The purpose of this article is to study the convergence of a few series with the Erdos function. The work is based on results concerning the convergence of some series with the Smarandache function

Tabirca, S.

Tabirca, T.

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NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

ON THE CONVERGENCE OF THE ERDOS HARMONIC SERIES

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ON THE CONVERGENCE OF THE ERDOS HA.R.tWONIC SERIES Tatiana Tabirca* Sabin Tabirca* "Transilvania Univenity of Brasov, Computer Science Department The purpose of this article is to study the convergence of a few series .... ith the Erdos function. The work is based on results concerning the convergence of some series with the Smarandache function. 1. INTRODUCTION The results used in this article are presented briefly in the following. These concern the relationship between the Smarandache and the Erdos functions and the convergence of some series. These two functions are important function in Number Theory. They are defined as follows: • The Smarandache function [Smarandache, 1980] is S: N* ~ N defined by Sen) = min{k E Nlk! =Mn} ('in EN *). (1) • The Erdos function is P : N* ~ N defined by Pen) = min{p E N In =Mp /\ pis prirn}('in E N*\{l}), P(1) = o. (2) The main properties of them are: ('ia,b E N*) (a,b) = l~S(a· b) = max{S(a),S(b)}, P(a· b) = max{P(a),P(b)}. (3) ('ia E N*) Pea) ~ Sea) ~ a and the equalities occur iifa is prim. (4) Erdos [1991] found the relationship between these two functions that is given by . I~ = l,n \ P(i) < Sci)} hm ' = o. (5) "-+(1) n This important result was extended by Ford [1999] to I~ = 1, n I P(i) < SCi)} = n· e-(J2 ... a.}~ , where lim an = o. I n~~ (6) Equations (5-6) are very important because allow us to translate convergence properties on the Smarandache function to convergence properties on the Erdos function. This translation represents the main technique that is used to obtain the convergence of some series .... ith the function P. 2. THE ERDOS HARMONIC SERIES 34 The Erdos hannonic series can be defined by L _1_. This is one of the important series with "":?2pQ(n) the Erdos function and its convergence is studied starting from the convergence of the Srnarandache hannonic series L _1_. Some results concerning series with the function S are ,,":?2 SQ(n) reviewed briefly in the following: • If (x"),,>o is an increasing sequence such that lim x" = 00, then the series L X,,+I - x" IS ,,~'" ,,>1 S(x,,) • • divergent. [Cojocaru, 1997]. (7) The series L-21- is divergent. [Tabirca, 1998] ,,":?2 S (n) (8) The series L_1- is divergent for all a>O. [Luca, 1999] ,,":?2 SQ(n) These above results are translated to the similar properties on the Erdos function. (9) Theorem 1. If (x,,),,> 0 is an increasing sequence such that lim x" = 00, then the series ,,~'" L X,,+I - x" is divergent. ,,>1 P(x,,) Proof The proof is obvious based on the equation P(x,.)::; S(x,,). Therefore, the equation x -x x -x "x -x ,,+1 " ~ ,,+1 " and the divergence of the series L..J ,,+1 " give that the series P(x,,) S(x,,) ,,>1 S(x,,) x -x l: ,,+1 " is divergent. 01 P(x,,) • 35 1 A direct consequence of Theorem 1 is the divergence of the series L ' where a,b>O 11>1 P(a·n+b) are positive numbers. This gives that L_l- is divergent and moreover that L_l- is II?2 pen) II?2 pQ(n) divergent for all a<1. Theorem 2. The series L _1_ is divergent for all a> 1. II?2 pQ(n) Proof The proof studies two cases. 1 Case 1. a ~ - . 2 1 In this case, the proof is made by using the divergence of L--' II?2 SQ(n) Denote A = f = 2,n 1 SCi) = P(i)} and B = f = 2,n 1 SCi) > P(i)} a partition of the set f = l,n }. We start from the following simple transfonnation (10) An i E B satisfies SQ (i) - PQ (i) ~ 1 and PCi) < SCi) :::;; n thus, (10) becomes II 1 II 1 1 II 1 1 ,,-> ,,-+ ,,-= "-+-'1 B 1 ~PQ(') - ~SQ(') ~ 2·Q ~SQ(') :'Q . ;=2 1 i=2 1 IEB n 1=2 1 n (11) IS divergent because the senes 1 . L -- IS divergent and II?2 SQ(n) I B I . -(J2+a.)-Jlnn·lnlnll 1 I· I' n e I' 0 Im-= 1m = 1m = """"" n2'Q """"" n2'Q II .... "" n2'Q-1 • e( J2 +Q. )·JIn II·1n In II . 1 Case 2. - > a > 1. 2 1 The first case gives that the series L-I- is divergent. II?2 p: (n) 36 n 1 n 1 Based on p2(n) > pa(n), the inequality ,,-- > ,,-- IS found. Thus, the series L. pa(i) L. !.. 1=2 1=2 p2 (i) 1 L -- is divergent. n~2 sa(n) • The technique that has been applied to the proof of Theorem 2 can be used in the both ways. Theorem 2 started from a property of the Smarandache function and found a property of the Erdos ilnS~i) function. Opposite, Finch [1999] found the property lim i=2 Inl = 1 based on the similar n-+'" n i InP(i) property lim i=2 In i = 1, where ),=0.6243299 is the Golomb-Dickman constant. Obviously, n-+'" n many other properties can be proved using this technique. Moreover, Equations (5-6) gives a very interesting fact - "the Smarandache and Erdos function may have the same behavior on the convergence problems." References Cojocaru, I. And Cojocaru, S. (1997) On a function in Number Theory, Smarandache Notions Journal, 8, No. 1-2-3, 164-169 Erdos, P. (1991) Problem 6674, Amer. Math. Monthly. 98, 965. Ford, K. (1999) The normal behaviours of the Smarandache function, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 81-86. Luca, F. (1999) On the divergence of the Smarandache harmonic series, Smarandache Notions Journal, 10, No. 1-2-3, 116-118. Tabirca, S. and Tabirca, T. (1998) The convergence of the Smarandache harmonic series, Smarandache Notions Journal, 9, No. 1-2,27-35. Smarandache, F. (1980) A Function in number theory, Analele Univ. Timisoara, XVIII. 37 

ON THE CONVERGENCE OF THE ERDOS HARMONIC SERIES

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