the average of the erdos function
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8/12/2019 THE AVERAGE OF THE ERDOS FUNCTION
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is not properly chosen for E S n)). Tabirca [1998] conjectured that the average E S n)) satisfiesnthe equation E S n)) . inally and the most important, Luca [1999] proposed the equationinn
1 r r] 5 1 31_ pr n)-Jr vn) < E S n)) < Jr(n) + l n l nn2 2 n 5 7)where ;r x) denotes the number of prim numbers less than or equal to x. Thus, the complexity orderfor the average E S n)) is indeed o _n_J .logn2. THE COMPLEXITY ORDER FOR THE ERDOS FUNCTIONSome of the above results are used to find the complexity order of E P n)). Based on the well-known formula lim Jr(n) =1, Equation 7) givesn-+ n
Inn. . lim inf E S n)) lim sup E S n)) 1.2 n-+ n n-+ n 8)
Inn InnTheorem 1.lim inf E S n)) = lim inf E P n)) , lim sup E S n)) = lim sup E P n)) 9)
n-+ n n-+ n n- oo n n-+ nInn Inn Inn Inn
Proof Let us denote A = i = ,n ISCi) > P i)} the set of the numbers that do not satisfy theequation S i)=P i). The cardinal of this set is 1A 1= n - . J 2 + a } ~ I v n . l n l n n , where lim an =o
n-+The proof is started from the following equation
IE S n)) - E P n ) ~ = t Ci) - t p i ) = { ~ S i ) - P i))] S ~ (10)Because we have (\1 i =1,n S i) n Equation 10) gives
iE S n))-E P n))1 IA = n.e-(.J2+a.}JIvn.lnlnn andE S n)) _ E P n)) Inn.e- .J2+a.}JIvn.lnlnn.n n
Inn Inn
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Because lim a =0 the equation lim In n - ( ~ + a . } ' / ' . I n l n =0~ < D
lim inf E(S(n)) = im inf E(P(n)) , lim sup E(S(n))n-+1Inn
ReferencesErdos, P. (1991) Problem 6674, Amer. Math. Monthly. 98, 965.Ford, K. (1999) The nonnal behaviours of the Smarandache function, Smarandache NotionsJournal Vol. 10, No. 1-3, 81-86.Luca, F. (1999) The average Smarandache function, Personal communication to S. Tabirca. [willappear in Smarandache Notion Journal].Tabirca, S. and Tabirca, T (1997) Some upper bounds for Smarandache s function, SmarandacheNotions Journal 8, 205-211.Tabirca, S. and Tabirca, T. (1998) Two new functions in number theory and some upper boundsfor Smarandache s function, Smarandache Notions Journal 9, No. 1-2,82-91.
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