Key words: Explicit upper bound for the function of sum of divisor σ(n), n 1.

Transcription

1 ISSN (Paer) ISSN (Online) Vol.4, No., 204 Exlicit uer bound for the function of sum of divisors σ(n) Dr. Saad A. Baddai, Khulood M. Hussein Det. Math., Collere of Science for Women, Univ. of Baghdad M-alsaedi )*( Abstract: we develoed the result roved by A.Evic^[6]where he roved the following theorem : Theorem:[6] let n 7,then : σ(n) < (2.59)n log log n,....() this roving required several Lemmas which are given to be : Lemma:[6] if n 39,then: Lemma2:[6] if n 3,then: Lemma3:[6] if n 3, n 42, n 20,then: log n < 7 5 log n ( + ) < 28 n log log n 5 n < 2.59 log log n. We a doe, the rocedure of Evic^by imroving the lemmas and get the following results: Lemma 2:If n 82, then: Lemma3; If n 3, then: log n < log n ( + n ) < log log n Lemma4: For n 82, n 42, n 20, n < log log n σ(n) which given in the following theorem: Where our lemma give above leads us to gut the new uer bound of the function Theorem: If n 82,then: σ(n) < n log log n Key words: Exlicit uer bound for the function of sum of divisor σ(n), n. 68

2 ISSN (Paer) ISSN (Online) Vol.4, No., 204 Introduction: In this aer, we discuss the uer bound of the multilicative number theoretic function σ(n), where σ(n) reresent the sum of all ositive divisors of n, n.this function is written to be, d n d, In Hardy and whright [4] the bound given to be exlicit which is Also, R.L. Duncan [3] showed that, σ(n) = 0(n log log n),..() σ(n) < n 6 [7ω(n) + 0].. (2) Where n = a. 2 a2 r ar and ω(n) = r and this is an exlicit bound interns of ω(n). this bound can not be determined for large n, in which ω(n) can not be counted. The best exlicit uer bound roved by Aleksander Ivic^[6]when he roved. where n 7. σ(n) < (2.59)n log log n,....(3) Our work dealing with Aleksander Ivic^ results in (3) and we imrove this bound by increasing n to be n 82,and roved the following result. σ(n) < ( )n log log n,.(4) Definition ω(n): Let n= a r ar.then ω(n) =r, reresent the number of rimes of n, if n= , then w(n)=3. Definition π(x) :the number of all rimes less them or equal to x denoted by π(x), from Definition,we can write: π(x) = For if, x=8.5, π(8.5)=4, x=2, π(2)=, x=2, π(2)=5. Theorem:[4]The number of rimes not exceeding x is asymtotic (x) ~ Theorem2:[6]there exist a constants c and c 2 such that, x log x, c 2x log x < π(x) < c x log x x logx i.e We shall imrove the bound given in [6]for the function σ(n).evic, roved the following theorem. 69

3 ISSN (Paer) ISSN (Online) Vol.4, No., 204 Theorem3:[4] If n 7, then σ(n) < (2.59) n log log n. Theorem4:[3]If n, Our imrovment will required Some estimates given by Rosser and schcoefeld in[5]. then σ(n) < n (7 w(n) + 0) 6 From J.Rosser and L.Schoefeld [5],we have the following estimates: Theorem5: [3] If n,then: σ(n)=0(log log n) Theorem6: [5] If e 3 2 < x and x 67, then: and: x log x 2 < π(x) π(x)< x log x 3 2 Corollary :[5] If x 7, then: x log x < π(x) Corollary 2:[5] If x>, then: π (x) < (.25556)x log x Theorem7: [5] If x>, and B= , then: Iog log x + B - < 2log 2 x and for x 280,then: x < log log x + B + 2log 2 x Corollary: [5]If x>, then: log log x < x Theorem8: [5]If x>,then: and when x 280, then; e γ (log x)( - 2log 2 x ) < x and when x 280, then: < eγ (log x )( + 2log 2 x ) x Corollary: [5] If x>,then: 70

4 ISSN (Paer) ISSN (Online) Vol.4, No., 204 x Where γ= is Euler, s constant. We need the following definition. < eγ (log x) ( + log 2 x ) Definition: Let n,then n denoted the n th rime, where: n=, = 2, n=2, 2 = 3, n=3, 3 = 5, n= 4, 4 = 7, Now, we shall give an imrovement of Evic, result by roving the following theorem. Theorem() : For n 82, then σ(n) < n log log Now, in order to rove theorem(4),we need the following Lemmas: Lemma(): [5]Let n 2, then: as n. this Lemme,we can written in the from: for some, > 0 and from, we can write where n n 0 ( ). some fixed number n 0. n ~ n Iog n n < (+ )n Iog n,.. () log n < (+ ) n Iog n Now, we shall give an imrovement of Evic, result by roving the following theorem. Lemma 2:If n 82, then: log n < log n Proof: we shall the bound given the lemma () to rove lemma(2)by using the inequality given in by G.Rosser and L.Schoenfeld[5]which given in the corollary ()of theorem(6) with x= n and π(x) = n, then: π(x) > x log x for x > 7 7

5 ISSN (Paer) ISSN (Online) Vol.4, No., 204 n > n log n for n 82 n < n log n () and so: 3 n < n n for n n < n, 2 log 3 n < log n, 2 3 log n < log n, and this gives us that; n log n < 3 2 n log n there fore, n < n log n < 3 n log n, 2 n log n < 3 2 n log n, n < n 3 2, for n n log n n7, for n 6 There fore, n < n 7 5, for n 6 log n < 7 log n, for 82 n 6 5 log n < 59 log n, for n 82 Lemma3; If n 3, then: 43 ( + ) n < 236 log log n 29 Proof: for x 286, and B= , we shall have that: log x > Iog 286 Iogx > Iog 2 > log 2 >

6 ISSN (Paer) ISSN (Online) Vol.4, No., 204 < = < log 2 x = < 0.06 (2Iog 2 x) (2log 2 286) x < loglog x + B + (2 log2 x) < log log x + log 4 3 x < loglog x x < loglog x x < log (4 log x) 3 There fore, log ( + ) = log [( + ) ( + r ) n ] log ( + n ) = [ log ( + ) log( + r )] = n log( + ) Also since, 2 then: log ( + ) log 3 2 log ( + ) n < (log 4 3 n k )log k By using Lemma() and k 286, where k=ω(n), then we have the following cases: Case (): (i) If ω(n)> 62, log ( + ) < log [ 4 n 3 log k ] < log [ 4 59 log k] 3 43 < log [ log k] log ( n ) < log [ log k] 29 log ( n ) < log [ log log n] 29 (ii) when ω(n) 6, and n < n log n,where n 82, then by calculation, we get: ( + ) n ( + ) n =

7 ISSN (Paer) ISSN (Online) Vol.4, No., 204 ( + n ) < < 236 log log n 29 Case(2): (i) If ω(n)>2,then: (ii) If ω(n) then, by calculation, we get : > , ( + ) n ( + ) < Case(3) ; ( + n ) < < 236 log log n 29 If n >6000,and n 6000,with ω(n) 5,then by calculation we get: ( + ) n < 236 log log n 29 Case(4); If n>200, and n 200,with ω(n) 3,then by calculation we get: Lemma4: For n 82, n 42, n 20, Proof: take x>455, then: since e γ < and ( + ) n 2 5 < 239 log log n 29 < log log n n log x > log 455 log 2 x > log 455 log 2 x > log 2 x < < log 2 x < () x < eγ log x ( + log 2 x) 74

8 ISSN (Paer) ISSN (Online) Vol.4, No., 204 x < ( )( ) log x < log x, Using Lemma (2) and k=ω(n),then if n> 82 with ω(n)<log n, n < log k. k log k log k 43 n n log k log log n log log n Case(): By taking k=ω(n) 82 and k 82 = 42,then n log log n So, if ω(n) 82,then: n < n 82 < < log log n Case(2): If n 5 0 7,then since > 0 8 7, ω(n) 6 and n in this case we get: < < log log n Case(3): n 6 If n 0 8, then since > , ω(n) 8 and n 0 8 in this case we get: < n 8 n 8 < < log log n Case(4): if n > 30000, then since k = 30030, ω(n) 5 and n in this case we get: 75

9 ISSN (Paer) ISSN (Online) Vol.4, No., = 77 6 n 77 n 6 < log log n Case(5): if n< 300, then ω(n) 3, excet for n=20 = and for n=20 then: n < log log n which is not true, If ω(n) 3, n = 5 = = 3.75 < log log n n Theorem: If n 82,then: σ(n) < n log log n a Proof: If n =. a2 2 ar r,then since σ(n) is multilicative function then: σ(n) = ( + + ai ) i n, ( i r) By lemma(4): [ = ( ai+ ) i n i If n 82, n 42, n 20 then: < log log n ] n σ(n) = i ai+ i = ai+ ( ai+ i ) i = ai.( ai+ i ) i =n i i ai i To show: Take L.H.S, than since: a - a+, 76

10 ISSN (Paer) ISSN (Online) Vol.4, No., 204 ( a+ ) a σ(n)= i i ai i σ(n) n for n 3, n 42, n 20. for n=42 and n=20 is not true in n σ(n) < ( )log log n. n [σ(n) < log log n. n ] Our result. declare for several values of n 82 which shows the validity of our bound which given in the table (). Table() n σ(n) n log log n References [] U.nnaurna, Inequalities for σ(n) and φ(n),math.magazine,(45) 972,

11 ISSN (Paer) ISSN (Online) Vol.4, No., 204 [2] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschrift,(74)962,-80. [3] R.L. Duncan, Some estimates for σ(n),amer. Math. Monthly,(74) 967, [4] G.H. Hardy and E.M. Wright, An introduction to the theory of number, 4 thed. oxford univ. ress, London,960. [5] B.J. Rosser and L. Schoenfeld,Aroximate formulas for some function of rime numbers, Illinois Journal of Math,(6) 962, [6] A. Ivic^,Two inequalities for the sum of divisor function.univ.u Novom Sadu, zbornik Radova.rirod mat.fak.7(977),

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