explicit upper bound for the function of sum of divisors

Mathematical Theory and Modeling www.iiste.org

ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)

Vol.4, No.11, 2014

68

Explicit upper bound for the function of sum of divisors 𝛔(𝐧)

Dr. Saad A. Baddai, Khulood M. Hussein

Dept. Math ., Collere of Science for Women, Univ. of Baghdad

M-alsaedi 87 [email protected]

[email protected])*(

following theorem : where he proved the 𝑐^[6]: we developed the result proved by A.EviAbstract

then : 𝑛 ≥ 7, let :[6]Theorem

σ(n) < (2.59)n log log n,…………………..…..(1)

this proving required several Lemmas which are given to be :

,then: ≥ 39 f n:[6] iLemma1

log 𝑝𝑛 <7

5 log n

,then: ≥ 31 :[6] if nLemma2

∏ (1 +1

𝑝) <

28

15𝑝|𝑛 log log n

,then: ≠ 42, 𝑛 ≠ 210, n≥ 31 :[6] if nLemma3

∏𝑝

𝑝−1< 2.59 log log np|n .

We a dope, the procedure of Evi𝑐^by improving the lemmas and get the following results:

, then: ≥ 82If n:Lemma 2

log pn <59

43 log n

then: ≥ 31,If n ;Lemma3

∏ (1 +1

p)p|n <

236

129 log log n

, ≥ 82, n ≠ 42, n ≠ 210For n :Lemma4

∏p

p−1< 2.509038354 log log 𝑛p|n

Where our lemma give above leads us to gut the new upper bound of the function 𝜎(𝑛) which given in the

following theorem:

then: ≥ 82,If n Theorem1:

σ(n) < 2.509038354 𝑛 log log n

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

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Vol.4, No.11, 2014

69

: Introduction

In this paper, we discuss the upper bound of the multiplicative number theoretic function σ(n), where σ(n)

represent the sum of all positive divisors of n, n≥ 1.this function is written to be,

∑ 𝑑𝑑|𝑛 ,

In Hardy and whright [4] the bound given to be explicit which is

σ(n) = 0(n log log n),…………………………..(1)

Also, R.L. Duncan [3] showed that ,

𝜎(𝑛) < 𝑛

6 [7𝜔(𝑛) + 10]……………………..…(2)

Where n = p1a1 . p2

a2 … … … prar and ω(n) = r and this is an explicit bound interns of ω(n) . this bound can

not be determined for large n , in which ω(n) can not be counted.

The best explicit upper bound proved by Aleksander Ivi𝑐^[6]when he proved .

σ(n) < (2.59)n log log n,…………………..…..(3)

where n≥ 7.

Our work dealing with Aleksander Ivi𝑐^ results in (3) and we improve this bound by increasing n to be n≥

82,and proved the following result .

σ(n) < (2.5090383)n log log n,……………….(4)

, then 32. 5, represent the number of primes of n, if n=2.r ω(n) =Then= p1a1 ⋯ pr

ar.: Let n)(n 𝛚Definition

w(n)=3.

(x), π:the number of all primes less them or equal to x denoted by (x)𝛑Definition

from Definition ,we can write:

π(x) = ∑ 1

For if, x=8.5, π(8.5)=4,

x=2 , π(2)=1,

x=12 , π(12)=5.

i.e x

logx The number of primes not exceeding x is asymptotic [4]eorem1:Th

(x) ~x

log x

such that, c2and c1there exist a constants [6]Theorem2:

, c2x

log x< π(x) <

c1x

log x

We shall improve the bound given in [6]for the function σ(n).Evi𝐜, proved the following theorem.




Vol.4, No.11, 2014

70

7, then ≥If n :[4]Theorem3

σ(n) < (2.59) n log log n .

ur improvment will required Some estimates given by Rosser and schcoefeld in[5].O 1, ≥If n[3]:Theorem4

then σ(n) < n

6 (7 w(n) + 10)

From J.Rosser and L.Schoefeld [5],we have the following estimates:

,then: ≥ 1If n: [3] Theorem5

σ(n)=0(log log n)

then: e3 2⁄ < x and x ≥ 67,If [5] :Theorem6

and:

x

log x − 12

< π(x)

π(x)<x

log x – 3

2

x ≥ 17, then:If :[5]Corollary 1

x

log x < π(x)

If x>1, then: :[5]Corollary 2

π (x) <(1.25556)x

log x

x>1, and B=0.261497 ………, then: If ][5 Theorem7:

Iog log x + B - 1

2log2 < ∑1

pp≤x

and for x≥280,then:

∑1

pp≤x < log log x + 𝐵 + 1

2log2x

1, then: If x>[5] :Corollary1

log log x < ∑1

pp≤x

≤ 280, then;and when xx>1,then: If[5] :orem8The

eγ(log x)(1 - 1

2log2x) < ∏

p

p−1 p≤x

and when x≤ 280, then:

∏p

p − 1< eγ (log x )(1 +

1

2log2x )

p≤x

1,then: If x> [5] Corollary1:




Vol.4, No.11, 2014

71

∏p

p − 1 < eγ(log x) (1 +

1

log2x)

p≤x

Where γ=0.5772157 is Euler,s constant .

We need the following definition.

denoted the n th prime , where: pn,then ≥ 1Let n Definition:

n=1 , p1 = 2,

n=2 , p2 = 3,

n=3 , p3 = 5,

n= 4 , p4 = 7,

Now, we shall give an improvement of Evi𝐜, result by proving the following theorem.

, then ≥ 82: For n 1)Theorem(

σ(n) < 2.509038354 n log log

Now, in order to prove theorem(4),we need the following Lemmas:

then: ≥ 2, Let n[5] Lemma(1):

pn~ n Iog n

as n→ ∞. this Lemme,we can written in the from:

pn < (1+∈)n Iog n, … … . . (1)

for some ∈, ∀∈> 0 and from, we can write

log pn < (1+∈) n Iog n

where n≥ n0(∈). some fixed number n0.

Now, we shall give an improvement of Evi𝐜 , result by proving the following theorem.

, then: ≥ 82If n:Lemma 2

log pn <59

43 log n

: we shall the bound given the lemma (1) to prove lemma(2)by Proof

using the inequality given in by G.Rosser and L.Schoenfeld[5]which given in the

corollary (1)of theorem(6)

π(x) > x log x ⁄ for x > 17

with x=pn and π(x) = n, then:




Vol.4, No.11, 2014

72

n > pn

logpn for n≥ 82

pn < 𝑛 log pn …………………………(1)

and so:

pn < n √pn3 for n≥ 82

pn2 3⁄

< 𝑛,

log pn2 3⁄

< log n,

2

3log pn < log n,

and this gives us that;

n log pn <3

2 n log n

there fore,

pn < n logpn <3

2 n log n,

n logpn <3

2 n log n ,

∴ pn < n3 2⁄ , for n≥ 82

3

2n log n ≤ n7 5⁄ , for n≥ 161

There fore,

pn < n7 5⁄ , for n ≥ 161

log pn <7

5 log n , for 82≤ n ≤ 161

∴ log pn <59

43log n , for n ≥ 82 ∎

then: ≥ 31,If n ;Lemma3

∏ (1 +1

p)

p|n

<236

129 log log n

, and B=0.261497……., we shall have that: x ≥ 286: for Proof

log x > Iog 286 → Iogx > 5.6559

Iog2 > 31.98920481

log2 > 2 × 31.98920481




Vol.4, No.11, 2014

73

1

2log2x<

1

63.97840962= 0.015630 < 0.016

1

(2Iog2x) =

1

(2log2286)< 0.016

∑1

p < loglog 𝑥 + 𝐵 + 1 (2 log2x) <⁄p≤x log log x + log

4

3

∴ ∑1

p < loglog x + 0.261497 + 0.016p≤x

∴ ∑1

p < loglog x + 0.277497p≤x

∴ ∑1

p< 𝑙𝑜𝑔 (

4

3log x)p≤x

There fore,

log∏ (1 +1

p) = log [(1 + 1 p1⁄ ) … … … … … (1 + 1 pr⁄ )p|n ]

log∏ (1 +1

p) = p|n [ log (1 + 1 p1⁄ ) … … log(1 + 1 pr⁄ )]

= ∑ log(1 + 1 p⁄ )p|n

Also since, p≥ 2 then:

1

p≤

1

2 → 1 +

1

p≤

3

2

log (1 +1

p) ≤ log

3

2≤

1

p

∴ ∑ log (1 +1

p)

p|n

≤ ∑1

p≤ ∑

1

p < (log

4

3p≤pk

)log pk

p|n

By using Lemma(1) and pk ≥ 286 , where k=ω(n), then we have the following cases:

Case (1):

(i) If ω(n)> 62,

log ∏ (1 +1

p) < log [

4

3 log pk]p|n

< log [ 4

3 ×

59

43 log k]

< log [ 236

129 log k]

∴ log ∏ (1 +1

p) < log [

236

129 log k]p|n

∴ log ∏ (1 +1

p) < log [

236

129 log log n]p|n

(ii) when ω(n)≤ 61, and pn < 𝑛 log pn ,where n≥ 82, then by calculation, we get:

∏ (1 +1

p)p|n ≤ ∏ (1 +

1

p)p≤pn=10.22575004




Vol.4, No.11, 2014

74

∴ ∏ (1 +1

p)p|n <10.22575004<

236

129 log log n

Case(2):

(i) If ω(n)>12 ,then:

∏ p > 7.42 × 1012p≤p12

,

(ii) If ω(n)≤ 11 then, by calculation, we get :

∏ (1 +1

p)p|n ≤ ∏ (1 +

1

p)p≤p11

< 6.542269709

∴ ∏ (1 +1

p)p|n < 6.542269709 <

236

129 log log n

Case(3) ;

If n >16000,and n≤ 16000,with ω(n)≤ 5 ,then by calculation we get:

∏ (1 +1

p) ≤

1152

385p|n

<236

129 log log n

Case(4);

If n>200, and n≤200,with ω(n)≤ 3,then by calculation we get:

∴ ∏ (1 +1

p) ≤

3

2∙

4

3∙

6

5p|n

≤12

5<

239

129log log n

,≥ 82, n ≠ 42, n ≠ 210For n :Lemma4

∏p

p − 1< 2.509038354 log log 𝑛

p|n

: take x>455, then: Proof

log x > log 455

log2x > log 455

log2x > 37.4580405

log−2x <1

37.4580405

< 0.026696537

1+log−2x < 1.026696537 ……………….(1)

since eγ < 1.7810727 and

∏p

p − 1 < eγ log x (1 + log−2x)

p≤ x




Vol.4, No.11, 2014

75

∏p

p−1 < (p≤x 1.7810727 )(1.026696537) log x

<1.828621173 log x ,

Using Lemma (2) and k=ω(n),then if n> 82 with ω(n)<log n ,

∏ p

p − 1 ≤ ∏

p

p − 1< 1.828621173 log pk.

p≤pkp|n

≤ 1.828621173 × 59

43 log k ≤ 2.509038354 log k

∴ ∏p

p−1 ≤ 2.509038354 log k ≤ 2.509038354 log log np|n

∴ ∏p

p−1 ≤ 2.509038354 log log n p|n

Case(1):

By taking k=ω(n)≥ 82 and pk ≥ p82 = 421,then

∏p

p − 1 ≤ 2.509038354 log log n

p|n

So, if ω(n)≤ 82,then:

∏p

p−1 ≤ ∏

p

p−1 < p≤p82 p|n 10.84851192

∴ ∏p

p − 1 ≤ ∏

p

p − 1 <

p≤p82 p|n

10.84851192 < 2.509038354 log log n

: Case(2)

If n ≥ 5 × 1017,then since ∏ p > 1018p≤p17

, ω(n) ≤ 16 and n≤ 5. 1017 in this

case we get:

∴ ∏p

p − 1 ≤ ∏

p

p − 1 < 7.348238613

p≤p16 p|n

< 2.509038354 log log n

: Case(3)

If n≥ 108, then since ∏ p > 2 × 108p≤p8

, ω(n) ≤ 8 and n ≤ 108 in this case we get:

∏p

p − 1 ≤ ∏

p

p − 1< 5.8471318

p ≤ p8 p|n

∴ ∏p

p−1 ≤ ∏

p

p−1< 5.8471318 <p≤p8 p|n 2.5090383 log log n

: Case(4)

if n > 30000, then since ∏ p = 30030,p≤pk𝜔(𝑛) ≤ 5 and n≤ 30000 in this case we get:




Vol.4, No.11, 2014

76

∏p

p − 1 ≤ 2 ∙

3

2∙

5

4∙

7

6∙

11

10=

77

16p|n

∴ ∏p

p−1 ≤

77

16p|n < 2.5090383 log log n

: Case(5)

if n< 300, 𝑡ℎ𝑒𝑛 𝜔(n) ≤ 3, except for n=210 =2∙3∙ 5 ∙7 and for n=210 then:

∏p

p−1 p|n < 2.5090383 log log n

which is not true, If ω(n)≤ 3,

∏p

p−1 ≤ 2 ∙

3

2∙

5

4=

15

4= 3.75p|n

∏p

p − 1 ≤

15

4= 3.75 <

p|n

2.5090383 log log n

then: ≥ 82,If n Theorem1:

σ(n) < 2.509038354 𝑛 log log n

is multiplicative function then: σ(n) ,then since p1a1 . p2

a2 ⋯ ⋯ prarn = If Proof:

σ(n) = ∏ (1 + p + ⋯ ⋯ pai) pi|n , (1 ≤ i ≤ r)

=∏ (pai+1−1

pi−1) pi|n

By lemma(4):

[ If n ≥ 82, n ≠ 42, n ≠ 210 then: ∏

p

p − 1p|n

< 2.509038354 log log n

]

σ(n) = ∏pi

ai+1

pi

=∏pai+1( 1−1 pi

ai+1⁄ )

pi−1

=∏pai.p(1−1 pi

ai+1⁄ )

pi−1

=n∏pi−pi

−ai

pi−1

To show:

p − p−a

p − 1≤

p

p − 1

Take L.H.S, than since:

1-1 pa+1 ≤ 1⁄ ,




Vol.4, No.11, 2014

77

p(1 − 1 pa+1⁄ )

p − 1≤

p

p − 1

p − p−a

p − 1≤

p

p − 1

σ(n)=∏pi−pi

−ai

pi−1

σ(n) ≤ n ∏p

p−1p|n

σ(n) < (2.5090383)log log n. n

for n≥ 31, n ≠ 42, n ≠ 210. for n=42 and n=210 is not true in

[σ(n) < 2.5090383log log n. n ] ∎

Our result . declare for several values of 𝑛 ≥ 82 which shows the validity of our bound which given in the table

(1).

Table(1)

2.5090383 n log log n 𝜎(𝑛) n

305.1409562 126 82

309.4342252 84 83

313.7327872 129 84

318.0365693 108 85

322.3455001 132 86

326.6595105 120 87

330.9785327 135 88

335.3025008 90 89

339.6313504 138 90

343.9650188 112 91

348.3034445 141 92

352.6465679 128 93

356.9943303 144 94

361.3466748 120 95

365.7035454 147 96

370.0648877 98 97

.

.

References

[1] U.nnapurna, Inequalities for σ(n) and φ(n),Math.Magazine,(45) 1972,pp.

187-190.




Vol.4, No.11, 2014

78

[2] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math.

Zeitschrift,(74)1962,pp-80.

[3] R.L. Duncan, Some estimates for σ(n),Amer. Math. Monthly,(74) 1967,pp.

713-715.

[4] G.H. Hardy and E.M. Wright, An introduction to the theory of number, 4 thed. oxford univ. press,

London,1960.

[5] B.J. Rosser and L. Schoenfeld ,Approximate formulas for some function of prime numbers, Illinois Journal

of Math,(6) 1962,pp.64-94

[6] A. Ivic^ ,Two inequalities for the sum of divisor function .Univ .u Novom Sadu, zbornik Radova .prirod

mat .Fak.7(1977),17-22.


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explicit upper bound for the function of sum of divisors

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