Applicable Aalysis ad Discrete Mathematics available olie at http://pefmath.etf.bg.ac.yu Appl. Aal. Discrete Math. 2 (2008), 27 22. doi:0.2298/aadm080227c SOME PROPERTIES OF THE SEQUENCE OF PRIME NUMBERS Vlad Copil, Laureţiu Paaitopol Let p be the -th prime umber ad = p + + /p. We show that the sequece () N is ot mootoic for ay iteger N > ad that the series + = / is diverget. Related series are studied as well.. INTRODUCTION We use the well-kow otatio π(x) the umber of prime umbers x, p the -th prime umber, d = p + p, for, f() g() if there exist 0 < c < c 2 such that c f() < g() < c 2 f() for large eough, f() f() g() if lim + g() =. The followig results are also well kow: () p log, (2) = log log + O(). k= Moreover, we eed the followig results. I. We have (3) lim sup + p + p log 2000 Mathematics Subject Classificatio. N05, A4. Keywords ad Phrases. Prime umbers, sequeces, series. = +. 27
28 Vlad Copil, Laureţiu Paaitopol This result ca be foud i [6], but [4] cotais sharper results, which were later proved. Erdős ad Prachar proved i [] the followig theorem: II. Let A(x) be the umber of idices k such that x/2 < x ad + < ( δ)log x, the (4) A(x) > c x log x for some δ (0, ) ad c > 0, ad for all x > 0 large eough. Erdős shows i [3] the followig fact: III. There exists c > such that the iequality (5) d > cd + holds for ifiitely may values of, ad the iequality (6) d + > cd holds for ifiitely may values of as well. The followig result is proved i [5]. IV. If the sequece (u ) is decreasig ad cosists oly of positive umbers, ad the sequece (α ) has the property that there exist M m > 0 such that α + α2 + + α M m for every, the M u k k= α k u k m u k, k= k= ad thus the series + u ad + α u are equicoverget. = We shall deote = p + + the sequece ( ). p = ad we are goig to poit out some properties of 2. THE MONOTONICITY OF THE SEQUENCE ( ) It immediately follows from Theorem III that the sequece (d ) is ot mootoic. It is also kow that the sequece (p + /p ) is ot mootoic. Thus the mootoicity problem for the sequece ( ) arises i a atural way. Sice > p +, it follows that lim = +, hece the sequece ( ) + caot be decreasig. The complete result i this coectio is give by Theorem. The sequece ( ) N is ot mootoic for ay iteger N.
Some properties of the sequece of prime umbers 29 Proof. It suffices to show that the sequece is oicreasig. To this ed, we show that + < for ifiitely may values of. We cosider oly the idices such that d > cd with c > ( see the theorem III above) ad moreover > c +. We have c (7) < p + + p < p 2. cp+ + p Sice d > cd we deduce p >. To prove (7), it suffices to ( ) c + cp+ + p 2 show that > p + c + + p p+. If we deote = x >, the it p ( ) cx + 2 remais to show that > +, that is, c + (8) cx (c + )+ 2 + > 0. For x> let f(x)= cx (c+)+ 2 +. The f (x) = c + 2 (c+)x 2 > 0 because x > implies x 2 while > c + ( + )(c + ) implies < c. c 2 Cosequetly, the fuctio f is icreasig for x >. Sice lim f(x) = 0, the x desired iequality (8) follows. The series + = 3. THE SERIES + = / is diverget, but (2) shows that the sequece p to ifiity fairly slowly. Sice <, the series + p + = Moreover we have (9) = ( ) p = p + p + p + k= teds could be coverget. ( ( + d /p ) p/d ) d/p. It ow follows by () ad the result i I that limsup ( ) p d Sice lim = 0, we have lim + d d p + p This could mislead us to coclude that the series + prove the opposite Theorem 2. The series + = is diverget. + = d p = +. = e, so limif + / /p + = 0. is coverget. But, we
220 Vlad Copil, Laureţiu Paaitopol We first eed the followig auxiliary result. Lemma. We have e kd k/. k= Proof. Let s = e kd k/. Sice kd k / < 0, we have s <. k= We put x = p i the theorem II, ad it follows that there exist A(p ) idices p k such that p /2 < p ad d k < ( δ)log p. We have A(p ) > c log p ad () implies that there exists c 2 > 0 such that A(p ) > c 2. For these idices k we have (0) e kd k/ > e ( δ)k log p/. We have that k log k k log ad, sice p /2 < p, it follows k log p k log p that log log p, hece. Cosequetly < c 3 ad the by (0) we have e kd k/ > e c3( δ) = c 4. This implies s A(p ) c 4 > c 2 c 4 ad, sice s <, we get s. Proof of Theorem 2. Let α k = e kd k/ ad u k = /+. Sice α k ad the series + is diverget, the property IV implies that the series + = p + = is diverget. Sice for x > 0 we have ( + x) /x < e, we get from (9) () ad the divergece of the series + > p + = e d/p follows. k= e d/p p + Remark. The above result ca be stated i a more precise form. With the above otatio we have s / > c 2c 4. It follows by IV that k= e kd k/ + > c 2c 4 + k= hece by (2) ad (0) we have S = > c 5 log log with c 5 > 0. k= x k Sice <, it follows that S < < c 6 log log. Thus we have x k + k= S log log. Remark 2. Sice p + p + > + p+, we coclude that the series = p + is diverget. We pk+ k deote σ = ad it follows that σ k= + > S > c 5 log log. I this this regard we k
Some properties of the sequece of prime umbers 22 may raise the followig Ope Problem. Is it true that σ log log? REFERENCES. P. Erdős, K. Prachar: Sätze ud Probleme über /k. Abh. Math. Sem. Uiv. Hamburg, 25 (96/962), 25 256. 2. P. Erdős, P. Turá: O some ew questio o the distributio of prime umbers. Bulleti Amer. Math. Soc., 54 (948), 37 378. 3. P. Erdős: O the differece of cosecutive primes. Bull. Amer. Math. Soc., 44 (948), 885 889. 4. D. S. Mitriović, J. Sádor, B. Crstici: Hadbook of Number Theory. Kluwer Academic Publishers, Dordrecht - Bosto - Lodo, (996). 5. L. Paaitopol: Geeralizatio of a iequality of Tchebysheff ad some applicatios. Gaz. Mat. XCVII, No. 4 (2000), 324 327 (i Romaia). 6. B. Westzythuis: Über die Verteilug der Zahle die zu de erste Primzahle teilerfremd. Comm. Phys. math. Soc. Sci. Fe. Helsigfors, 5 (93), 37. Uiversity Spiru Haret, (Received April 8, 2008) Departmet of Mathematics, (Revised Jue 27, 2008) Str. Io Ghica, 3, 030045, Bucharest, Romaia E mail: vladcopil@gmail.com Uiversity of Bucharest, Departmet of Mathematics, Str. Academiei 4, 0004, Bucharest, Romaia, E mail: pa@fmi.uibuc.ro