor. 3.4: R E = μ2 ν{ x } + O p j <<x, L j x σ +i = 2 s 2 x s ds [ 2πi σ i ζ s 2<p p j 2 p s Ds s ζ s ] s ζ 3 σ x σ + O + O x log3 x. 33 By Chapt. 3, formula 3..8 of Ref. [6] this also follows from the sharper estimates i Lemma 2.3 of Ref. [7] there is a absolute costat c>0 so that ζ s = O log t +2, 0 <t c 0 t, δ t σ + ε, δ t = log t +2 34

10 0 H. J. Weber ad ζ s = O for t <t 0, σ δ t. Let R be the rectagle joiig the vertices + a log i,+ a log + i, δ + i, δ i, δ = c log. 35 We move the lie segmet of itegratio from σ = + a/log, a>0 to the left o the lie σ = b/ log with a>0, 0 <b<cto be chose later. he the bouds of ζ s i Eq. 34 ad below hold o the boudary of R. he itegrad is holomorphic iside ad o the rectagle because ζ s does ot vaish there ad o the vertical lie o the left it is of order at most x log b log s p p j p + log b. 36 Sice p p j log x we kow that p +b/ log 0asx provided grows with x faster tha a power of log x, which will be the case. he a lower boud for the product will be at least of order / log p j α for ay <α<2. So the product is at most of order 2<p p j 2 p s = O log p j α = O log x α. 37 Itegratio over s gives a factor log.hus, the itegral over the vertical segmet is at most of order Ox b/ log log 3 log x α. Similarly, the itegrals over the horizotal segmets are at most of order O x + log a log 2 log x α Puttig all this together we obtai the middle sectio of Eq. 32. Now we choose = exp τ log x ad optimize with respect to a, b, τ uder the coditios a > 0, 0 <b<c.i the limit we ca set b = c, a = 0, τ = c ad coclude with the boud o the rhs of Eq. 32. A commet o the choice a = 0isi order. he extra covergece factor s/s ζ s /ζ s at large t is the Dirichlet series 2 μ s that coverges o σ = ad oscillates rapidly at large t. his is how the real reductio from [x/] i heor. 3.4 to {x/} i heor. 3.7 plays out aalytically. Its presece allows reachig the optimizatio poit a = 0 of the error terms represetig the Perro itegral i heor More precisely, heor..8b for ay σ> ad heor..0 for 2 <α<σ < β< i Ref. [6] imply that the umber of solutios of ζ s = + ε for σ > ad ζ s = ε for <σ < ad arbitrarily small ε>0have positive desity i the 2 38

11 wi Primes ad the Zeros of the Riema Zeta Fuctio iterval [0, ] o vertical lies that may be chose as close as oe likes to σ =. Note that s = + σ 2 + O, σ> 2, s t 2 s s = + σ t 2 i t + O t 4 t 3, s =. hus the covergece factor is as small as oe likes for large eough t i may areas of the regio eclosed by the cotour ad o it. his is ot the case with the Perro itegral i heor. 3.4, where the fluctuatig x s has large absolute value x s =x σ. his completes the proof. his proves that the miimal asymptotic law obtaied i Ref. [] is valid with the remaider smaller tha it by ay positive power of log x. 4. Summary ad Discussio Whe the Legedre-type formula for π 2 is reworked ito a Perro itegral ivolvig ζ s, the otrivial zeta zeros are see to be liked to the twi prime coutig fuctio π 2. he asymptotic law of its leadig term R 0 = Lp j 2 p Cx, log log x 39 log log x 2 with x = Lp j Mj +, p j log Lp j log x requires log log x to become large. I cotrast, oly log x is large i the prime umber theorem [3], [4]. herefore, the true asymptotic regio of twi primes starts much higher up tha for primes. Such values of x>x 0 = e eee with log log x 0 = e e = are larger tha followed by more tha.6565 millio zeros, i.e. are truly titaic umbers. It is clear that preset umerical results of otrivial zeta zeros are far from reachig the asymptotic twi prime realm. his is valid whether or ot there are ifiitely may twi primes, because R 0 is the umber of remats icludig twi pairs i.e. twi raks ad o-raks to primes p j <p<x.he Perro itegral i heor. 3.4 represets the latter s cotributios that will reduce R 0 to π 2 6x +, R 0 beig much larger tha kow bouds from sieve theory [3], [5] o π 2 that are due to V. Bru, A. Selberg ad others. Oly otrivial zeta zeros i the Perro itegral ca produce terms that reduce R 0 to the proper magitude. Our first result is that the zeros o the critical lie caot do the job. Despite trillios of iitial zeros o the critical lie that are relevat for the prime umber distributio without asymptotic twi prime attributes, oce twi prime asymptotics matter some zeta zeros must move off the critical lie toward the borders of the critical strip. Fially, from the poit of view of our twi prime formulas 3, 20 a fiite umber of twi primes is either a simple or atural case, as it would require the cacellatio of

12 2 H. J. Weber the leadig ad all subleadig asymptotic terms ivolvig fie-tuig of the large primes > p j that orgaize the o-raks. But this ever happes because, whe the Perro itegral i Cor. 3.6 is developed for μ2 ν {x/} usig P j s i the kow zero-free regio of the Riema zeta <x, L j x fuctio, the twi prime theorem ear primorial argumets follows, our secod result. Refereces [] H. J. Weber, wi Prime Sieve, to appear i Id. J. Math. Res [2] H. Riesel, Prime Numbers ad Computer Methods for Factorizatio, 2d ed., Birkhäuser, Bosto, 994. [3] M. Ram Murty, Problems i Aalytic Number heory, Spriger, New York, 200. [4] G. H. Hardy ad E. M. Wright, A Itroductio to the heory of Numbers, 5th ed., Claredo Press, Oxford, 988. [5] R. Friedlader ad H. Iwaiec, Opera Cribro, Amer. Math. Soc. Colloq. Publ. 57, Prov. RI, 200. [6] E. C. itchmarsh, he heory of the Riema Zeta-Fuctio, 2d ed. edited by D. R. Heath-Brow, Oxford, Claredo Press, 986. [7] A. Ivić, he Riema Zeta-Fuctio, Dover, Mieola, N.Y., 2003.