E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
On Number Theoretic Functions of Positive Integers
Prof. Aloysius Aurelio and Dr. Elsie M. Pacho
Abstract
This paper tried to investigate characteristic patterns in the number theoretic function
values of some consecutive positive integers, namely 1 to 70. These number theoretic
functions include the tau function τ(n), the sigma function σ(n), the phi function φ(n), and the
Möbius function µ(n). The respective values of the number theoretic function of these
numbers were then generated using definitions and theorems on the said functions.
The following characteristics, specifically for prime numbers were arrived at :For any
prime p, τ ( p ) = 2. ; For any prime p, σ ( p ) = p + 1. ;For any prime p, ϕ ( p ) = p − 1. ;For any
prime p, µ ( p ) = − 1. ;and For any prime p, σ ( p ) = ϕ ( p ) + τ ( p ). Using the above
mathematical statements, one could easily determine the four Number Theoretic function
values of a given prime number without having to go through tedious mathematical
computation using some theorem-supported formula. The computed values of the Number
Theoretic functions of the integers used were also plotted on line graphs then analyzed. It was
observed that the four functions arrange themselves into some layer-like pattern, with the
sigma function on top, followed by the phi, the tau, and the Möbius Function or mu at the
bottom.It was also observed that the graphs of the sigma and phi functions almost touch each
other through points corresponding to the sigma and phi function values of the primes.
Introduction
This paper focused on certain functions of integers particularly on the possibility of
occurrence of patterns. It is to be noted that laws, theorems, or principles, particularly in the
natural sciences of which mathematics is included are statements of generalities governing
definitive and predictive outcomes of events. The Pythagorean theorem for example applies to
any plane right triangle. It has been proven correct over and over with constant use. The laws
of Physics, as formulated by Archimedes, Newton, Einstein, etc. were made the foundation of a
number of practical applications both down here on earth and the space beyond as well.
A number theoretic function or simply arithmetic function is any function whose
domain of definition is the set of positive integers (Burton, 2002). Two such functions, the τ
and the σ, are herein defined:
Definition 1. Given a positive integer n, let τ (n ) denote the number of positive divisors
of n and σ (n ) denote the sum of these divisors.
Consider n = 12. Because 12 has for its divisors 1, 2, 3, 4, 6, and 12,
τ (12) = 6 and σ (12) = 1 + 2 + 3 + 4 + 6 + 12 = 28
Other examples are
τ (1) = 1 τ (2) = 2 τ (3) = 2 τ (4 ) = 3 τ (5) = 2
and
σ (1) = 1 σ (2) = 3 σ (3) = 4 σ (4) = 7 σ (5) = 6
To facilitate computation of τ (n ) and σ (n ) , the following theorem is considered:
Theorem 1. If n = p1k1 p 2k 2 ... p rk r is the prime factorization of n > 1, then
(a) τ (n ) = (k1 + 1)(k 2 + 1)... (k r + 1) , and
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
(b) σ (n ) =
p1k1 + 1 − 1 p 2k 2 + 1 − 1 p rk r + 1 − 1
...
.
p1 − 1 p 2 − 1
pr − 1
Examples:
n = 8 = 23 where p = 2 and k = 3. Then
τ (8) = τ 2 3 = 3 + 1 = 4 and
( )
23 +1 − 1
= 15 .
2 −1
Another number theoretic function was introduced by the Swiss mathematician
Leonhard Euler (1707-1783).
Definition 2. For n ≥ 1 , let ϕ (n ) denote the number of positive integers not exceeding
n that are relatively prime to n.
Illustrations:
If n = 30, the numbers that are less than 30 which are also relatively prime to 30 are 1,
7, 11, 13, 17, 19, 23, and 29. Thus,
ϕ (n ) = 8
Likewise, the following could be checked:
ϕ (1) = 1 , ϕ (2) = 1 , ϕ (3) = 2 , ϕ (4) = 2 , ϕ (5) = 4
Like in the cases of the τ and σ functions, computing for φ is made more convenient
through the next theorem:
Theorem 2. If p is a prime and k > 0 , then
σ (8) =
ϕ ( p k ) = p k − p k − 1 = p k 1 − 
p


1
Illustrations:
1. ϕ (9 ) = ϕ 3 2 = 3 2 − 3 = 6 . The six numbers that are less than and relatively prime
to 9 are 1, 2, 4, 5, 7, 8.
2. ϕ (16 ) = ϕ 2 4 = 2 4 − 2 3 = 8
The fourth Number Theoretic Function to be tackled is the Möbius function or µ. This
function was named after the German mathematician August Ferdinand Möbius, who first
introduced it in 1831 (http://en.wikipedia.org).
Definition 3. For a positive integer n, define µ by the rules
1
if n = 1

µ (n ) = 0
if p 2 n for some prime p

(− 1)r if n = p1 p 2 ... p r , where p i are distinct primes
Illustrations:
3
1. µ (30 ) = µ (2 ⋅ 3 ⋅ 5) = (− 1) = − 1
2. µ (1) = 1 , µ (2 ) = − 1 , µ (3) = − 1 , µ (4 ) = 0 , µ (5) = − 1
( )
( )
Objectives
This research dealt on some characteristics of the Number Theoretic Function values of
positive numbers.
Specifically, it tried to:
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
1. Determine the existence of patterns in the values of the Number Theoretic Functions
of prime numbers;
2. Predict the value of the Number Theoretic Functions of prime numbers; and
3. Identify salient characteristics of the values of the Number Theoretic Functions of
prime numbers.
Symbols and Notations
τ (n ) − number of positive divisors of n
σ (n ) − sum of the positive divisors of n
ϕ (n ) − number of positive integers not exceeding n that are relatively prime to n
µ (n ) − Möbius function of n equal to 1, -1 or 0
Definition of Terms
Divisor. It is a number that can divide another number without leaving any remainder.
Function. This is a rule that associates or pairs the elements of one set with another.
Every member of the first set should be paired to exactly one element coming from the second
set.
Positive integer. This is a whole number having a positive sign affixed before it.
Oftentimes, leaving a number without a sign before it makes the number positive.
Prime number. It is a number having no other divisor except itself and 1. The number
2 is considered the least of the primes.
Proof. It is a procedure to show that a mathematical statement is true or false.
Relatively prime. This is the relationship existing between integers wherein the only
number that can possibly divide them without any remainder is 1. That is, two or more
integers are said to be relatively prime to each other if the greatest common divisor is one.
Theorem. It is a mathematical statement that still needs to be proven for its validity.
In this paper, the characteristics formulated regarding the Number Theoretic Functions
have been provided with their respective proofs. The end of each proof is indicated by the
symbol ◄.
Results and Discussion
From the definitions and related theorems on the number theoretic functions τ(n), σ(n),
φ(n), and µ(n), some positive integers were treated using the above functions. Table 1 shows
the results.
Note that the Tau function of a prime number is always equal to 2. In fact, it is only the
prime numbers which have τ(n) equal to 2.
Another interesting observation is that the prime numbers have σ(n) equal to n + 1, and
φ(n) equal to n – 1. Put in another way,
σ(n) = φ(n) + τ(n).
In the case of the Möbius Function of n, denoted by µ(n), it is always -1 for all primes.
The following statements will formalize the above observations:
1. For any prime p, τ ( p ) = 2.
Proof:
Since τ p α = α + 1 , and with α always equal to 1 for any prime p, it follows that
τ ( p ) = 1 + 1 = 2. ◄
2. For any prime p, σ ( p ) = p + 1.
Proof:
( )
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
From σ ( p ) =
σ ( p) =
pα +1 − 1
, and again with α = 1, then
p −1
p 1 + 1 − 1 ( p + 1)( p − 1)
=
= p + 1. ◄
p −1
p −1
Table 1. Number Theoretic Function Values of Numbers 1-70
n
τ(n)
σ(n)
φ(n)
µ(n)
n
τ(n)
σ(n)
φ(n)
µ(n)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
1
2
2
3
2
4
2
4
3
4
2
6
2
4
4
5
2
6
2
6
4
4
2
8
3
4
4
6
2
8
2
6
1
3
4
7
6
12
8
15
13
18
12
28
14
24
24
31
18
39
20
42
32
36
24
60
31
42
40
56
30
72
32
63
1
1
2
2
4
2
6
4
6
4
10
4
12
6
8
8
16
6
18
20
12
10
22
8
20
12
18
12
28
8
30
16
1
-1
-1
0
-1
1
-1
0
0
1
-1
0
-1
1
1
0
-1
0
-1
0
1
1
-1
0
0
1
0
0
-1
-1
-1
0
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
9
2
4
4
8
2
8
2
6
6
4
2
10
3
6
2
6
2
8
4
8
2
4
2
12
2
4
6
7
4
8
2
91
38
60
56
90
42
96
44
84
78
72
48
124
57
93
52
98
54
120
72
104
58
90
60
168
62
96
104
63
84
144
68
12
36
18
24
16
40
12
42
20
24
22
46
16
42
20
50
24
52
18
40
24
56
28
58
16
60
30
36
32
48
20
66
0
-1
1
1
0
-1
-1
-1
0
0
1
-1
0
0
0
-1
0
-1
0
1
0
-1
1
-1
0
-1
1
0
0
1
-1
-1
33
4
48
20
1
68
6
126
32
0
34
35
4
4
54
48
16
24
1
1
69
70
4
8
96
144
44
24
1
-1
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
3. For any prime p, ϕ ( p ) = p − 1.
Proof:
By Theorem, ϕ p α = p α − p α − 1 . As before, with α = 1, then
( )
ϕ ( p ) = p − p 0 = p − 1. ◄
4. For any prime p, µ ( p ) = − 1.
1
Proof:
By definition, µ (n ) = (− 1) for n = p1 p 2 ... p r ,
Since p is a distinct prime, then
µ ( p ) = (− 1)1 = − 1. ◄
r
where p i is any distinct prime.
The above first three statements can be further lumped into one:
5. For any prime p, σ ( p ) = ϕ ( p ) + τ ( p ).
Proof:
Since, for any prime p, τ ( p ) = 2 , σ ( p ) = p + 1 , and µ ( p ) = − 1 , then
p +1= p −1+ 2
p +1= p +1
which translates into σ ( p ) = ϕ ( p ) + τ ( p ). ◄
Thus, if p = 101,
τ (101) = 2 ,
ϕ (101) = 100 ,
σ (101) = 102 ,
µ (101) = − 1 .
Furthermore, if ϕ ( p ) = 210 , and τ ( p ) = 2 , then σ ( p ) = 212 , that is, by using
σ ( p ) = ϕ ( p ) + τ ( p ) . Likewise, p is finally determined to be 211.
Another example would be if it is known that a certain prime p has σ ( p ) = 31314 , then
it is easy to figure out that τ ( p ) = 2 , ϕ ( p ) = 30312 , and the unknown prime is 30313.
Thus, a very convenient way to determine the values of the four Number Theoretic
Functions of any prime has been evolved.
When all the computed number theoretic function values of the given positive integers
were plotted on line graphs, the figures in the following pages came out.
There are visibly some interesting patterns seen from the individual graphs. The
generally somewhat erratic rise and fall of the sigma, phi, and tau function graphs are
punctuated by regular increase in value at some definite intervals. The sigma and phi function
graphs increase as one as they fluctuate while the tau function graph, although rising at its top
always touches a base value of 2. That corresponds to the constant tau value of the primes.
As could be expected, the Möbius function graph is just an oscillation between 1
and -1 and the intermediate value of 0.
A more interesting graph is generated when all four graphs are plotted in one plane.
Figure 5 shows the combination of the graphs. It is worthwhile to note the somewhat clear
boundaries between the graphs from each other.
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
180
160
140
100
80
60
40
20
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
Positive Integers
Figure 1. Graph of the Sigma Function Values of Numbers 1-70
14
12
10
T au F u nctio n
Function Values
120
8
6
4
2
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
Positive Integers
Figure 2. Graph of the Phi Function Values of Numbers 1-70
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ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
70
60
40
30
20
10
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
Positive Integers
Figure 3. Graph of the Tau Function Values of Numbers 1-70
1.5
1
0.5
Function Values
Function Values
50
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
-0.5
-1
-1.5
Positive Integers
Figure 4. Graph of the Mu Function Values of Numbers 1-70
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
180
160
140
Function Values
120
100
tau
sigma
phi
mu
80
60
40
20
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
-20
Positive Integers
Figure 5. Combined Graphs of the Number Theoretic Function
Values of the Numbers 1-70
The sigma function graph stays on top, followed by the phi function, then the tau
function. The Möbius function provides some kind of a base line on which the three other
number theoretic functions seem to perch.
The points where the graphs of the sigma and phi functions almost touch each other are
the number theoretic function values for the prime numbers, determined by σ ( p ) = p + 1 and
ϕ ( p ) = p − 1.
Summary
This paper tried to investigate possible patterns in the number theoretic function values
of some consecutive positive integers namely 1 to 70. These number-theoretic functions are
the tau function τ(n), the sigma function σ(n), the phi function φ(n), and the Möbius Function
µ(n). The respective values of the numbers were then generated using definitions and theorems
on the said functions.
Peculiar to this investigation are the number – theoretic function values of the prime
numbers. This may be attributed to the fact that any positive integer can be written
as a product of prime numbers. The following statements were arrived at:
1. For any prime p, τ ( p ) = 2.
2. For any prime p, σ ( p ) = p + 1.
3. For any prime p, ϕ ( p ) = p − 1.
4. For any prime p, µ ( p ) = − 1.
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E-International Scientific Research Journal
ISSN: 2094-1749 Volume: 2 Issue: 1, 2010
5. For any prime p, σ ( p ) = ϕ ( p ) + τ ( p ).
Using the above mathematical statements, one could easily determine the four Number
Theoretic function values of a given prime number without having to go through tedious
mathematical computations.
The computed values of the Number Theoretic functions of the integers used were also
plotted on line graphs then analyzed. It was observed that the four functions arrange
themselves into some layer-like pattern, with the sigma function on top, followed by the phi,
the tau, and the Möbius Function or mu at the bottom.
Conclusion
This research established some interesting characteristics of the Number Theoretic
function values positive integers particularly to prime numbers. Through these properties or
characteristics, the process of determining the Number Theoretic function values of prime
numbers is greatly facilitated.
Moreover, the respective graphs of these functions for positive integers show clearly
how they relate to each other and further reveal the likelihood of establishing the maximum
and minimum function values for given integers.
Recommendations
The results of this investigation can be incorporated in the study of the Number
Theoretic functions of positive integers. Although the derived statements / properties pertain
only to prime numbers, they may somehow give students additional insights into the nature of
these functions especially regarding prime numbers.
In addition, it is strongly recommended that a follow-up study be conducted concerning
the non-prime numbers as regards to the possibility of establishing properties and relationships
concerning their Number Theoretic function values.
REFERENCES
Burton, David M. Elementary Number Theory. Boston: McGraw-Hill Co., Inc. 2002.
Cayabyab, Vanessa P. Properties of Large Prime Numbers. Thesis. DMMMSU-MLUC
Graduate Studies, San Fernando City, La Union. October 2003.
Vance, Elbridge P. Modern College Algebra. 3rd ed. Massachusetts: Addison-Wesley
Publishing Co., Inc. 1977.
http://en.wikipedia.org/wiki/M%C3%B6bius_function, October 23, 2007.
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