LOWER ESTIMATOR OF A NUMBER OF POSITIVE INTEGERS h, FOR WHICH EXIST INFINITELY MANY. Tatiana Liubenova Todorova

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1 МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2002 MATHEMATICS AD EDUCATIO I MATHEMATICS, 2002 Proceedigs of Thirty First Sprig Coferece of the Uio of Bulgaria Mathematicias Borovets, April 3 6, 2002 LOWER ESTIMATOR OF A UMBER OF POSITIVE ITEGERS h, FOR WHICH EXIST IFIITELY MAY PRIMES h-twis * Tatiaa Liubeova Todorova I this paper the upper estimator for module of sum of complex fuctios of iteger argumet is obtaied The umber of positive itegers h, for which there exist ifiitely may primes p such that p + 2h is prime, is derived too Itroductio Oe of the aciet problems i umber theory is the hypothesis for the umber of twi primes It states that exist ifiitely may primes p, for which p+2 is prime, too I 99, W Bru proved, that there are ifiitely may atural umbers, such that ad + 2 possess ot more tha 9 prime divisors The positive itegers p ad p + 2h are called h-twis, if they are simultaeously primes I 923 G Hardy ad J Litlewood itroduced the hypothesis, that umber of primes p,, such that p + 2h h ) is prime, is give by the formula: Z, 2h) 2 pp 2) p ) 2 p p 2 l 2, where: Z, 2h) = Card{p : p, p, p + 2h primes}; p/h deotes that p divides h; if h = 2 k, k the secod product is equal to I 96, A Lavrik [3] proved that the equality ) Z, 2h) = 2 pp 2) p ) 2 ) p p 2 l 2 + O l 3 ) γ is fulfilled for each h 0, except ot more tha 2 l l) M, γ R+ \ {0} umbers; where: M is a costat, M, ); the umber γ ad) remaider term i equality ) deped oly o M The costrait h 0, is due to the fact 2 l there ot eough good estimate for the umber of primes i arithmetic progressios with 06 * 2000 Math Subject Classificatio: A4, 05, 80

2 differece greater tha Usig sieve methods H Halberstam, H Richert [] Th l 3, ch 3, 7, []) received for Z, 2h) the followig estimator: 2) pp 2) p ) 2 Z, 2h) 8 pp 2) p ) ; which is the best oe till ow We shall use the followig otatios: m, m 2, k, h, positive itegers; p prime umber; π) = the umber of primes p ; p p p 2 l 2, p l l l3h) + O), p 2 [a] the iteger part of a, resultig i the greatest iteger less or equal to a; λ) is Magold s fuctio: { l p, for = p k λ) = 0, for p k Z, 2h) = = Card{p : p, p, p + 2h primes} umber of p p,p+2h primes primes p, such that p + 2h is a prime, too The asymptotic law of primes distributio π) l is applied, too 2 Mai results We shall prove the followig geeralizatio of Lemma 3, ch, 3, [2]: Theorem Let the followig coditios hold: ϕ : a, b] C is a complex fuctio; 2 q, q The a<b ϕ) q sup a<b ϕ) + 2) q q r= a<b r ϕ)ϕ + r) Proof We set ϕ) = 0 for Z \ a, b] The S = ϕ) = ϕ + m), for m Z a<b I this sum we have fiitely may summads, so we ca write: S = q ϕ + m) q m= 07

3 Usig Cauchy s iequality, we obtai: S 2 ) q ϕ + m) m= From the fact that ϕ) 0 for umbers it follows S 2 q 2 ϕ + m) 3) { q m= q ϕm + ) m= m m= = ϕm + ) 2 + 2q m<sq 2 ) ϕ + m)ϕ + s) } q r ϕ)ϕ + r) Further the proof of theorem is similar to the proof of Lemma 3, ch, 3, [2] For ϕ) = e 2πif) we obtai the statemet of quote above Lemma: Corollary Let the followig coditios be fulfilled: The f : a, b] R is a real fuctio; 2 g) = f + r) f), a, b], r [, q ], q [2, ] ; a<b e 2πif) r= = 2) + q q Corollary 2 Let the followig coditios be valid: ; ) 2 ε 2, ; 3 h 0, ) 2 q r= a<b r e 2πig) [ ] 2ε ) The there exist at least umbers h, such that Z, 2h) 2 ll l3) λ) Proof We cosider the sum S = The followig equality is valid: l 08 S = < = λ) l = + p p 2 + p = = π) + 2 π ) + 3 π 3 ) + ε) l 2

4 Evidetly 4) S π) Let q [, ] For ϕ) = λ) l S 2 { q m= m = applyig iequality 3) oe obtais the estimate: λ 2 + m) l 2 + m) + 2q From the above iequality ad 4) we have: 5) where S = q π) m m= = ) 2 We shall evaluate the sums S ad S 2 The followig iequality is valid: 6) S q m= = ) 2 λ) = l q r r= = S + 2qS 2 ), λ 2 + m) l 2 + m), S 2 = q m= qπ) + qπ ) ) q r= = } λ)λ + r) ll + r) r λ)λ + r) ll + r) π) π ) π 3 ) ) + qπ) + 2qπ ) The defiitio of Magold s fuctio yields λ) 0 for = p k, p 2 The: 7) S 2 = S 2 + S 22, where S 2 = q r= =2 k r λ)λ + r) ll + r), S 22 = [ q r= =p k 2r λ)λ + 2r) ll + 2r) The umber of itegers = 2 k < is [log I accordace with Euler s costat ) C = lim ν l = 05772, we obtai for S 2 the followig iequality: ν= 8) S 2 q r= k[log k 2q l log 2 4q l l We are goig to show that: 9) =p k 2r λ)λ + 2r) Z 2r, 2r) + 2 l ll + 2r) 09

5 For p > 2 ad p, p + 2r simultaeously primes the followig equality is valid: 0) + =p k 2r + λ)λ + 2r) ll + 2r) = p,p+2r) 2 p 2,p+2r) 2 p 2r + p 2,p+2r p 3,p+2r p,p+2r) p 3,p+2r) 2 p 2,p+2r) 3 The greatest k, for which p k, 2 < p is [log 3 ] There are k summads correspodig to ay power k, k [2, [log 3 ]] of the followig form: p m,p+2r) m 2 m m 2, where m + m 2 = k; m, m 2 [, k) For k 3 we have: ) [log 3 ] k=3 p m,p+2r) m 2 m +m 2 =k ) m m 2 [log 3 ] k=3 k k < 2 l Applyig the defiitio of fuctio Z, 2h), 0) ad ) we obtai the iequality 9) Hece for S 22 we derive the followig estimator: [ q S 22 q l + Z 2r, 2r) r= The above iequality ad 8) yield: 2) S 2 4q l l + q l + 3) r[ q Z 2r, 2r) Usig 5), 6) ad 2) we obtai: ) 2 π) { qπ) + 2qπ ) + 8q l l+ +2 l + 2q r[ q Z 2r, 2r) Let s be a umber of positive itegers 2r, 2r <, such that Z 2r, 2r) I accordace with 2) we have: 0 max Z 2r, 2r) r[ } ε) l 2

6 56 sup r[ 2r l 2 2r) max{2, sup 3<r[ where A) < 56 ll l3) Therefore, it follows from the iequality 3) that: 4) ) 2 π) q l l l3r)} A) l 2, 2q s) ε) π) + q l q A l 2 π)l From the fact that lim =, dividig both sides of 4) by 2 l 2 ad performig the limitig coversio as we have: l lim q + lim 2 s q ) ε) + lim 2As q Let q = The for eough large the followig iequality is valid: s 2ε ) 2A 2ε ) 2 ll l3) The choice of umber s ad the above iequality yield the statemet of Corollary 2 REFERECES [] H alberstam, H Richert Sieve Methods, Lodo, Academic Press Ic, 974 [2] А А Карацуба Основы аналитической теории чисел, Москва, Наука, 983 [3] А Ф Лаврик К бинарным проблемам аддитивной теории простых чисел в связи с методом тригонометрических сумм И М Виноградова Вестник Ленинградского Университета, 3 96), 27 Departmet of Algebra Faculty of Mathematics ad Iformatics St Klimet Ohridski Uiversity of Sofia 5 JBaurchier Blvd 64 Sofia, Bulgaria tlt@fmiui-sofiabg ОЦЕНКА ОТДОЛУ НА БРОЯ НА ЧИСЛАТА h, ЗА КОИТО СЪЩЕСТВУВАТ БЕЗБРОЙ МНОГО ПРОСТИ ЧИСЛА h-близнаци Татяна Любенова Тодорова В статията е получена оценка отгоре за модула на сума от стойностите на комплексните функции от целочислен аргумент Намерена е оценка отдолу за броя на естествените числа h, за които съществуват безброй много прости числа p, такива че p + 2h също е просто